Toward a Unified Framework for Analysis of Multi-RAT...

Research ArticleToward a Unified Framework for Analysis of Multi-RATHeterogeneous Wireless Networks

Murk Marvi 1 Adnan Aijaz 2 and Muhammad Khurram1

1Computer and Information Systems Engineering Department NED University of Engineering and Technology Pakistan2Telecommunications Research Laboratory Toshiba Research Europe Ltd Bristol BS1 4ND UK

Correspondence should be addressed to Adnan Aijaz adnanaaijazieeeorg

Received 15 August 2018 Revised 29 November 2018 Accepted 16 December 2018 Published 3 January 2019

Academic Editor Antonio De Domenico

Copyright copy 2019 Murk Marvi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The increased penetration of different radio access technologies (RATs) and the growing trend towards their convergencenecessitates the investigation of wireless heterogeneous networks (HetNets) from coverage and capacity perspective This paperdevelops a unified framework for signal-to-interference-plus-noise ratio and rate coverage analysis ofmulti-RATHetNets with eachRAT employing either a contention-free or a contention-based channel access strategyThe proposed framework adopts tools fromstochastic geometry with the location of APs and mobile users modeled through independent Poisson point processes (PPPs) Wespecifically focus on a two-RAT scenario (ie cellular andWi-Fi) where for multi-tier Wi-Fi RAT with contention-based channelaccess like CSMACA the location dependent distribution of interfering APs has been approximated through a homogeneous PPPMoreover by using some simple yet realistic set of assumptions the distance to nearest active AP has been defined which results insimplified expressions The medium access probability for a random and a tagged AP under a multi-tier Wi-Fi RAT has also beenderived and discussed By keeping in view the tremendous effect of temporal domain on overall network performance the stablequeue probability has been derived by assuming a non-saturated traffic model The results have been validated through extensivesimulations and compared with existing approaches Some useful insights have also been presented that shed light on design andanalysis of multi-RAT HetNets and provide motivation for further research in this direction

1 Introduction

According to Ciscorsquos forecasts [1] 83 billion hand-helddevices and 33 billion machine-to-machine (M2M) deviceswill be connected by 2021 The number of connected deviceswould clearly exceed the expected global population of 78billion by that time The monthly global mobile data trafficis expected to reach 49 exabytes and the annual trafficwill exceed half a zettabyte by 2021 Extreme densificationand offloading in wireless heterogeneous networks (Het-Nets) is the key technique for meeting the ever increasingcapacity requirements which can potentially bring the 1000timesimprovement in aggregate data rate as envisioned for 5Gnetworks [2] With networks becoming denser more trafficcan be offloaded to small cells especially because a significantproportion of traffic is generated by users in indoor locations(home offices etc) Offloading toWi-Fi networks is a naturaland most effective technique for reducing congestion oncellular networks [3] The integration of cellular and Wi-Fi

networks has been a topic of interest in both industrial andacademic communities since early 2000sWi-Fi is experienc-ing a trend towards ubiquity with operator-deployed Wi-Finetworks becoming a norm According to recent projections[1] 66 of mobile data traffic would be offloaded from 4Gnetworks to Wi-Fi by 2021

Witnessing these technological trends it can be easilyinferred that understanding the fundamental limits of Het-Nets comprising different radio access technologies (RATs)and employing different channel access techniques becomesparticularly important Analytical tools for HetNets canprovide valuable insights to system designers in meeting thefuture coverage and capacity demands through integration ofdifferent RATs with multiple tiers and by exploiting licensedand unlicensed resources in an effective manner

11 Related Work and Motivation Over the last few yearstools from stochastic geometry have been used for analysis

HindawiWireless Communications and Mobile ComputingVolume 2019 Article ID 6918637 19 pageshttpsdoiorg10115520196918637

2 Wireless Communications and Mobile Computing

of single-tier and multi-tier cellular networks [4 5] wheremostly Poisson point process (PPP) is used for modeling thespatial location of access points (APs) The PPP assumptionprovides a lower bound whereas the traditional grid modelprovides an upper bound on coverage Other point processeslike Gibbs [6] Strauss [7] Ginibre [8] and Detriminantal [9]have also been used for coverage analysis of cellular networksSuch processes provide better accuracy as compared to PPPbut at the cost of limited tractability Therefore PPP hasbeen a popular choice for analysis of cellular networksOn the other hand when it comes to Wi-Fi RAT PPPcannot be used for spatial modeling of active APs due tocontention-based nature of carrier sense multiple accesswith collision avoidance (CSMACA) scheme Howeverapproximated solution for SINR coverage has been presentedin [10] by exploiting Modified Matern Hard Core Process(MMHCP) for estimating the set of active APs where theAPs are originally distributed using a PPPThework is furtherextended from spatial averages to spatial distributions in [11]and throughput analysis has also been conducted againstvarious parameters of interest Due to approximation of theset of interfering APs through a non-homogeneous PPP theresulting expressions in [10 11] are extremely complicatedThe asymptotic expression for outage probability of generalad-hoc networks has been obtained in [12] however theresults are limited to high signal-to-interference ratio (SIR)cases

Although closed-form expressions for analysis of cellularRAT under special cases are available in literature [4 13]no such results are reported for Wi-Fi RAT due to difficultyin characterizing the interference effect of active APs [1011] This limits the in-depth analysis of multi-RAT HetNetswhich could be an important tool for coverage and capacityplanning of future wireless networks A general model formulti-RAT HetNets has been presented in [14] by assumingindependent PPPs for distribution of APs and users in agiven region However the effect of channel access schemesassociated with different RATs has not been taken intoaccount which directly affects the interference and hence thecoverage analysis Thus such a framework cannot provideaccurate insights into the different characteristics exhibitedby multi-RAT HetNets

Recently the coexistence problemofmulti-RATnetworkshas received considerable attention In [15 16] the coverageand capacity analysis has been presented by assuming thecoexistence of cellular and Wi-Fi RAT in unlicensed spec-trum After investigating various transmission mechanismthe authors in [15] reported that the LTE can coexist withWi-Fi but under certain conditions On the other hand in [16] asimilar issue has been addressed by assuming the operation ofLTE users in both licensed and unlicensed band Two differ-ent user association schemes ie crossing and non-crossing-RAT have been considered where the users of licensedRAT can access unlicensed band by exploiting opportunisticCSMACA scheme The crossing-RAT user association isshown to provide better performance as compared to non-crossing-RAT user association However the coexistenceof LTE and Wi-Fi users in unlicensed band cannot bringsignificant gain in performance because increase in the

capacity of LTE RAT is achieved at the cost of decrease inthe capacity of Wi-Fi RAT [16] Therefore in this researchthe coexistence of different RATs in unlicensed band has notbeen covered

Some recent studies have investigated the fundamentallimits on densification of cellular network by assumingdifferent path loss models and fading distributions [17ndash20]According to [17] the SINR coverage decreases after a certainthreshold due to increased interference with smaller pathloss exponents which is in contrast to widely used results asreported in [4] A detailed investigation has been presentedin [18] by exploring four different performance regimes whiletransitioning from sparse to dense networks Results similarto [17] are reported where after a certain threshold theSIR coverage no longer remains constant however the areaspectral efficiency increases linearly According to [19] theSIR coverage can be increased by exploiting the idle modecapability of APs under dense scenarios Further in [20]energy efficiency analysis has been presented by defining theoptimal transmission power for APs as a function of RATdensity However in all of the mentioned investigations asingle-tier dense cellular network with small cells has beenconsidered Moreover in few of the existing studies [21 22]multi-RAT networks have been investigated by assumingoperation of APs in different bands In [21] the sub-6GHzmacrocells are overlaid with mm-wave small cells and in[22] the small cells can operate on both bands Various cellassociation schemes considering both uplink and downlinkchannels have been investigated in [21] Following a similarthought in [22] a biasing based strategy has been proposedfor load balancing across a multi-RAT network In contrastto traditional approaches in [22] two biasing thresholds areexploited one for offloading users from macrocells to smallcells and other for offloading users from sub-GHz bandto mm-wave band In a nutshell the integration of mm-wave communication into existing infrastructure is one ofthe potential contributors for increasing capacity of futurewireless networks However the communication in mm-wave exhibits the characteristics which are different fromsub-GHz band and hence its realization requires a lot ofenhancements both at component and architecture level [23]

Densemulti-RATHetNetswould be a key aspect of futurewireless networks [2] According to some studies [17 18]the degradation in coverage provided by dense small cellnetworks after a certain limit is expectedTherefore in orderto meet the demands in coming future the focus must beshifted from single-RAT to multi-RAT HetNets Although infew of the recent studies [21 22] the multi-RAT networkshave been analyzed by exploiting sub-GHz and mm-waveband this work is focused around the analysis of multi-RATHetNets where each RAT operates on a different pool ofresources and can use either contention-based or contention-free channel access schemes As cellular contention-free andWi-Fi contention-based RATs are already deployed at a widescale and hence their integration can be considered as apotential contributor for improving the capacity of futurewireless networks without many modifications into existinginfrastructure Thus the key motivating factor behind thiswork is the growing convergence of the two RATs and as

Wireless Communications and Mobile Computing 3

opposed to standalone RATs their integration can lead tobetter network performance

12 Contributions and Outline With the aforementionedbackground and motivation the key contributions of thiswork can be summarized as follows

(i) A unified framework for multi-RAT HetNetsusing tools from stochastic geometry we developa unified framework for SINR and rate coverageanalysis of multi-RAT HetNets where RATs canoperate on either contention-free or contention-based (CSMACA) channel access schemes It differsfrom existing framework [14] due to incorporationof contention-based channel access scheme Morespecifically we focus on a two-RAT HetNet scenariowhich includes a cellular and a Wi-Fi RAT TheLaplace transform of interference for cellular RATcan be derived easily and it is available in existingliterature [4 13]Themain difficulty ariseswhilemod-eling the cumulative interference effect under Wi-FiRAT [10 15] which operates on a contention-basedchannel access scheme Thus by exploiting a fewapproximations we derive the Laplace transform ofinterference for Wi-Fi RAT which provides accuracycomparable to existing studies [15]

(ii) Analysis of heterogeneous Wi-Fi RAT we present atractable solution for SINR and rate coverage analysisof a multi-tier Wi-Fi RAT by exploiting a few approx-imations To the best of the authors knowledge thenotion of multi-tier Wi-Fi RAT has not been studiedin existing literature However as new techniques likedynamic carrier sensing and extreme densificationare emerging it is important to analyze the effectof heterogeneity in Wi-Fi RAT We have derived themedium access probability (MAP) for a random anda tagged AP under multi-tier Wi-Fi RAT and resultsshow that under dense network conditions the MAPfor a typical AP approaches that of a tagged AP

(iii) Stable queue probability by assuming a non-saturated traffic model we derive the stable queueprobability for a user under an AP of a RAT In orderto avoid the problem of interacting queues similarto [24 25] we assume a dominant and a modifiedsystemwhere results for each case have been reportedand analyzed It has been found that for low packetarrival rate the stable queue probability of a userunder Wi-Fi RAT is slightly higher compared tocellular RAT However for higher packet arrival ratethe stable queue probability of a user under cellularRAT is better

(iv) Various insights we provide various insights byanalyzing different HetNet scenarios with the aid ofproposed framework It has been shown that theintegration of femto-tier with Wi-Fi tier providesreasonable SIR coverage as compared to multi-tiercellular or Wi-Fi RAT however the rate coverage

starts declining as the user association with Wi-Fi RAT exceeds the cellular RAT Further the SIRcoverage increases and gradually approaches unity asa function of Wi-Fi RAT density this insight is incontrast to existing results reported in [14] Althoughthe SIR coverage provided byWi-Fi RAT is better thancellular RAT the stable queue probability of a userunder cellular RAT is overall better than Wi-Fi RATWe also explore the trade-off between user and APdensity and the results show that the rate coveragedecreases by increasing the AP density of Wi-Fi RATwhile maintaining a constant average load per APUnder such circumstances it has been suggested toincrease the number of non-overlapping channels forWi-Fi RAT as it can greatly improve the rate coverage

The rest of the paper has been organized as follows Sec-tion 2 introduces the underlying systemmodel in detail alongwith the considered channel access schemes and performancemetrics The MAP metric has been covered under Section 3Themain results of the paper have been covered in Section 4where a unified framework has been presented for SINRand rate coverage analysis of multi-RAT HetNets and stablequeue probability for a user under a RAT has been derivedVarious results have been reported and discussed in Section 5Finally Section 6 concludes the paper

Thenotation used in the paper and associated details havebeen provided in Table 1 The general parameters consideredfor generating various results under Sections 3 4 and 5 havebeen provided in Table 2

2 System Model

We consider a 119872-RAT 119873-tier HetNet scenario whereinRATs can employ either contention-free (OFDMA TDMACDMA etc) or contention-based (CSMACA) channelaccess scheme We specifically consider a two-RAT scenario(119872 = 2)which includes a cellular and aWi-Fi RAT each with119873 ge 1number of tiersMoreRATs (119872 gt 2) can be consideredprovided that each RAT operates on a different pool ofresources and the user equipment supports connection to allconsidered RATs We adopt a homogeneous PPP Φ119898119899 withdensity 120582119898119899 for drawing the locations of APs belonging tothe 119899119905ℎ tier of the 119898119905ℎ RAT whereas 119898 isin 119888119908 and 119899 isin1 2 119873 Another independent PPPΦ119906 with density 120582119906has been considered for the distribution of users in a givenregion We assume that Φ119888 = cup119873

119895=1Φ119888119895 and Φ119908 = cup119873119895=1Φ119908119895

denote the set of all APs under cellular and Wi-Fi RATrespectively Moreover all APs provide open access ie thereis no closed subscriber group andΦ119886 = cup119894isin119888119908Φ119894 denotes theset of all APs deployed in the given region

We consider a downlink channel wherein single resourceblock (ie time frequency and code) is utilized in everycell of cellular network For Wi-Fi RAT we assume sin-gle downlink channel A saturated traffic model has beenconsidered where APs transmit continuously even withoutany packet in queue for transmission Further APs of oneRAT cannot interfere with those of the other RATs as theyoperate in different pools of wireless resources However APs


Table 1 Notation summary

Notation Description119872 Total number of RATs119873 Total number of tiers under a RATΦ119894119895 120582119894119895PPP for APs belonging to 119895119905ℎ tier of 119894119905ℎ RAT and itsdensityΦ119906 120582119906 PPP for users and its density119875119894119895 Transmit power of APs belonging to pair (119894 119895)120572119895 Path loss exponent for 119895119905ℎ tier1205902

119894 Thermal Noise power associated with 119894119905ℎ RAT120579119894119895 SINR threshold for pair (119894 119895)120588119894119895 Rate threshold for pair (119894 119895)Γ119908119899 Carrier sensing for an AP pf Wi-Fi RAT119877119908119899 Carrier sensing range for an AP of Wi-Fi RAT119890119905119909 119890119905119909119900 Medium access indicator for a random and taggedAP under Wi-Fi RAT

P119905119908 P

119905119908

Medium access probability for a random andtagged AP under Wi-Fi RAT

S119894119895 S SINR coverage of pair (119894 119895) and overall for Φ119886

R119894119895R Rate coverage of pair (119894 119895) and overall for Φ119886

B119894119895 Effective bandwidth of serving APN119894119895 Load under serving AP119898119899 119899 Normalized Power and path loss exponentΦ119886 PPP for all APs deployed in the regionΦ119888 Φ119908 PPP for cellular and Wi-Fi RAT respectively119861(119909 119903)119861119888(119909 119903) A ball of radius 119909 with center at origin and its

compliment120600119888120600119908

Stable queue probability for a user under cellularand Wi-Fi RAT respectively120585 Packet arrival rate for a user during a time-slot119902 Active probability of an AP during a time-slot

Table 2 General parameters and settings

119875119898119899 forall(119898) and 119899 = 1 2 and 3 46 33 and 23 dBm120582119898119899 forall(119898) and 119899 = 1 2 and 3 1 100 and 1000APkm2forall(119899) 120572119899 4forall(119894) 1205902119894 0 dBforall(119894 119895) 120579119894119895 0 dBforall(119894 119895) 120588119894119895 2Mbpsforall(119898 119899) Γ119898119899 minus82 dBmforall(119894 119895) B119894119895 10MHz119891119888 119891119908 19 GHz 5GHz

of different tiers under the same RAT interfere with eachother due to shared resources All APs of (119898 119899) transmit atthe same power 119875119898119899 over the bandwidth B119898119899 We considerboth large-scale path loss and small-scale fading Free spacepath loss (FSPL) model with reference distance of 1 meteras given by 119897[dB](119889) = 20 log10(4120587120582119888) + 10120572119899 log10(119889) hasbeen assumed for all links here 120582119888 and 120572119899 gt 2 denote theoperating wavelength and path loss exponent respectively

The fading channels are Rayleigh distributed with averagepower of unity ie ℎ sim exp(1) The noise is assumed additivewith power1205902

119898 corresponding to the119898119905ℎ RATWe assume thatuser association is based on the maximum average receivedsignal strengthHowever it can be easily extended to a genericuser association scheme as given in [14] by just introducinga weight or bias variable For simplification normalizedparameters for a pair (119898 119899) with respect to serving pair (119894 119895)have been defined as 119898119899 ≜ 119875119898119899119875119894119895 119899 ≜ 120572119899120572119895 and 119891119898119899 ≜11989121198981198991198912

119894119895 Similar to [13 14] the probability density function(PDF) 119891119884119894119895

(119910) of the distance 119884119894119895 between a typical user andthe tagged AP is given by

119891119884119894119895(119910)= 2120587120582119894119895

A119894119895

expminus120587 sum

(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899119910 (1)

whereA119894119895 is the probability that a typical user associates withan AP of pair (119894 119895) and it can be given as

A119894119895 = 2120587120582119894119895 intinfin

0exp

minus120587 sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899sdot 119910119889119910(2)

Due to assumption of FSPLmodel the association of a user toan AP of pair (119894 119895) is dependent on the operating frequenciesof RATs as clear from (1) and (2)Thenormalized component(119891119898119899) for standalone RATs becomes unity as we have assumedthat all tiers under a single-RAT share the same resourceshence for such cases we get simplified expressions for userassociation which are similar to those in [4 13]

21 Channel Access Contention-free channel access schemesare employed by cellular RAT where some of the operatorsdeploy frequency reuse factor of unity and others go forfractional frequency reuse Under contention-based channelaccess schemes like CSMACA used by Wi-Fi RAT onlythe APs with different contention domains are allowed totransmit simultaneously and therefore the set of activeAPs can be less than the deployed one Under such ascheme for channel contention each APmaintains a randomback-off timer and waits for its expiry when the channelis sensed as free Meanwhile the transmission starts if noother AP accesses the channel Otherwise it freezes thetimer and repeats the procedure Due to various reasons acollision may occur when two APs in the same contentiondomain transmit simultaneously However there are definedprocedures in Wi-Fi for handling such situations

Under cellular RAT with contention-free channel accessall deployed APs are active therefore the original PPP(Φ119888) can be used for capturing the cumulative interferenceeffect However under Wi-Fi RAT with CSMACA channelaccess APs sharing the same contention domains are notallowed to transmit simultaneously Therefore the original


homogeneous PPP (Φ119908) used for drawing the location ofWi-Fi APs across a given region cannot be used for interferencemodeling In literature Modified Matern Hard Core Process(MMHCP) also known as MHCP-2 is widely used forestimating the set of active APs [10 15] MMHCP is basicallyobtained by mark (119905119909) dependent thinning of original PPP(Φ119908) where 119905119909 represents the back-off timer of an AP locatedat 119909Thus any point (119909119900) of the original PPP (Φ119908) is retainedonly if it has a mark (119905119909119900) smaller than all marks associatedwith the APs in its contention domain ie Φ119905

119908 = 119909119900 isin Φ119908 119905119909119900 lt 119905119909 forall119911 119875119909119909119900gt Γ119908119899 MMHCP does not take into account

the effect of variable back-off timer window size or collisionsHowever in [10 11] it has beenproved that themodel providesa reasonable conservative representation of active APs bycomparing it against an actual CSMACA networks

22 Performance Metrics We consider four performancemetrics described as follows

221 Medium Access Probability For cellular RAT the MAPdenoted by P119905

119888 is unity as all APs are allowed to transmitsimultaneously On the other hand due to contention-basedchannel access the MAP forWi-Fi RAT denoted byP119905

119908 canbe less than unity According to MMHCP defined in [10 15]a random AP under Wi-Fi RAT can access medium only ifit has the smallest mark among all the APs in its contentiondomain (3) Hence the medium access indicator 119890119905119895 for an AP119909119895 is given by

119890119905119895 = prod119909119899isinΦ119908119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899) (3)

For further details please refer to Section 3

222 SINR Coverage A typical user is said to be undercoverage if the received SINR from a tagged AP of pair (119894 119895)located at 119909119900 = (119910 0) is greater than some defined threshold120579119894119895 and it is given by

S119894119895 (120579119894119895)= E119910 P [SINR119894119895 (119910) gt 120579119894119895 | 119909119900 = (119910 0) 119890119905119909119900 = 1] (4)

where

SINR119894119895 (119910) = 119875119894119895ℎ119910119897 (10038171003817100381710038171199101003817100381710038171003817)sum119899isinΦ119894I119894119899 + 1205902

119894

(5)

ℎ119910 denotes the channel gain from a tagged AP located at 119910distance from the user andI119894119899 = 119875119894119899sum119909isinΦ119899119909119900

119890119905119909ℎ119909119897(119909) isthe cumulative interference from all APs of serving RAT-tierpair (119894 119895) outside the disk of radius 119910 with center at originBy using total probability theorem the overall SINR coverageprovided to a randomly located user can be given as

S = sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895) (6)

Further details are covered under Section 41

223 Rate Coverage The probability that a user which isassociated with anAP of pair (119894 119895) receives a rate greater thana certain threshold (120588119894119895) is given by

R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (7)

where

C119894119895 = B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 (8)

represents the rate of a userN119894119895 denotes the number of usersserved by an AP of pair (119894 119895) and P119905

119894119895 represents the MAPfor a tagged AP By exploiting total probability theorem theoverall rate coverage provided to a randomly located user canbe given as

R = sum(119894119895)isinΦ119886

A119894119895R119894119895 (120588119894119895) (9)


224 Stable Queue Probability The stable queue probabilityhas been defined as the probability that a user queue under anAP of a RAT is stable A queue is stable only if the providedservice rate (120583) is greater than the arrival rate of packets (120585)during a time-slot

120583 gt 120585 (10)

However the service rate provided by the network is depen-dent on the queues status and vice versa is also true Thiscreates the problem of interacting queues and it becomesdifficult to analyze the combined effect of spatial and tem-poral domain on overall performance of the network Thusin order to avoid this issue the concept of dominant andmodified systems has been exploited in existing literature[24 25] Where the dominant system provides a lower boundonperformance by assuming full buffermodel for interferingAPs and modified system provides an upper bound byassuming that the active probability of APs is equal to thepacket arrival rate of users hence the packets not transferredsuccessfully are dropped Further details are included underSection 43

3 Medium Access Probability

According to the given definition (3) for MAP a Wi-Fi APcannot transmit whenever any of its contender AP has asmaller back-off timer which is similar to one in [10 15] Aswe have assumed a multi-tier Wi-Fi RAT the APs operate atdifferent power levels (119875119908119899) based on the tier to which theybelong to hence it is possible that theAPs operating at higherpower levels do not sense the presence of low power APs intheir vicinity This effect needs to be captured carefully inorder to derive theMAP for amulti-tierWi-Fi RAT For betterillustration a two-tier Wi-Fi RAT scenario has been shownin Figure 1 where ldquo119877119899lt119873rdquo and ldquo119877119873rdquo represent the sensingradius for APs operating at high (119899 lt 119873) and low (119899 =


lt

Figure 1 Contention domains of APs under a two-tier Wi-Fi RAT

119873) power levels respectively The sensing radius has beenobtained by using (14) which does not include small-scalefading however this is just an illustration of possible effectson contention domains while considering multi-tier Wi-FiRAT The contention domain of each AP for scenario shownin Figure 1 is AP-0[1] AP-1[0] AP-2[0] AP-3[3] AP-4[3] It must be clear that AP-2 is not part of the contentiondomain of AP-0 as the received signal strength at AP-0 isbelow the required threshold (Γ119908119899) On the other hand AP-0is in the contention domain of AP-2 AP-1 is sufficiently closeto AP-0 and the required threshold is maintained hence itbelongs to the contention domain of AP-0 AP-3 and AP-4are at a far distance from AP-0 such that the received signalstrength is less than the required threshold If AP-0 get achance to access medium AP-1 and AP-2 remain in silentmode On the other hand if AP-2 access the medium thenAP-0 can also transmit given it has a smaller back-off timerthan AP-1 as it cannot detect the presence of AP-2 Thusthe MAP under multi-tier case can easily be obtained byexploiting the given definition (3)

Lemma 1 Given a Wi-Fi RAT with 119873-tiers each with trans-mission power (119875119908119899) and sensing threshold (Γ119908119899) then theMAPfor a typical AP is given by

P119905119908 = 1 minus exp (minussum119899isinΦ119908

120582119908119899119860119908n)sum119899isinΦ119908120582119908119899119860119908119899

(11)

where

119860119908119899 = intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 (12)

Proof See Appendix A

Remark 2 If either sum119899isinΦ119908120582119908119899 or 119860119908119899 997888rarr infin P119905

119908 997888rarr1sum119899isinΦ119908120582119908119899119860119908119899 Furthermore P119905

119908 decays at a faster ratewith respect to 119860119908119899 as compared to 120582119908119899

Remark 3 TheMAP for any randomAP is the same irrespec-tive of the tier to which it belongs As clear from Figure 1 thecontention domain of an AP operating at either high or lowpower level includes both low and high power APs within thesensing range ldquo119877119873rdquo and ldquo119877119899lt119873rdquo respectively

The obtained expression (11) can be approximated byfollowing expression

250 500 750 1000 1250 1500 1750 2000

AP density (w) in APkＧ2

SimulatedNumericalAnalytical

w3 = w

w2 = 02w w3 = 08w

w2 = 05w w3 = 05w

w2 = w

00

02

04

06

08

10

Med

ium

acce

ss p

roba

bilit

y (

t w)

Figure 2 Comparison of numerical analytical and simulationresults for the MAP of a random AP against Wi-Fi RAT density

119875119905119908 asymp 1 minus exp (minussum119899isinΦ119908

120582119908119899119860119908119899)sum119899isinΦ119908120582119908119899119860119908119899

(13)

which provides a lower boun on MAP where 119860119908119899 = 1205871198772119908119899

and

119877119908119899 = ( 1205821198884120587radic119875119908119899Γ119908119899

)(2120572119899)

(14)

is the sensing radius of APs belonging to the 119899119905ℎ tier Basedon the parameters listed in Table 2 the MAP for a single-tier and a two-tier Wi-Fi RAT has been plotted in Figure 2against density parameter The numerical and analyticalresults are obtained by using (11) and (13) respectivelywhereas the simulation results are generated by using givendefinition (3) It must be noted that the simulation results areclosely following the numerical onesThe results of analyticalexpression (13) are fairly close and providing a lower boundAs tier-3 operates at a lower power as compared to tier-2under single-tier scenario the MAP for tier-3 is higher ascompared to tier-2 In accordance to Remark 2 it must beclear from the reported results that with gradual increase in120582119908 or 119877119908119899 the MAP approaches 1sum119899isinΦ119908

120582119908119899119860119908119899

Remark 4 The approximated expression (13) provides alower bound on P119905

119908 therefore it is reasonable to say that119860119908119899 lt 119860119908119899 here 119860119908119899 takes into account only large-scalepath loss whereas 119860119908119899 also considers the effect of small-scale fading This implies that the expected sensing area orequivalently sensing radius for an AP is small when fadingeffects are taken into account hence the expected numberof contenders are less which results in improved MAP ieP119905

119908 gt 119875119905119908


(a)

(b)

(c)

Figure 3 The relationship between approximated sensing range of a tagged AP and its distance to user

According to Corollary 1 of [15] the MAP of a tagged APis the biased version of the MAP for a typical AP Howeverwe argue that as the density or power of tier increases theMAP for a tagged AP approaches the MAP for a typicalAP For better illustration please refer to Figure 3 wherethree different cases are considered ie low moderate andhigh density by assuming single-tier scenario Part (119886) showsmoderate density case because the distance between a userand its tagged AP is 119903119900 le 119877119908119899 As the user associates withthe nearest AP the shaded region does not include any APother than the tagged one That is why in [15] it has beensuggested that the MAP for a tagged AP is the biased versionof MAP for a random AP Now let us consider the sparse casein part (119888) of Figure 3 where 119903119900 gtgt 119877119908119899 Although the MAPis high in this case the link between user and its tagged AP isof no use because the received signal strength is less than therequired threshold (Γ119908119899) assuming that the received signalstrength required for user is the same as that for the taggedAP Thus under sparse condition the MAP for a tagged APand even for a random AP approaches unity but at the cost ofdecrease in received signal strength Finally moving to densecase part (119888) of Figure 3 where 119903119900 ltlt 119877119908119899 it must be clearthat as density of the RAT increases 119903119900 decreases hence theshaded region starts shrinking and the MAP for a tagged APapproaches that of a typical AP

Lemma 5 The MAP for a tagged AP belonging to the 119895119905ℎtier of Wi-Fi RAT with transmission power (119875119908119895) and sensingthreshold (Γ119908119895) is given by

P119905119908119895

= intinfin

0

1 minus exp minussum119899isinΦ119908120582119908119899 [119860119908119899 minus 119860119908119899 (119910)]sum119899isinΦ119908

120582119908119899 [119860119908119899 minus 119860119908119899 (119910)] 119891119884119908119895(119910) (15)

where

119860119908119899 (119910)= 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909 (16)

119860119908119899 is defined in (12) and 119891119884119908119895(119910) is given by (1)

250 500 750 1000 1250 1500 1750 2000AP density (w) in APkＧ2

tagged APrandom AP

10

08

06

04

02

00

Med

ium

acce

ss p

roba

bilit

y (

t w)

w3 = w

w2 = 02w w3 = 08w

w2 = 05w w3 = 05w

w2 = w

Figure 4 Comparison of MAP for a typical and a tagged AP againstWi-Fi RAT density for single-tier and multi-tier scenarios

Proof See Appendix B

Remark 6 By using total probability theorem the overallMAP for a tagged AP in Φ119908 can be given as P119905

119908 =sum119895isinΦ119908A119908119895P

119905119908119895

As we have assumed a multi-tier Wi-Fi RAT scenarioLemma 5 provides theMAP for a tagged APwhich belongs tothe 119895119905ℎ tier of Wi-Fi RAT It is an extension of Lemma 2 from[26] in which the retention probability for an associated APhas been defined when LTE APs coexist with single-tier Wi-Fi RAT in unlicensed band In Figure 4 the numerical resultshave beenplotted for a tagged and a randomAP under single-tier and multi-tier scenarios against density parameter Itmust be clear that under low density with smaller power oftransmission (119899 = 3) the MAP for a tagged AP is slightlyhigher than random AP However as the density or powerof transmission increases (119899 = 2) the MAP for a taggedAP approaches that of a random AP Further in Figure 5 thevoid probability given in [4] for no AP within a region of


P[N

o A

P clo

ser t

hanR

]

Distance (R) in meters

w = 100 ０ＥＧ2

w = 500 ０ＥＧ2

w = 1000 ０ＥＧ2

w = 1500 ０ＥＧ2

Rw2 asymp 51 mRw3 asymp 29 m

10

08

06

04

02

000 20 40 60 80 100 120

Figure 5 Probability that the distance froma typical user to a taggedAP is greater than approximated sensing radius of an AP

radius119877 has been plotted and the approximated sensing radiifor tier-2 and tier-3 are also denoted with markers It mustbe clear that under sparse case when 120582119908 = 100APkm2 theprobability that the distance between a user and the taggedAP is greater than the corresponding sensing radius is around80 for tier-3 and 40 for tier-2 As already mentionedwhile discussing Figure 3 such an event does not provide asuccessful connection to a user because of low received signalstrength As density increases to 1500 APkm2 the probabilityof such an event approaches zero and the MAP for a taggedAP approaches that of a random AP which is evident fromFigure 4

Remark 7 Under dense network scenario it is reasonableto approximate P119905

119908 by P119905119908 whereas by dense here we

mean that the probability of no AP within the approximatedsensing region approaches zero hence the required receivedsignal strength for a successful connection is fulfilled acrossthe region This can be achieved by either increasing thetransmission power of APs or density of the RAT

4 Coverage

Under this section we cover the rest of the three performancemetrics namely SINR coverage rate coverage and stablequeue probability The key factor which plays an importantrole for derivation of each of the mentioned metric isthe Laplace transform of cumulative interference We haveassumed amulti-RATHetNet scenario where APs can accesschannel by using either contention-free or contention-basedschemes therefore the interference distribution vary undereach RAT and hence the corresponding Laplace transformMoreover it is also important to consider if the user equip-ment can support multi-RAT connection Thus in this workwe specifically focus on a two-RAT scenario by assuming a

cellular and aWi-Fi RAT each with119873-tiers such that the APsof tier-1 have maximum and tier-119873 have minimum power oftransmission Please note that the framework is generalizedand can be extended to more RATs

41 SINR Coverage Cellular RAT is deeply investigated inexisting literature by using tools from stochastic geometrytherefore we refer to [4 13] for the Laplace transformof cumulative interference under cellular RAT Due tocontention-based nature of channel access in Wi-Fi RATit is hard to characterize the cumulative interference effectAs the distribution of interfering APs is non-independentthinning of Φ119908 the Laplace transform of interference is notknown in closed-form [10 15] Therefore in [15] the setof interfering APs under Wi-Fi RAT is approximated bynon-homogeneous PPP with certain density which has beendefined by exploiting the conditional MAP and Bayesrsquo ruleOn the other hand in [26] the set of interfering APs hasbeen approximated by a homogeneous PPP (Φ119905

119908)with density120582119905119908 = P119905

119908120582119908 and it has been assumed that the repulsionamong APs is captured by P119905

119908 which is reasonable as perdiscussions in [10 27] Two main factors for capturing thecumulative interference effect are (1) the density of active APsand (2) the distance to those APs In this work similar to [26]we approximate the conditional MAP for an interfering AP(P[119890119905119909 = 1 | 119890119909119900 = 1]) by the conditional MAP of a tagged AP(P[119890119905119909119900 = 1 | 119909119900 = (119910 0)]) As per an alternative definitiongiven in [10] theMAP represents the probability of successfulsimultaneous transmissions This implies that if a taggedAP transmits then on average the number of simultaneoustransmissions and hence the number of active APs in agiven region remain constant Thus we can approximatethe set of interfering APs by a PPP (Φ119905

119908119899) with density120582119905119908119899 = P119905

119908119895120582119908119899 The other important factor in modeling theinterference effect is the distance to nearest active AP Asheavy portion in interference is mainly contributed by theclosest active APs the distance to nearest interfering AP hasbeen approximated by using some simple yet effective set ofassumptionsThe following lemmaprovides an approximatedLaplace transform of cumulative interference for Wi-Fi RATAlthough our framework is based on a few approximations itprovides reasonable accuracy when compared with simulatedand existing results

Lemma 8 The Laplace transform of cumulative interferencefor Wi-Fi RAT with119873-tiers is approximated by

LI119908119899(119904119908119895)

asymp expminus120587P119905119908119895120582119908119899 [(Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119886

) | 119910 lt 1198771199081198734 )+ (Z (119904119908119895119875119908119899 120572119899 119911120572119899

119908119899119887) | 119910 gt 1198771199081198734 )]

(17)

where 119877119908119873 represents the mean sensing radius for a tier withlowest power of transmission (119873) and 119911119908119899119886

and 119911119908119899119887are defined

in (C7) and (C8) respectively


Proof See Appendix C

Following Lemma 8 and existing studies [4 13] forLaplace transform of cumulative interference under cellularRAT the SINR coverage for a typical user has been defined inthe following theorem

Theorem 9 The SINR coverage of a randomly located userunder a multi-RAT HetNet as defined in Section 2 is approx-imated by

S asymp sum119895isinΦ119888

2120587120582119888119895 intinfin

0exp(minus1199041198881198951205902

119888 minus 120587 sum119899isinΦ119888

120582119888119899Z (119904119888119895119875119888119899 120572119899 119911120572119899119888119899119887) + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)119910119889119910+ sum

119895isinΦ119908

2120587120582119908119895 intinfin

0exp(minus1199041199081198951205902

119908 minus 120587P119905119908119895

sum119899isinΦ119908

120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119886) | 119910 lt 1198771199081198734 ] + 120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119887

) | 119910 gt 1198771199081198734 ] + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)sdot 119910119889119910(18)

where 119904119894119895 = 120579119894119895119897(119910)119875minus1119894119895 120579119894119895 is the SINR threshold for the 119895119905ℎ

tier of the 119894119905ℎ RAT andZ (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (19)

Proof By following given definition (4) for SINR coveragewe get

S119894119895 (120579119894119895) 119886= int119910gt0

E[[expminus

120579119894119895119897 (10038171003817100381710038171199101003817100381710038171003817)119875119894119895

(sum119899isinΦ119894

119875119894119899 sum119909isinΦ119899cap119861119888(01199101015840)

119890119905119909ℎ119909119897 (119909) + 1205902119894 ) | 119890119905119909119900 = 1119891119884119894119895

(119910) 119889119910]] (20)

119887asymp int119910gt0

E[[expminus119904119894119895(sum

119899isinΦ119905119894

I119894119899 + 1205902119894 )119891119884119894119895

(119910) 119889119910]] (21)

119888= int119910gt0

119890minus1199041198941198951205902119894 prod119899isinΦ119905119894

LI119894119899(minus119904119894119895) 119891119884119894119895

(119910) 119889119910 (22)

where (119886) is the result of deconditioning with respect to 119909119900 =(119910 0) and assumption that ℎ119910 sim exp(1) (119887) follows from anapproximation P[119890119905119909 = 1 | 119890119905119909119900 = 1] asymp P[119890119905119909119900 = 1 | 119909119900 = (119910 0)]for 119894 isin 119908 and an assumption that P[119890119905119909 = 1] = 1 for119894 isin 119888 (119888) follows from independent random variableI119894119899 andLI119894119899

(119904119894119895) is the Laplace transform of interference We refer toexisting results from [4 13] forLI119888119899

(119904119888119895) By using Lemma 8we get an approximated LI119908119899

(119904119908119895) for Wi-Fi RAT and thefinal expression (18) is obtained by using total probabilitytheorem (6) which completes the proof

Corollary 10 By assuming an interference-limited scenarioie 1205902

119908 = 0 with 120572119895 = 4 and P119905119908119895 asymp P119905

119908119895 the SIR coverageof a randomly located user under a single-tier (119895119905ℎ)Wi-Fi RATis given by

S119908119895 asymp 1 minus exp (minus120587120582119908119895 (1198772

11990811989516) 120574119908119895 (3))120574119908119895 (3)+ exp (minus120587120582119908119895 (1198772

11990811989516) 120574119908119895 (1))120574119908119895 (1) (23)

where

120574119908119895 (V) = 1 +P119905119908119895radic120579119908119895 arctan(radic120579119908119895

V2) (24)

Proof Substituting given parameters in (18) performingsome mathematical operations and re-arranging variablesproof the given corollary

In Figure 6 the numerical results obtained through (18)are compared against the simulated ones for two single-tier(120582119908 = 1205821199082 120582119908 = 1205821199083) and two multi-tier cases under Wi-FiRAT The simulation environment was created by randomlydeploying APs of given density in a region of size 1 km times1 km The results were averaged over number of iterationsand under each iteration the SIR was evaluated for 2000 ran-domly chosen points It must be clear that the approximatedexpression (18) is closely following the simulated results andprovides a lower bound on coverage which is according todiscussions under Lemma 8 and Theorem 9 Although theinterfering APs are very close to the tagged one under highdensity regime the distance between a user and tagged AP is


SIR

cove

rage

(w

)w = 1000 ０ＥＧ2

SimNum

SIR threshold () in dB

10

08

06

04

02

00

w3 = w

w2 = 02w w3 = 08w

w2 = 052 w3 = 05ww2 = w

minus10 minus5 0 5 10 15 20

Figure 6 Comparison of numerical results with simulated ones forsingle-tier and two-tier Wi-Fi RAT only

also very less as compared to the sensing radius of APs thatis why in Figure 6 the numerical results provide an upperbound on SIR coverage for tier-2 as density of APs increasesFurther in Figure 7 the numerical results are plotted forvarious network configurations including both standaloneand multi-RAT HetNets Standalone cellular (Φ119888) and Wi-Fi (Φ119908) RAT each with two tiers have been consideredwhere Φ119888 is providing a lower bound and it is according toreported results [4 13] On the other hand Φ119908 is providingbetter coverage as some of the APs are prohibited to transmitbecause of the contention domains The results for two multi-RAT HetNets are also reported where in Φ119898

119886 a macro-tier(119899 = 1) has been overlaid with a Wi-Fi tier (119899 = 2)and in Φ119891

119886 a femto-tier (119899 = 2) is overlaid with a Wi-Fitier (119899 = 3) Although the power of tier-2 gtgt tier-3 theconsidered density for tier-3 gtgt tier-2 which reduces theMAP and hence improves the SIR coverage that is whyall configurations which include tier-3 of Wi-Fi RAT areproviding better coverage as compared to those with tier-2

42 Rate Coverage Under this section in the following the-orem we derive the rate coverage probability of a randomlylocated user

Theorem 11 The probability that a randomly located user ina network setting as defined in Section 2 receives a rate greaterthan some defined threshold (120588119894119895) is approximated by

R asymp sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895 = 120591119894119895 (N119894119895)) (25)

where N119894119895 denotes expected load under the serving AP and120591119894119895(N119894119895) = 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1

00

02

04

06

08

10

SIR

cove

rage

()

Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

minus5 0 5 10 15 20minus10


Figure 7 Numerical results for SIR coverage under various net-work configurations obtained through (18)

Proof The proof simply follows from [14] however forreadability the details are included in Appendix D

Remark 12 The rate coverage is function of four parametersincluding rate threshold (120588119894119895) average load under serving AP(N119894119895) MAP (P119905

119894119895) and bandwidth (B119894119895) Under cellular RATthe relation of rate coverage with the mentioned parameterscan be explained with the help of the following expression

120588119888119895N119888119895

P119905119888119895B119888119895

= 120588119888119895B119888119895

+ 128120588119888119895120582119906A119888119895120582119888119895B119888119895

(26)

where P119905119888119895 = 1 It must be clear that the rate coverage of a user

under cellular RAT is directly proportional toB119888119895 and 120582119888119895 ofthe tier whereas it is inversely proportional to 120588119888119895 and 120582119906 Incase of Wi-Fi RAT by using an approximation P119905

119908119895 asymp P119905119908119895

we get

120588119908119895N119908119895

P119905119908119895B119908119895

asymp 120588119908119895120582119908119895119860119908119895 + 128120588119908119895120582119906A119908119895[1 minus exp (minus120582119908119895119860119908119895)]B119908119895

(27)

Similar to cellular RAT the rate coverage under Wi-Fi RATis inversely proportional to 120588119908119895 and 120582119906 and it is directlyproportional toB119908119895 Moreover the rate coverage is indirectlyproportional to the product 120582119908119895119860119908119895 and at the same timedirectly proportional to the negative exponent of it For lowervalues of 120582119908119895 the negative exponential effect dominates andtherefore the rate coverage increases On the other hand as120582119908119895 997888rarr infin the term [1 minus exp(minus120582119908119895119860119908119895)] approaches unityand hence the rate coverage starts declining

Remark 13 The rate coverage under Wi-Fi RAT is inverselyproportional to 119860119908119895 and directly proportional to the negativeexponent of it please see (27) Therefore for lower values


u = 2000 usersＥＧ2Ra

te co

vera

ge (ℛ

)

Rate threshold () in Mbps

Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

10

08

06

04

02

000 2 4 6 8 10

Figure 8 Numerical results for rate coverage under variousnetwork configurations obtained through (25)

of 119860119908119895 the term in denominator of (27) dominates andhence the rate coverage improves As 119860119908119895 997888rarr infin theterm [1 minus exp(minus120582119908119895119860119908119895)] approaches unity and hence therate coverage starts declining Thus in either case the tiersoperating at low power levels provide better rate coverageas compared to high power tiers Equivalently we can alsoconclude that the rate coverage increases as a function ofsensing threshold (Γ119908119895)

In Figure 8 the numerical results obtained through (25)have been plotted by considering network configurationssimilar to those of Figure 7 It must be noted that in Figure 7the SIR coverage was slightly affected by the changes inconfiguration as compared to the rate coverage in Figure 8which is significantly varying for various network configu-rations The reason behind such a result is the dependenceof rate coverage over four different parameters as clearfrom Theorem 9 and Remark 12 Moreover for all thoseconfigurations the rate coverage is high which include tier-3 of Wi-Fi RAT because of its high density and lowerpower of transmission please see Remarks 12 and 13 forfurther details In Figure 9 the rate coverage for differentnetwork configurations has been plotted and the results arein accordance with Remarks 12 and 13 The rate coverageincreases for standalone cellular RAT however for Wi-Fi RAT it initially increases and then it starts decliningSimilarly under multi-RAT case as the user association withWi-Fi RAT exceeds the cellular RAT the rate coverage startsdeclining Moreover the rate of low powerWi-Fi tier is betterthan high power tier which is in accordance with Remark 13

43 Stable Queue Most of the existing studies assume asaturation model for traffic which do not capture the ran-domness introduced by the temporal domain In few of the

Rate

cove

rage

(ℛ)

Φa = Φc2

Φa = Φw3

Φa = Φc2 + Φw3

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

06

05

04

03

02

01

000 250 500 750 1000 1250 1500 1750 2000

u = 4000 APkm2 Γwj = minus92 dBm

０＞ＨＭＣＮＳ (wj) ＣＨ０ＥＧ2

Figure 9 Rate coverage as a function of Wi-Fi RAT density or incase of standalone cellular RAT it is function of cellular RATdensity

recent works [24 25 28] both the temporal and spatialdomains have been analyzed by exploiting tools fromqueuingtheory and stochastic geometry In [25 29] the conditionsfor a network to be stable have been derived by assuming adominant and a modified system In [28] the probability fora user queue to be unstable has been derived by assuming aPoisson and a uniform distribution for arrival rate of packetswhere PPP and Poisson cluster process (PCP) have beenused for the distribution of APs across a given region Inall of the aforementioned works single-tier cellular RATand a downlink channel have been assumed As the PPPrealization is random and irregular there are some APswith good and others with poor transmission environmentresulting in some users near APs with good experience andothers at the edge under outage [25] In [30] the outageprobability has been derived as a function of distance froma user to the tagged AP and it has been shown that the outageincreases as the distance increases By exploiting the givenconcepts mainly from [25 29 30] we derive the stable queueprobability for a user under an AP of a given RAT

In this section for simplified analysis we follow a differ-ent set of assumptions [24 28]We assume standalone single-tier cellular and Wi-Fi RAT and an interference-limitedscenario ie 1205902

119894 = 0 and 120572119899 = 4 A non-saturated trafficmodel has been considered where packets arrive at a userwith probability 120585 isin [0 1] during a time-slot Further weassume that 119902 represents the probability that an AP is activeduring a time-slot For avoiding interacting queues problemsimilar to [24 29] we assume a dominant and a modifiedsystem Under a dominant system the interfering APs havefull buffers and transmit continuously ie 119902 = 1 whereasunder modified system the interfering APs are active withprobability 119902 = 120585 the packets not delivered successfullyare hence assumed to be droppedWith the aforementioned


Packet arrival probability ()

Stab

le q

ueue

pro

babi

lity

()

= 15 dB

tw=046

10

08

06

04

02

0000 02 04 06 08 10

Φc -domΦw -dom

Φc -modΦw-mod

Figure 10 Stable queue probability (120600) as a function of packetarrival rate (120585) by assuming a dominant and a modified systems

assumptions the following theorem provides the probabilitythat a user queue is stable

Theorem 14 The stable queue probability of a user under asingle-tier cellular (Φ119888) or a Wi-Fi (Φ119908) RAT with a packetarrival rate of 120585 is given by

120600119888 (120579 120585 119902) = 1 minus exp( log 120585119902radic120579 arctanradic120579) (28)

120600119908 (120579 120585 119902) = [1 minus exp( log (120585P119905119908)119902P119905

119908radic120579 arctan (radic1205799))]

sdot 11199031199003lt1198771199084+ [1 minus exp(minus120587120582119908

1198772

11990816 )]11199031199003ge1198771199084

+ [exp(minus120587120582119908

1198772

11990816 )minus exp( log (120585P119905

119908)119902P119905119908radic120579 arctanradic120579)] 11199031199001gt1198771199084

(29)

respectively where

1199032119900V = minus log (120585P119905119908)120587119902P119905

119908120582119908radic120579 arctan (radic120579V2) (30)

Proof See Appendix E

Remark 15 From the given condition (E5) for a stable queueof a user under an AP of Wi-Fi RAT it is clear that the MAPfor an AP must be greater than the arrival rate of packetsduring a time-slot Hence (30) is valid only when P119905

119908 gt 120585By assuming a dominant (119902 = 1) and a modified (119902 =120585) system the numerical results for stable queue probability

Φc -domΦw -dom

Φc -modΦw-mod

Stab

le q

ueue

pro

babi

lity

()

10

08

06

04

02

00

SIR threshold () in dBminus10 minus5 0 5 10 15 20

= 03

Figure 11 Stable queue probability (120600) as a function of SIR threshold(120579) by assuming a dominant and a modified system

000

005

010

015

020

025

030

035

040St

able

que

ue p

roba

bilit

y (

) = 15 dB = 03

tw=029

tw=031

AP density (w or w) in APkＧ2

250 500 750 1000 1250 1500 1750 2000

Φc -domΦw -dom

Φc -modΦw-mod

Figure 12 Stable queue probability (120600) as a function of AP densityby assuming a dominant and a modified system

have been reported in Figures 10 11 and 12 against differentparameters of interest The dominant system in each resultis providing a lower bound whereas the modified system isproviding an upper bound [24 25] It must be clear fromFigure 10 that the stable queue probability for a user underWi-Fi RAT (120600119908) is slightly better than cellular RAT (120600119888)whenthe packet arrival rate is low As 120585 increases 120600119908 decreasesand eventually approaches zero when 120585 997888rarr P119905

119908 which is inaccordance with Remark 15 please see Figures 10 and 12 forclarification Moreover it must also be noted that the decayin 120600 as a function of 120585 is faster as compared to 120579 in Figure 11and 120582119908 in Figure 12 which is in agreement with the resultsreported in [29] 120585 as a function of AP density is constantfor cellular RAT because under interference-limited scenario


Φcj

Φwj

Wi-Fi RAT density (wj) in APkＧ2

0 250 500 750 1000 1250 1500 1750 2000

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

Φa = Φc2 + Φw3

00

02

04

06

08

10A

ssoc

iatio

n pr

obab

ility

(ij

)

Figure 13 Association probability as a function of Wi-Fi RATdensity

with 120572119888 = 4 the SIR coverage becomes independent ofdensity of the RAT [4] On the other hand underWi-Fi RAT120600119908 first decreases because P119905

119908 decreases as a function of 120582119908After that it increases slightly as the probability of distance119903119900 gt 1198771199084 between a user and its tagged AP approacheszero hence the second indicator function in (29) becomesactive as all other factors are constant thus increase in120582119908 results in an increase in 120600119908 Finally when the distancebetween a user and its tagged AP is 119903119900 le 1198771199084 the veryfirst indicator function in (29) becomes active and hence 120600119908

starts declining and finally approaches zeros as P119905119908 997888rarr 120585

5 Numerical Results and Discussions

Under this section various numerical results for differentperformance metrics have been discussed An interference-limited scenario 120590119894 = 0 with 120572119899 = 4 has been assumed forall RAT-tier pairs (119898 119899) The parameters have been carefullychosen by considering dense HetNet scenario [18 31] andsummarized in Table 2 In general if not specified theparameters mentioned in Table 2 have been used for all theresults reported in this paper

The association probability as a function of Wi-Fi RATdensity for multi-RAT HetNets has been plotted in Figure 13Initially most of the users are associated with cellular RATand as the density of Wi-Fi RAT increases the user asso-ciation (A119908119895) increases For a two-RAT scenario each withsingle-tier as assumed for Figure 13 the AP density at whichthe association probability of Wi-Fi RAT becomes equal tothe cellular RAT can be obtained by the following relation

120582119908119895 = 120582119888119895

119891119908119895119891119888119895

radic 119875119888119895119875119908119895

(31)

Exs[14] Φa = Φc1 + Φw3

Theo 1 Φa = Φc1 + Φw3



200 400 600 800 1000 1200 1400 1600 1800 2000

AP density (w3) in APkm2

c2 = w3 = 3 ＞

SIR

cove

rage

()

Exs[14] Φa = Φc1

Exs[14] Φa = Φc2

Exs[14] Φa = Φw3

080

075

070

065

060

055

050

045

Figure 14 SIR coverage as a function of Wi-Fi RAT density whenoverlaid with macro- or femto-tier

For the case when Φ119886 = Φ1198882 + Φ1199082 in Figure 13 thepower of Wi-Fi and cellular tier is the same ie 119899 = 2However in order to get equal association ie A1199082 = A1198882the required 1205821199082 gt 1205821198882 as 1198911199082 gt 1198911198882 and this is evident from(31) In Figure 14 the SIR coverage of two different HetNetshas been analyzed against Wi-Fi tier density (1205821199083) When1205821199083 lt 200 most of the users are associated with cellularRAT as clear from Figure 13 and the overall SIR coverage(S) of multi-RAT HetNet becomes equal to the single-tiercellular RAT (S1198882) which is function of the chosen thresholdsonly (1205791198882) According to the results of Theorem 9 as 1205821199083

increases the association of users with Wi-Fi RAT increasesand hence the coverage On the other hand according to[14] the SIR coverage keeps on decreasing and at last it meetsS1198881 as the same thresholds are used (ie 1205791198881 = 1205791198882 = 1205791199083)S1198881 = S1198882 = S1199083 each denoting the SIR coverage ofstandalone cellular tiers (macro femto) and the Wi-Fi tier(119899 = 3) It is because of the fact that the framework givenin [14] for multi-RAT HetNets does not capture the effectof different channel accessing schemes Thus addition ofa new RAT is simply another cellular RAT which operateson a different pool of resources hence it does not causeinterference to existing RATs The proposed framework inthis work captures the effect of both the contention-free andthe contention-based channel accessing schemes thereforeit provides generalization and ease of analysis for variousnetwork configurations

In Figure 15 the SIR coverage has been analyzed againstsensing threshold (Γ1199083) and 1205821199083 By increasing Γ1199083 the SIRcoverage decreases because of the increase in density ofactive APs (120582119905

1199083) It must also be noted that after a certainsensing threshold the SIR coverage becomes almost constantas P119905

1199083 997888rarr 1 1205821199051199083 997888rarr 1205821199083 Similarly in Figure 16 the

rate coverage has been analyzed against Γ1199083 and 1205821199083 Initially


SIR

cove

rage

()

070

075

080

085

minus90 minus80 minus70 minus60 minus50

Sensing threshold (Γw3) in dBm

w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3

Figure 15 SIR coverage as a function of sensing threshold and APdensity

Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2

u = 2000 ＯＭＬＭＥＧ2

minus130 minus120 minus110 minus100 minus90 minus80 minus70 minus60

Figure 16 Rate coverage as a function of sensing threshold undervarious user and AP density

the rate coverage improves by increasing Γ1199083 because thedensity of active APs increases and hence the average loadper AP decreases After a certain limit it becomes constantas 120582119905

1199083 997888rarr 1205821199083 Please see Remark 13 for an alternativeand detailed description of the results reported in Figure 16The rate coverage has been analyzed against users density andbandwidth of Wi-Fi RAT in Figure 17 which shows that theincrease inB1199083 greatly affects the rate coverage Apart fromthat as 120582119906 increases the rate coverage decreases because theaverage load per AP increases

In Figure 18 an interesting result has been reported bykeeping the density ratio of users and APs constant Although

Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20

Bandwidth (ℬw3) (MHz)

040

045

050

055

060

065

070

u = 2000 ＯＭＬskＧ2



Rate

cove

rage

(ℛ)

Figure 17 Variation in rate coverage as a function of bandwidth ofWi-Fi tier (B1199083) and AP density

Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3

Γw3 = minus92 dBmΓw3 = minus82 dBm

08

07

06

05

04

03

02

01

00

Figure 18 Rate coverage against constant user to AP density ratiowhen femto-tier is overlaid with Wi-Fi tier

the average load per AP has been kept fixed the rate coveragedeclines as the density increases and the sensing thresholddecreases This is due to the fact that the rate coveragedepends on four factors which include both the averageload and the MAP of a serving AP By increasing the APdensity and reducing the sensing threshold under a constantload the MAP decreases hence the overall rate coveragedeclines Please see Remark 12 for further details Undersuch situations increasing the number of non-overlappingchannels can improve the rate coverage


6 Conclusion

In this paper we have proposed a unified framework forSINR and rate coverage analysis of multi-RAT HetNets byconsidering different channel access schemes By assuming amulti-tierWi-Fi RAT we have derived theMAP for a randomand a tagged AP where the results show that the MAP for atypical AP approaches that of a tagged AP as density of Wi-Fi RAT approachesinfin It has been shown that by increasingthe density of Wi-Fi RAT the SIR coverage of multi-RATHetNet increases and gradually approaches unity Moreovermulti-RAT HetNets specifically with small cell tiers providebetter SIR coverage however as the user association withWi-Fi RAT increases the rate coverage starts declining Wehave also derived the stable queue probability of a user undercellular and Wi-Fi RAT by assuming a non-saturated trafficmodel The results show that the stable queue probabilityof a user under cellular RAT is better as compared to Wi-Fi RAT when packet arrival rate is high Although Wi-FiRAT provides better SIR coverage it is hard to maintainthe stability of a queue as the medium access probabilityof an AP is less than unity This result suggests that theun-bounded increase in the density of Wi-Fi RAT cannotbring significant improvement in users experience hencecare must be taken while planning the deployment of Wi-FiRAT

Recently research on ultra-dense small cell networkshas received significant attention Various tools and tech-niques like multi-slope path loss models LOS and non-LOSchannels and different shadowing effects have been usedto provide new insights However such investigations arelimited to single-tier single-RAT scenario A straightforwardextension of the proposed work is to incorporate such toolsfor the analysis of multi-RAT HetNets Another potentialarea for future work is the incorporation of queuing theoryevaluating the impact of traffic variations on the performancebounds of multi-RAT HetNets

Appendix

A Proof of Lemma 1

The proof is an extension of existing studies [10 15] TheMAP of an AP 119909119895 is the Palm probability that its mediumaccess indicator is 1 Given the timer of a typical AP 119905119895 = 119905the MAP can be derived as

119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)

119887= exp[minus119905 sum119899isinΦ119908

120582119908119899119860119908119899] (A3)

where (119886) follows from small-scale fading which is expo-nentially distributed with mean unity and the fact that thereceived signal strength from APs with timers less than 119905is of concern (119887) follows from Slyvniakrsquos theorem and theprobability generating functional (PGFL) of homogeneousPPP and finally we get (11) by deconditioning with respectto ldquo119905rdquo where 119905 sim 119880(0 1)B Proof of Lemma 5

Association of users based on the maximum average receivedsignal strength has been considered in this work Given thatthe tagged AP belonging to the 119895119905ℎ tier of Wi-Fi RAT islocated at 119909119900 = (119910 0) then the MAP can be given as

P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)

where 1199101015840 = 1120572119899119908119899 1199101120572119899 (119886) follows from deconditioning with

respect to 119909119900 = (119910 0) and (119887) is based on PGFL of PPPand cosine rule the PPP Φ119908 has been translated in such away that the tagged AP is located at origin for further detailsplease refer to Lemma 2 in [26] As shown in Figure 19 due

tomulti-tiers and association based on themaximum averagereceived signal strength it is possible that the tagged AP isnot the nearest one However it is the closest among APs ofthe tier to which it belongs to Thus 1199101015840 distance from a userto the tagged AP has been defined for properly locating the


ΦnltN

Φn=N

TaggedActive

Figure 19 Illustration for the scaling of distance in order to obtainthe radius of circle around the user when there is not any interferingAP

exclusion region around the user which does not include anyother AP This completes the proof and we get the final result(15)

C Proof of Lemma 8

For simplification here we drop the notation ldquo119908rdquo whichis used to denote the Wi-Fi RAT By following the givendefinition for cumulative interference under Section 222 weget

LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp

minus119904119895119875119899( sum119909isinΦ119905119899cap119861119888(01199101015840)

ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)

119887asymp exp [minus2120587P119905119895120582119899 intinfin

119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)

119888asymp expminus120587P119905

119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)

where (119886) follows from the independence of Φ119905119899 and ℎ119909 due

to PPP assumption for the set of interfering APs where119904119895 = 120579119895119910120572119895119875119895 (119887) is obtained using PGFL of PPP and (119888) isobtained through Laplace transform of exponential randomvariable with unit mean By assuming 119906 = (119904119895119875119899)minus21205721198991199092 thesimplified expressions are obtained Moreover for compactrepresentation a general expression given in [14] has beenused as

Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)

asymp expminus120587P119905119895120582119899 [(Z (119904119895119875119899 120572119899 119911120572119899119899119886 ) | 119910 lt 1198771198734 )

+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)

We have approximated the distance to nearest interfering APas given in (C7) and (C8) by using simple yet effective setof assumptions For better illustration let us assume a two-tier scenario as shown in Figure 20 where ldquo119877rdquo represents themean sensing radius for respective tiers and ldquo119910rdquo denotes thedistance from a user to the tagged AP Here the mean sensingradius (119877) has been obtained by using (12) Due to contentiondomains we assume that not any AP is allowed to transmitwithin an approximated region of mean sensing radius 119877119873

around the tagged AP which provides a lower bound onthe expected number of contending APs as discussed underRemark 4 The approximation is reasonable as the nearestactive AP can severely degrade the signal by causing excessiveinterference Further as clear from (C6) based on thedistance from a user to the tagged AP two different cases havebeen considered where the mean sensing radius of APs withminimum power level (119899 = 119873) is exploited as a referenceDue to 119873-tiers the tagged AP may not be the nearest onehowever it is the closest among APs of the tier to which itbelongs to That is the reason we are using 119877119873 as a referencefor defining two cases in (C6)

In part (119886) of Figure 20 a user is associated with an APof tier having minimum power of transmission (119895 = 119873)such that 119910 lt 1198771198734 Within approximately 119877119873 distancearound the tagged AP there cannot be any other active APTherefore the nearest interfering AP of any tier (119899 le 119873)is at least 3119910 distance apart from the user Further in part(119887) a user is associated with an AP of a tier having higherpower of transmission (119895 lt 119873) such that 119910 lt 1198771198734Under such situation due to differences in power levels theAPs of tiers with power less than the tagged AP (119899 gt 119873)can be closer to the user Therefore by exploiting the 119873119905ℎ

tier as a reference a generalized formula for approximatingthe distance to nearest interfering AP of any tier has beenobtained as 119911119899 asymp (4119877119899le119873119877119873 minus 1)119910 When interfering APbelongs to the 119873119905ℎ tier the expression simplifies to 119911119899 asymp 3119910Furthermore for 119899 lt 119873 assuming that 119877119873 asymp 4119910 theexpression simplifies to (119877119899lt119873minus119877119873)119910which is approximatelyequivalent to the nearest interfering AP as clear from part


lt

=

(a)

lty

lt (lt

minus )

(b)

Figure 20 Illustration for approximated distance to the nearest interfering AP under multi-tier Wi-Fi RAT

(119887) of Figure 20 It must be noted that the given formula isgeneralized enough and applicable to part (119886) as well

If the distance between a user and the tagged AP 119910 gt1198771198734 then we assume that the distance to nearest interferingAP is simply function of association [14] and is given by (C8)This approximation provides an upper bound on interferenceas some of the interfering APs within expected sensingregion of the tagged AP may not detect its presence due torandom fading effects Hence the supposed approximationsare tight and provide an upper bound on interference forWi-Fi RAT and this completes the proof

D Proof of Theorem 11

As defined in (7) the probability that a typical user receives arate greater than some defined threshold (120588119894119895) from the taggedAP is

R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)

where 120591119894119895(N119894119895) = 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1 C119894119895 is given in (8) andN119894119895 is the load under serving AP It must be noted herethat the rate coverage is function of rate threshold (120588119894119895) loadunder serving AP (N119894119895) transmission probability (P119905

119894119895) andbandwidth (B119894119895) of the AP By increasing P119905

119894119895 or B119894119895 anddecreasing 120588119894119895 or N119894119895 the rate coverage improves Howeverin case of Wi-Fi RAT higher P119905

119894119895 and lower N119894119895 cannot beachieved at the same time As for higher P119905

119894119895 lower density ofWi-Fi RAT is required whereas for lowerN119894119895 higher density

of RAT is required By using Lemma 3 of [32] the probabilitymass function (PMF) for 119899 number of users other than thetypical user under a tagged AP can be given as

P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)

where 119889 = 45 is a constant and the load under serving AP isgiven asN119894119895 =N119900119894119895 + 1

EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)

By following a procedure similar to [14] we use an approx-imation EN119894119895

[S119894119895120591119894119895(N119894119895)] asymp S119894119895[120591119894119895E(N119894119895)] where theexpected load under a serving AP is given as N119894119895 = 1 +128(120582119906A119894119895120582119894119895) Finally simplification of (D7) completes theproof

E Proof of Theorem 14

By assuming that single user is connected to each AP of aRAT [25] the conditional SIR or equivalently the service rateof a typical user at 119903119900 distance from the tagged AP has beendefined in [30] as

120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)

Please note that 120583Φ119894 is a random variable as it is conditionedon a particular PPP realization (Φ119894) therefore it can be


analyzed through a statistical distribution [24 25] In order toobtain a simplified solution by following an approach similarto [28] we approximate the service rate for cellular RAT by(E2) and for Wi-Fi RAT by (E3) however the presentedwork can be extended by following the given approaches in[24 25]

120583119888 asymp exp (minus1205871199021205821198881199032119900radic120579 arctanradic120579) (E2)

120583119908

asymp exp(minus120587119902P119905

1199081205821199081199032119900radic120579 arctan radic1205799 ) if 119903119900 lt 1198771199084exp (minus120587119902P119905

1199081205821199081199032119900radic120579 arctanradic120579) otherwise(E3)

Assuming that packet arrives at a user with rate 120585 during atime-slot then on average for a queue to be stable under acellular RAT the minimum required service rate is given by

120583119888 ge 120585 (E4)

and for Wi-Fi RAT it is given by

120583119908P119905119908 ge 120585 (E5)

This implies that under cellular RAT when the distancebetween a user and its tagged AP obeys the relation

1199032119900 le minus log 120585120587119902120582119888radic120579 arctanradic120579 (E6)

then the queue is stable given that the packet arrival rate is120585 Thus by exploiting the void probability [4] we obtain theprobability that the distance between a user and its tagged APis less than 119903119900 as

120600119888 (120579 120585 119902) = 1 minus exp( log 120585119902radic120579 arctanradic120579) (E7)

or equivalently it can be interpreted as the probability thatthe queue of a user under cellular RAT is stable as it iswithin a critical distance 119903119900 from the tagged AP For Wi-FiRAT depending on the distance (119903119900)with respect to expectedsensing radius of APs (119877119908) the distance to the nearestinterfering AP changes and hence the service rate Thus byusing (E3) and (E5) we obtain the following relation

1199032119900V le minus log (120585P119905119908)120587119902P119905

119908120582119908radic120579 arctan (radic120579V2) (E8)

where V = 3 when 119903119900 le 1198771199084 otherwise V = 1Further by exploiting the void probability [4] the stablequeue probability of a user under a Wi-Fi RAT dependingon the distance (119903119900) with respect to expected sensing radiusof APs (119877119908) can be given as

120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905

119908radic120579 arctan (radic1205799)) 1199031199003 lt 1198771199084

1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905

119908radic120579 arctanradic120579) 1199031199001 gt 1198771199084

(E9)

As a result with the help of indicator function we obtain thefinal expression (29)

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] Cisco ldquoGlobal mobile data traffic forecast update 2016-2021rdquo2017

[2] J G Andrews S Buzzi W Choi et al ldquoWhat will 5G berdquo IEEEJournal on Selected Areas in Communications vol 32 no 6 pp1065ndash1082 2014

[3] A Aijaz H Aghvami andM Amani ldquoA survey on mobile dataoffloading technical and business perspectivesrdquo IEEE WirelessCommunications Magazine vol 20 no 2 pp 104ndash112 2013

[4] J G Andrews F Baccelli and R K Ganti ldquoA tractable approachto coverage and rate in cellular networksrdquo IEEE Transactions onCommunications vol 59 no 11 pp 3122ndash3134 2011

[5] H S Dhillon R K Ganti F Baccelli and J G Andrews ldquoMod-eling and analysis of K-tier downlink heterogeneous cellular


networksrdquo IEEE Journal on Selected Areas in Communicationsvol 30 no 3 pp 550ndash560 2012

[6] D B Taylor H S Dhillon T D Novlan and J G AndrewsldquoPairwise interaction processes for modeling cellular networktopologyrdquo in Proceedings of the 2012 IEEE Global Communica-tions Conference GLOBECOM rsquo12 pp 4524ndash4529 December2012

[7] A Guo andM Haenggi ldquoSpatial stochasticmodels andmetricsfor the structure of base stations in cellular networksrdquo IEEETransactions on Wireless Communications vol 12 no 11 pp5800ndash5812 2013

[8] N Deng W Zhou and M Haenggi ldquoThe ginibre pointprocess as a model for wireless networks with repulsionrdquo IEEETransactions onWireless Communications vol 14 no 1 pp 107ndash121 2015

[9] Y Li F Baccelli H S Dhillon and J G Andrews ldquoStatisticalmodeling and probabilistic analysis of cellular networks withdeterminantal point processesrdquo IEEE Transactions on Commu-nications vol 63 no 9 pp 3405ndash3422 2015

[10] H Q Nguyen F Baccelli and D Kofman ldquoA stochasticgeometry analysis of dense IEEE 80211 networksrdquo in Proceed-ings of the 26th IEEE International Conference on ComputerCommunications INFOCOM rsquo07 pp 1199ndash1207 IEEE May2007

[11] GAlfanoMGaretto andE Leonardi ldquoNewdirections into thestochastic geometry analysis of dense CSMA networksrdquo IEEETransactions on Mobile Computing vol 13 no 2 pp 324ndash3262014

[12] RGiacomelli R KGanti andMHaenggi ldquoOutage probabilityof general ad hoc networks in the high-reliability regimerdquoIEEEACM Transactions on Networking vol 19 no 4 pp 1151ndash1163 2011

[13] H-S Jo Y J Sang P Xia and J G Andrews ldquoHeterogeneouscellular networks with flexible cell association a comprehensivedownlink SINR analysisrdquo IEEE Transactions on Wireless Com-munications vol 11 no 10 pp 3484ndash3494 2012

[14] S Singh H S Dhillon and J G Andrews ldquoOffloading in het-erogeneous networks modeling analysis and design insightsrdquoIEEE Transactions on Wireless Communications vol 12 no 5pp 2484ndash2497 2013

[15] Y Li F Baccelli J G Andrews T D Novlan and J CZhang ldquoModeling and analyzing the coexistence of Wi-Fi andLTE in unlicensed spectrumrdquo IEEE Transactions on WirelessCommunications vol 15 no 9 pp 6310ndash6326 2016

[16] C-H Liu and H-C Tsai ldquoOn the limits of coexisting coverageand capacity in multi-RAT heterogeneous networksrdquo IEEETransactions on Wireless Communications vol 16 no 5 pp3086ndash3101 2017

[17] J G Andrews X Zhang G D Durgin and A K Gupta ldquoArewe approaching the fundamental limits of wireless networkdensificationrdquo IEEE Communications Magazine vol 54 no 10pp 184ndash190 2016

[18] B Yang G Mao M Ding X Ge and X Tao ldquoDense small cellnetworks from noise-limited to dense interference-limitedrdquoIEEE Transactions on Vehicular Technology 2018

[19] M Ding D Lopez-Perez G Mao and Z Lin ldquoPerformanceimpact of idle mode capability on dense small cell networksrdquoIEEE Transactions on Vehicular Technology vol 66 no 11 pp10446ndash10460 2017

[20] B Yang G Mao X Ge M Ding and X Yang ldquoOn the energy-efficient deployment for ultra-dense heterogeneous networks

with NLoS and LoS transmissionsrdquo IEEE Transactions on GreenCommunications and Networking vol 2 no 2 pp 369ndash3842018

[21] H Elshaer M N Kulkarni F Boccardi J G Andrews and MDohler ldquoDownlink and uplink cell association with traditionalmacrocells and millimeter wave small cellsrdquo IEEE TransactionsonWireless Communications vol 15 no 9 pp 6244ndash6258 2016

[22] G Ghatak A De Domenico and M Coupechoux ldquoCoverageanalysis and load balancing in HetNets with millimeter wavemulti-RAT small cellsrdquo IEEE Transactions on Wireless Commu-nications vol 17 no 5 pp 3154ndash3169 2018

[23] F Boccardi R W Heath A Lozano T L Marzetta and PPopovski ldquoFive disruptive technology directions for 5Grdquo IEEECommunications Magazine vol 52 no 2 pp 74ndash80 2014

[24] Y Zhong T Q S Quek and X Ge ldquoHeterogeneous cellu-lar networks with spatio-temporal traffic delay analysis andschedulingrdquo IEEE Journal on SelectedAreas in Communicationsvol 35 no 6 pp 1373ndash1386 2017

[25] H H Yang and T Q Quek ldquoSIR coverage analysis in cel-lular networks with temporal traffic a stochastic geometryapproachrdquo 2018 httpsarxivorgabs180109888

[26] X Wang T Q S Quek M Sheng and J Li ldquoThroughput andfairness analysis ofWi-Fi and LTE-U in unlicensed bandrdquo IEEEJournal on Selected Areas in Communications vol 35 no 1 pp63ndash78 2017

[27] M Haenggi ldquoMean interference in hard-core wireless net-worksrdquo IEEE Communications Letters vol 15 no 8 pp 792ndash794 2011

[28] Y Zhong GWang R Li T Han X Ge and T Q Quek ldquoEffectof spatial and temporal traffic statistics on the performance ofwireless networksrdquo 2018 httpsarxivorgabs180406754

[29] Y Zhong M Haenggi T Q S Quek and W Zhang ldquoOn thestability of static poisson networks under random accessrdquo IEEETransactions on Communications vol 64 no 7 pp 2985ndash29982016

[30] H ElSawy A Sultan-Salem M-S Alouini and M Z WinldquoModeling and analysis of cellular networks using stochasticgeometry a tutorialrdquo IEEE Communications Surveys amp Tutori-als vol 19 no 1 pp 167ndash203 2017

[31] D Lopez-Perez M Ding H Claussen and A H JafarildquoTowards 1 GbpsUE in cellular systems understanding ultra-dense small cell deploymentsrdquo IEEE Communications Surveys ampTutorials vol 17 no 4 pp 2078ndash2101 2015

[32] S M Yu and S-L Kim ldquoDownlink capacity and base stationdensity in cellular networksrdquo in Proceedings of the 11th Inter-national Symposium on Modeling and in Mobile Ad Hoc andWireless Networks (WiOpt rsquo13) pp 119ndash124 IEEE 2013

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of single-tier and multi-tier cellular networks [4 5] wheremostly Poisson point process (PPP) is used for modeling thespatial location of access points (APs) The PPP assumptionprovides a lower bound whereas the traditional grid modelprovides an upper bound on coverage Other point processeslike Gibbs [6] Strauss [7] Ginibre [8] and Detriminantal [9]have also been used for coverage analysis of cellular networksSuch processes provide better accuracy as compared to PPPbut at the cost of limited tractability Therefore PPP hasbeen a popular choice for analysis of cellular networksOn the other hand when it comes to Wi-Fi RAT PPPcannot be used for spatial modeling of active APs due tocontention-based nature of carrier sense multiple accesswith collision avoidance (CSMACA) scheme Howeverapproximated solution for SINR coverage has been presentedin [10] by exploiting Modified Matern Hard Core Process(MMHCP) for estimating the set of active APs where theAPs are originally distributed using a PPPThework is furtherextended from spatial averages to spatial distributions in [11]and throughput analysis has also been conducted againstvarious parameters of interest Due to approximation of theset of interfering APs through a non-homogeneous PPP theresulting expressions in [10 11] are extremely complicatedThe asymptotic expression for outage probability of generalad-hoc networks has been obtained in [12] however theresults are limited to high signal-to-interference ratio (SIR)cases

Although closed-form expressions for analysis of cellularRAT under special cases are available in literature [4 13]no such results are reported for Wi-Fi RAT due to difficultyin characterizing the interference effect of active APs [1011] This limits the in-depth analysis of multi-RAT HetNetswhich could be an important tool for coverage and capacityplanning of future wireless networks A general model formulti-RAT HetNets has been presented in [14] by assumingindependent PPPs for distribution of APs and users in agiven region However the effect of channel access schemesassociated with different RATs has not been taken intoaccount which directly affects the interference and hence thecoverage analysis Thus such a framework cannot provideaccurate insights into the different characteristics exhibitedby multi-RAT HetNets

Recently the coexistence problemofmulti-RATnetworkshas received considerable attention In [15 16] the coverageand capacity analysis has been presented by assuming thecoexistence of cellular and Wi-Fi RAT in unlicensed spec-trum After investigating various transmission mechanismthe authors in [15] reported that the LTE can coexist withWi-Fi but under certain conditions On the other hand in [16] asimilar issue has been addressed by assuming the operation ofLTE users in both licensed and unlicensed band Two differ-ent user association schemes ie crossing and non-crossing-RAT have been considered where the users of licensedRAT can access unlicensed band by exploiting opportunisticCSMACA scheme The crossing-RAT user association isshown to provide better performance as compared to non-crossing-RAT user association However the coexistenceof LTE and Wi-Fi users in unlicensed band cannot bringsignificant gain in performance because increase in the

capacity of LTE RAT is achieved at the cost of decrease inthe capacity of Wi-Fi RAT [16] Therefore in this researchthe coexistence of different RATs in unlicensed band has notbeen covered

Some recent studies have investigated the fundamentallimits on densification of cellular network by assumingdifferent path loss models and fading distributions [17ndash20]According to [17] the SINR coverage decreases after a certainthreshold due to increased interference with smaller pathloss exponents which is in contrast to widely used results asreported in [4] A detailed investigation has been presentedin [18] by exploring four different performance regimes whiletransitioning from sparse to dense networks Results similarto [17] are reported where after a certain threshold theSIR coverage no longer remains constant however the areaspectral efficiency increases linearly According to [19] theSIR coverage can be increased by exploiting the idle modecapability of APs under dense scenarios Further in [20]energy efficiency analysis has been presented by defining theoptimal transmission power for APs as a function of RATdensity However in all of the mentioned investigations asingle-tier dense cellular network with small cells has beenconsidered Moreover in few of the existing studies [21 22]multi-RAT networks have been investigated by assumingoperation of APs in different bands In [21] the sub-6GHzmacrocells are overlaid with mm-wave small cells and in[22] the small cells can operate on both bands Various cellassociation schemes considering both uplink and downlinkchannels have been investigated in [21] Following a similarthought in [22] a biasing based strategy has been proposedfor load balancing across a multi-RAT network In contrastto traditional approaches in [22] two biasing thresholds areexploited one for offloading users from macrocells to smallcells and other for offloading users from sub-GHz bandto mm-wave band In a nutshell the integration of mm-wave communication into existing infrastructure is one ofthe potential contributors for increasing capacity of futurewireless networks However the communication in mm-wave exhibits the characteristics which are different fromsub-GHz band and hence its realization requires a lot ofenhancements both at component and architecture level [23]

Densemulti-RATHetNetswould be a key aspect of futurewireless networks [2] According to some studies [17 18]the degradation in coverage provided by dense small cellnetworks after a certain limit is expectedTherefore in orderto meet the demands in coming future the focus must beshifted from single-RAT to multi-RAT HetNets Although infew of the recent studies [21 22] the multi-RAT networkshave been analyzed by exploiting sub-GHz and mm-waveband this work is focused around the analysis of multi-RATHetNets where each RAT operates on a different pool ofresources and can use either contention-based or contention-free channel access schemes As cellular contention-free andWi-Fi contention-based RATs are already deployed at a widescale and hence their integration can be considered as apotential contributor for improving the capacity of futurewireless networks without many modifications into existinginfrastructure Thus the key motivating factor behind thiswork is the growing convergence of the two RATs and as











2 System Model


119895=1Φ119888119895 and Φ119908 = cup119873119895=1Φ119908119895







P119905119908 P

119905119908





compliment120600119888120600119908









119891119884119894119895(119910)= 2120587120582119894119895

A119894119895

expminus120587 sum

(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899119910 (1)


A119894119895 = 2120587120582119894119895 intinfin

0exp

minus120587 sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899sdot 119910119889119910(2)












119890119905119895 = prod119909119899isinΦ119908119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899) (3)



S119894119895 (120579119894119895)= E119910 P [SINR119894119895 (119910) gt 120579119894119895 | 119909119900 = (119910 0) 119890119905119909119900 = 1] (4)

where

SINR119894119895 (119910) = 119875119894119895ℎ119910119897 (10038171003817100381710038171199101003817100381710038171003817)sum119899isinΦ119894I119894119899 + 1205902

119894

(5)



S = sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895) (6)



R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (7)

where

C119894119895 = B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 (8)



R = sum(119894119895)isinΦ119886

A119894119895R119894119895 (120588119894119895) (9)



120583 gt 120585 (10)





lt





120582119908119899119860119908n)sum119899isinΦ119908120582119908119899119860119908119899

(11)

where

119860119908119899 = intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 (12)







250 500 750 1000 1250 1500 1750 2000



w3 = w

w2 = 02w w3 = 08w

w2 = 05w w3 = 05w

w2 = w

00

02

04

06

08

10

Med

ium

acce

ss p

roba

bilit

y (

t w)



120582119908119899119860119908119899)sum119899isinΦ119908120582119908119899119860119908119899

(13)


and

119877119908119899 = ( 1205821198884120587radic119875119908119899Γ119908119899

)(2120572119899)

(14)


120582119908119899119860119908119899



119908 gt 119875119905119908


(a)

(b)

(c)




P119905119908119895

= intinfin

0


120582119908119899 [119860119908119899 minus 119860119908119899 (119910)] 119891119884119908119895(119910) (15)

where

119860119908119899 (119910)= 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909 (16)



tagged APrandom AP

10

08

06

04

02

00

Med

ium

acce

ss p

roba

bilit

y (

t w)

w3 = w

w2 = 02w w3 = 08w

w2 = 05w w3 = 05w

w2 = w




119908 =sum119895isinΦ119908A119908119895P

119905119908119895



P[N

o A

P clo

ser t

hanR

]


w = 100 ０ＥＧ2

w = 500 ０ＥＧ2

w = 1000 ０ＥＧ2

w = 1500 ０ＥＧ2


10

08

06

04

02

000 20 40 60 80 100 120






4 Coverage




119908)with density120582119905119908 = P119905



119908119899) with density120582119905119908119899 = P119905



LI119908119899(119904119908119895)


) | 119910 lt 1198771199081198734 )+ (Z (119904119908119895119875119908119899 120572119899 119911120572119899

119908119899119887) | 119910 gt 1198771199081198734 )]

(17)


and 119911119908119899119887are defined







2120587120582119888119895 intinfin

0exp(minus1199041198881198951205902

119888 minus 120587 sum119899isinΦ119888

120582119888119899Z (119904119888119895119875119888119899 120572119899 119911120572119899119888119899119887) + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)119910119889119910+ sum

119895isinΦ119908

2120587120582119908119895 intinfin

0exp(minus1199041199081198951205902

119908 minus 120587P119905119908119895


120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119886) | 119910 lt 1198771199081198734 ] + 120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119887

) | 119910 gt 1198771199081198734 ] + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)sdot 119910119889119910(18)



(119888119886)2119887

1198891199061 + 1199061198872 (19)


S119894119895 (120579119894119895) 119886= int119910gt0

E[[expminus

120579119894119895119897 (10038171003817100381710038171199101003817100381710038171003817)119875119894119895

(sum119899isinΦ119894

119875119894119899 sum119909isinΦ119899cap119861119888(01199101015840)

119890119905119909ℎ119909119897 (119909) + 1205902119894 ) | 119890119905119909119900 = 1119891119884119894119895

(119910) 119889119910]] (20)


E[[expminus119904119894119895(sum

119899isinΦ119905119894

I119894119899 + 1205902119894 )119891119884119894119895

(119910) 119889119910]] (21)

119888= int119910gt0

119890minus1199041198941198951205902119894 prod119899isinΦ119905119894

LI119894119899(minus119904119894119895) 119891119884119894119895

(119910) 119889119910 (22)






119908 = 0 with 120572119895 = 4 and P119905119908119895 asymp P119905



11990811989516) 120574119908119895 (3))120574119908119895 (3)+ exp (minus120587120582119908119895 (1198772

11990811989516) 120574119908119895 (1))120574119908119895 (1) (23)

where


V2) (24)




SIR

cove

rage

(w

)w = 1000 ０ＥＧ2

SimNum


10

08

06

04

02

00

w3 = w

w2 = 02w w3 = 08w

w2 = 052 w3 = 05ww2 = w

minus10 minus5 0 5 10 15 20







R asymp sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895 = 120591119894119895 (N119894119895)) (25)


00

02

04

06

08

10

SIR

cove

rage

()

Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

minus5 0 5 10 15 20minus10






120588119888119895N119888119895

P119905119888119895B119888119895

= 120588119888119895B119888119895

+ 128120588119888119895120582119906A119888119895120582119888119895B119888119895

(26)



119908119895 asymp P119905119908119895

we get

120588119908119895N119908119895

P119905119908119895B119908119895


(27)





te co

vera

ge (ℛ

)


Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

10

08

06

04

02

000 2 4 6 8 10





Rate

cove

rage

(ℛ)

Φa = Φc2

Φa = Φw3

Φa = Φc2 + Φw3

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

06

05

04

03

02

01

000 250 500 750 1000 1250 1500 1750 2000









Stab

le q

ueue

pro

babi

lity

()

= 15 dB

tw=046

10

08

06

04

02

0000 02 04 06 08 10

Φc -domΦw -dom

Φc -modΦw-mod





120600119908 (120579 120585 119902) = [1 minus exp( log (120585P119905119908)119902P119905



1198772

11990816 )]11199031199003ge1198771199084

+ [exp(minus120587120582119908

1198772

11990816 )minus exp( log (120585P119905


(29)

respectively where

1199032119900V = minus log (120585P119905119908)120587119902P119905





Φc -domΦw -dom

Φc -modΦw-mod

Stab

le q

ueue

pro

babi

lity

()

10

08

06

04

02

00


= 03


000

005

010

015

020

025

030

035

040St

able

que

ue p

roba

bilit

y (

) = 15 dB = 03

tw=029

tw=031


250 500 750 1000 1250 1500 1750 2000

Φc -domΦw -dom

Φc -modΦw-mod





Φcj

Φwj


0 250 500 750 1000 1250 1500 1750 2000

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

Φa = Φc2 + Φw3

00

02

04

06

08

10A

ssoc

iatio

n pr

obab

ility

(ij

)








120582119908119895 = 120582119888119895

119891119908119895119891119888119895

radic 119875119888119895119875119908119895

(31)





200 400 600 800 1000 1200 1400 1600 1800 2000


c2 = w3 = 3 ＞

SIR

cove

rage

()

Exs[14] Φa = Φc1

Exs[14] Φa = Φc2

Exs[14] Φa = Φw3

080

075

070

065

060

055

050

045









SIR

cove

rage

()

070

075

080

085



w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3


Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2







Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20


040

045

050

055

060

065

070




Rate

cove

rage

(ℛ)


Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3


08

07

06

05

04

03

02

01

00




6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












2 System Model


119895=1Φ119888119895 and Φ119908 = cup119873119895=1Φ119908119895







P119905119908 P

119905119908





compliment120600119888120600119908









119891119884119894119895(119910)= 2120587120582119894119895

A119894119895

expminus120587 sum

(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899119910 (1)


A119894119895 = 2120587120582119894119895 intinfin

0exp

minus120587 sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899sdot 119910119889119910(2)












119890119905119895 = prod119909119899isinΦ119908119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899) (3)



S119894119895 (120579119894119895)= E119910 P [SINR119894119895 (119910) gt 120579119894119895 | 119909119900 = (119910 0) 119890119905119909119900 = 1] (4)

where

SINR119894119895 (119910) = 119875119894119895ℎ119910119897 (10038171003817100381710038171199101003817100381710038171003817)sum119899isinΦ119894I119894119899 + 1205902

119894

(5)



S = sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895) (6)



R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (7)

where

C119894119895 = B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 (8)



R = sum(119894119895)isinΦ119886

A119894119895R119894119895 (120588119894119895) (9)



120583 gt 120585 (10)





lt





120582119908119899119860119908n)sum119899isinΦ119908120582119908119899119860119908119899

(11)

where

119860119908119899 = intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 (12)







250 500 750 1000 1250 1500 1750 2000



w3 = w

w2 = 02w w3 = 08w

w2 = 05w w3 = 05w

w2 = w

00

02

04

06

08

10

Med

ium

acce

ss p

roba

bilit

y (

t w)



120582119908119899119860119908119899)sum119899isinΦ119908120582119908119899119860119908119899

(13)


and

119877119908119899 = ( 1205821198884120587radic119875119908119899Γ119908119899

)(2120572119899)

(14)


120582119908119899119860119908119899



119908 gt 119875119905119908


(a)

(b)

(c)




P119905119908119895

= intinfin

0


120582119908119899 [119860119908119899 minus 119860119908119899 (119910)] 119891119884119908119895(119910) (15)

where

119860119908119899 (119910)= 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909 (16)



tagged APrandom AP

10

08

06

04

02

00

Med

ium

acce

ss p

roba

bilit

y (

t w)

w3 = w

w2 = 02w w3 = 08w

w2 = 05w w3 = 05w

w2 = w




119908 =sum119895isinΦ119908A119908119895P

119905119908119895



P[N

o A

P clo

ser t

hanR

]


w = 100 ０ＥＧ2

w = 500 ０ＥＧ2

w = 1000 ０ＥＧ2

w = 1500 ０ＥＧ2


10

08

06

04

02

000 20 40 60 80 100 120






4 Coverage




119908)with density120582119905119908 = P119905



119908119899) with density120582119905119908119899 = P119905



LI119908119899(119904119908119895)


) | 119910 lt 1198771199081198734 )+ (Z (119904119908119895119875119908119899 120572119899 119911120572119899

119908119899119887) | 119910 gt 1198771199081198734 )]

(17)


and 119911119908119899119887are defined







2120587120582119888119895 intinfin

0exp(minus1199041198881198951205902

119888 minus 120587 sum119899isinΦ119888

120582119888119899Z (119904119888119895119875119888119899 120572119899 119911120572119899119888119899119887) + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)119910119889119910+ sum

119895isinΦ119908

2120587120582119908119895 intinfin

0exp(minus1199041199081198951205902

119908 minus 120587P119905119908119895


120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119886) | 119910 lt 1198771199081198734 ] + 120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119887

) | 119910 gt 1198771199081198734 ] + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)sdot 119910119889119910(18)



(119888119886)2119887

1198891199061 + 1199061198872 (19)


S119894119895 (120579119894119895) 119886= int119910gt0

E[[expminus

120579119894119895119897 (10038171003817100381710038171199101003817100381710038171003817)119875119894119895

(sum119899isinΦ119894

119875119894119899 sum119909isinΦ119899cap119861119888(01199101015840)

119890119905119909ℎ119909119897 (119909) + 1205902119894 ) | 119890119905119909119900 = 1119891119884119894119895

(119910) 119889119910]] (20)


E[[expminus119904119894119895(sum

119899isinΦ119905119894

I119894119899 + 1205902119894 )119891119884119894119895

(119910) 119889119910]] (21)

119888= int119910gt0

119890minus1199041198941198951205902119894 prod119899isinΦ119905119894

LI119894119899(minus119904119894119895) 119891119884119894119895

(119910) 119889119910 (22)






119908 = 0 with 120572119895 = 4 and P119905119908119895 asymp P119905



11990811989516) 120574119908119895 (3))120574119908119895 (3)+ exp (minus120587120582119908119895 (1198772

11990811989516) 120574119908119895 (1))120574119908119895 (1) (23)

where


V2) (24)




SIR

cove

rage

(w

)w = 1000 ０ＥＧ2

SimNum


10

08

06

04

02

00

w3 = w

w2 = 02w w3 = 08w

w2 = 052 w3 = 05ww2 = w

minus10 minus5 0 5 10 15 20







R asymp sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895 = 120591119894119895 (N119894119895)) (25)


00

02

04

06

08

10

SIR

cove

rage

()

Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

minus5 0 5 10 15 20minus10






120588119888119895N119888119895

P119905119888119895B119888119895

= 120588119888119895B119888119895

+ 128120588119888119895120582119906A119888119895120582119888119895B119888119895

(26)



119908119895 asymp P119905119908119895

we get

120588119908119895N119908119895

P119905119908119895B119908119895


(27)





te co

vera

ge (ℛ

)


Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

10

08

06

04

02

000 2 4 6 8 10





Rate

cove

rage

(ℛ)

Φa = Φc2

Φa = Φw3

Φa = Φc2 + Φw3

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

06

05

04

03

02

01

000 250 500 750 1000 1250 1500 1750 2000









Stab

le q

ueue

pro

babi

lity

()

= 15 dB

tw=046

10

08

06

04

02

0000 02 04 06 08 10

Φc -domΦw -dom

Φc -modΦw-mod





120600119908 (120579 120585 119902) = [1 minus exp( log (120585P119905119908)119902P119905



1198772

11990816 )]11199031199003ge1198771199084

+ [exp(minus120587120582119908

1198772

11990816 )minus exp( log (120585P119905


(29)

respectively where

1199032119900V = minus log (120585P119905119908)120587119902P119905





Φc -domΦw -dom

Φc -modΦw-mod

Stab

le q

ueue

pro

babi

lity

()

10

08

06

04

02

00


= 03


000

005

010

015

020

025

030

035

040St

able

que

ue p

roba

bilit

y (

) = 15 dB = 03

tw=029

tw=031


250 500 750 1000 1250 1500 1750 2000

Φc -domΦw -dom

Φc -modΦw-mod





Φcj

Φwj


0 250 500 750 1000 1250 1500 1750 2000

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

Φa = Φc2 + Φw3

00

02

04

06

08

10A

ssoc

iatio

n pr

obab

ility

(ij

)








120582119908119895 = 120582119888119895

119891119908119895119891119888119895

radic 119875119888119895119875119908119895

(31)





200 400 600 800 1000 1200 1400 1600 1800 2000


c2 = w3 = 3 ＞

SIR

cove

rage

()

Exs[14] Φa = Φc1

Exs[14] Φa = Φc2

Exs[14] Φa = Φw3

080

075

070

065

060

055

050

045









SIR

cove

rage

()

070

075

080

085



w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3


Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2







Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20


040

045

050

055

060

065

070




Rate

cove

rage

(ℛ)


Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3


08

07

06

05

04

03

02

01

00




6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






P119905119908 P

119905119908





compliment120600119888120600119908









119891119884119894119895(119910)= 2120587120582119894119895

A119894119895

expminus120587 sum

(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899119910 (1)


A119894119895 = 2120587120582119894119895 intinfin

0exp

minus120587 sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899sdot 119910119889119910(2)












119890119905119895 = prod119909119899isinΦ119908119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899) (3)



S119894119895 (120579119894119895)= E119910 P [SINR119894119895 (119910) gt 120579119894119895 | 119909119900 = (119910 0) 119890119905119909119900 = 1] (4)

where

SINR119894119895 (119910) = 119875119894119895ℎ119910119897 (10038171003817100381710038171199101003817100381710038171003817)sum119899isinΦ119894I119894119899 + 1205902

119894

(5)



S = sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895) (6)



R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (7)

where

C119894119895 = B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 (8)



R = sum(119894119895)isinΦ119886

A119894119895R119894119895 (120588119894119895) (9)



120583 gt 120585 (10)





lt





120582119908119899119860119908n)sum119899isinΦ119908120582119908119899119860119908119899

(11)

where

119860119908119899 = intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 (12)







250 500 750 1000 1250 1500 1750 2000



w3 = w

w2 = 02w w3 = 08w

w2 = 05w w3 = 05w

w2 = w

00

02

04

06

08

10

Med

ium

acce

ss p

roba

bilit

y (

t w)



120582119908119899119860119908119899)sum119899isinΦ119908120582119908119899119860119908119899

(13)


and

119877119908119899 = ( 1205821198884120587radic119875119908119899Γ119908119899

)(2120572119899)

(14)


120582119908119899119860119908119899



119908 gt 119875119905119908


(a)

(b)

(c)




P119905119908119895

= intinfin

0


120582119908119899 [119860119908119899 minus 119860119908119899 (119910)] 119891119884119908119895(119910) (15)

where

119860119908119899 (119910)= 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909 (16)



tagged APrandom AP

10

08

06

04

02

00

Med

ium

acce

ss p

roba

bilit

y (

t w)

w3 = w

w2 = 02w w3 = 08w

w2 = 05w w3 = 05w

w2 = w




119908 =sum119895isinΦ119908A119908119895P

119905119908119895



P[N

o A

P clo

ser t

hanR

]


w = 100 ０ＥＧ2

w = 500 ０ＥＧ2

w = 1000 ０ＥＧ2

w = 1500 ０ＥＧ2


10

08

06

04

02

000 20 40 60 80 100 120






4 Coverage




119908)with density120582119905119908 = P119905



119908119899) with density120582119905119908119899 = P119905



LI119908119899(119904119908119895)


) | 119910 lt 1198771199081198734 )+ (Z (119904119908119895119875119908119899 120572119899 119911120572119899

119908119899119887) | 119910 gt 1198771199081198734 )]

(17)


and 119911119908119899119887are defined







2120587120582119888119895 intinfin

0exp(minus1199041198881198951205902

119888 minus 120587 sum119899isinΦ119888

120582119888119899Z (119904119888119895119875119888119899 120572119899 119911120572119899119888119899119887) + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)119910119889119910+ sum

119895isinΦ119908

2120587120582119908119895 intinfin

0exp(minus1199041199081198951205902

119908 minus 120587P119905119908119895


120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119886) | 119910 lt 1198771199081198734 ] + 120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119887

) | 119910 gt 1198771199081198734 ] + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)sdot 119910119889119910(18)



(119888119886)2119887

1198891199061 + 1199061198872 (19)


S119894119895 (120579119894119895) 119886= int119910gt0

E[[expminus

120579119894119895119897 (10038171003817100381710038171199101003817100381710038171003817)119875119894119895

(sum119899isinΦ119894

119875119894119899 sum119909isinΦ119899cap119861119888(01199101015840)

119890119905119909ℎ119909119897 (119909) + 1205902119894 ) | 119890119905119909119900 = 1119891119884119894119895

(119910) 119889119910]] (20)


E[[expminus119904119894119895(sum

119899isinΦ119905119894

I119894119899 + 1205902119894 )119891119884119894119895

(119910) 119889119910]] (21)

119888= int119910gt0

119890minus1199041198941198951205902119894 prod119899isinΦ119905119894

LI119894119899(minus119904119894119895) 119891119884119894119895

(119910) 119889119910 (22)






119908 = 0 with 120572119895 = 4 and P119905119908119895 asymp P119905



11990811989516) 120574119908119895 (3))120574119908119895 (3)+ exp (minus120587120582119908119895 (1198772

11990811989516) 120574119908119895 (1))120574119908119895 (1) (23)

where


V2) (24)




SIR

cove

rage

(w

)w = 1000 ０ＥＧ2

SimNum


10

08

06

04

02

00

w3 = w

w2 = 02w w3 = 08w

w2 = 052 w3 = 05ww2 = w

minus10 minus5 0 5 10 15 20







R asymp sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895 = 120591119894119895 (N119894119895)) (25)


00

02

04

06

08

10

SIR

cove

rage

()

Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

minus5 0 5 10 15 20minus10






120588119888119895N119888119895

P119905119888119895B119888119895

= 120588119888119895B119888119895

+ 128120588119888119895120582119906A119888119895120582119888119895B119888119895

(26)



119908119895 asymp P119905119908119895

we get

120588119908119895N119908119895

P119905119908119895B119908119895


(27)





te co

vera

ge (ℛ

)


Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

10

08

06

04

02

000 2 4 6 8 10





Rate

cove

rage

(ℛ)

Φa = Φc2

Φa = Φw3

Φa = Φc2 + Φw3

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

06

05

04

03

02

01

000 250 500 750 1000 1250 1500 1750 2000









Stab

le q

ueue

pro

babi

lity

()

= 15 dB

tw=046

10

08

06

04

02

0000 02 04 06 08 10

Φc -domΦw -dom

Φc -modΦw-mod





120600119908 (120579 120585 119902) = [1 minus exp( log (120585P119905119908)119902P119905



1198772

11990816 )]11199031199003ge1198771199084

+ [exp(minus120587120582119908

1198772

11990816 )minus exp( log (120585P119905


(29)

respectively where

1199032119900V = minus log (120585P119905119908)120587119902P119905





Φc -domΦw -dom

Φc -modΦw-mod

Stab

le q

ueue

pro

babi

lity

()

10

08

06

04

02

00


= 03


000

005

010

015

020

025

030

035

040St

able

que

ue p

roba

bilit

y (

) = 15 dB = 03

tw=029

tw=031


250 500 750 1000 1250 1500 1750 2000

Φc -domΦw -dom

Φc -modΦw-mod





Φcj

Φwj


0 250 500 750 1000 1250 1500 1750 2000

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

Φa = Φc2 + Φw3

00

02

04

06

08

10A

ssoc

iatio

n pr

obab

ility

(ij

)








120582119908119895 = 120582119888119895

119891119908119895119891119888119895

radic 119875119888119895119875119908119895

(31)





200 400 600 800 1000 1200 1400 1600 1800 2000


c2 = w3 = 3 ＞

SIR

cove

rage

()

Exs[14] Φa = Φc1

Exs[14] Φa = Φc2

Exs[14] Φa = Φw3

080

075

070

065

060

055

050

045









SIR

cove

rage

()

070

075

080

085



w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3


Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2







Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20


040

045

050

055

060

065

070




Rate

cove

rage

(ℛ)


Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3


08

07

06

05

04

03

02

01

00




6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










119890119905119895 = prod119909119899isinΦ119908119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899) (3)



S119894119895 (120579119894119895)= E119910 P [SINR119894119895 (119910) gt 120579119894119895 | 119909119900 = (119910 0) 119890119905119909119900 = 1] (4)

where

SINR119894119895 (119910) = 119875119894119895ℎ119910119897 (10038171003817100381710038171199101003817100381710038171003817)sum119899isinΦ119894I119894119899 + 1205902

119894

(5)



S = sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895) (6)



R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (7)

where

C119894119895 = B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 (8)



R = sum(119894119895)isinΦ119886

A119894119895R119894119895 (120588119894119895) (9)



120583 gt 120585 (10)





lt





120582119908119899119860119908n)sum119899isinΦ119908120582119908119899119860119908119899

(11)

where

119860119908119899 = intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 (12)







250 500 750 1000 1250 1500 1750 2000



w3 = w

w2 = 02w w3 = 08w

w2 = 05w w3 = 05w

w2 = w

00

02

04

06

08

10

Med

ium

acce

ss p

roba

bilit

y (

t w)



120582119908119899119860119908119899)sum119899isinΦ119908120582119908119899119860119908119899

(13)


and

119877119908119899 = ( 1205821198884120587radic119875119908119899Γ119908119899

)(2120572119899)

(14)


120582119908119899119860119908119899



119908 gt 119875119905119908


(a)

(b)

(c)




P119905119908119895

= intinfin

0


120582119908119899 [119860119908119899 minus 119860119908119899 (119910)] 119891119884119908119895(119910) (15)

where

119860119908119899 (119910)= 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909 (16)



tagged APrandom AP

10

08

06

04

02

00

Med

ium

acce

ss p

roba

bilit

y (

t w)

w3 = w

w2 = 02w w3 = 08w

w2 = 05w w3 = 05w

w2 = w




119908 =sum119895isinΦ119908A119908119895P

119905119908119895



P[N

o A

P clo

ser t

hanR

]


w = 100 ０ＥＧ2

w = 500 ０ＥＧ2

w = 1000 ０ＥＧ2

w = 1500 ０ＥＧ2


10

08

06

04

02

000 20 40 60 80 100 120






4 Coverage




119908)with density120582119905119908 = P119905



119908119899) with density120582119905119908119899 = P119905



LI119908119899(119904119908119895)


) | 119910 lt 1198771199081198734 )+ (Z (119904119908119895119875119908119899 120572119899 119911120572119899

119908119899119887) | 119910 gt 1198771199081198734 )]

(17)


and 119911119908119899119887are defined







2120587120582119888119895 intinfin

0exp(minus1199041198881198951205902

119888 minus 120587 sum119899isinΦ119888

120582119888119899Z (119904119888119895119875119888119899 120572119899 119911120572119899119888119899119887) + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)119910119889119910+ sum

119895isinΦ119908

2120587120582119908119895 intinfin

0exp(minus1199041199081198951205902

119908 minus 120587P119905119908119895


120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119886) | 119910 lt 1198771199081198734 ] + 120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119887

) | 119910 gt 1198771199081198734 ] + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)sdot 119910119889119910(18)



(119888119886)2119887

1198891199061 + 1199061198872 (19)


S119894119895 (120579119894119895) 119886= int119910gt0

E[[expminus

120579119894119895119897 (10038171003817100381710038171199101003817100381710038171003817)119875119894119895

(sum119899isinΦ119894

119875119894119899 sum119909isinΦ119899cap119861119888(01199101015840)

119890119905119909ℎ119909119897 (119909) + 1205902119894 ) | 119890119905119909119900 = 1119891119884119894119895

(119910) 119889119910]] (20)


E[[expminus119904119894119895(sum

119899isinΦ119905119894

I119894119899 + 1205902119894 )119891119884119894119895

(119910) 119889119910]] (21)

119888= int119910gt0

119890minus1199041198941198951205902119894 prod119899isinΦ119905119894

LI119894119899(minus119904119894119895) 119891119884119894119895

(119910) 119889119910 (22)






119908 = 0 with 120572119895 = 4 and P119905119908119895 asymp P119905



11990811989516) 120574119908119895 (3))120574119908119895 (3)+ exp (minus120587120582119908119895 (1198772

11990811989516) 120574119908119895 (1))120574119908119895 (1) (23)

where


V2) (24)




SIR

cove

rage

(w

)w = 1000 ０ＥＧ2

SimNum


10

08

06

04

02

00

w3 = w

w2 = 02w w3 = 08w

w2 = 052 w3 = 05ww2 = w

minus10 minus5 0 5 10 15 20







R asymp sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895 = 120591119894119895 (N119894119895)) (25)


00

02

04

06

08

10

SIR

cove

rage

()

Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

minus5 0 5 10 15 20minus10






120588119888119895N119888119895

P119905119888119895B119888119895

= 120588119888119895B119888119895

+ 128120588119888119895120582119906A119888119895120582119888119895B119888119895

(26)



119908119895 asymp P119905119908119895

we get

120588119908119895N119908119895

P119905119908119895B119908119895


(27)





te co

vera

ge (ℛ

)


Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

10

08

06

04

02

000 2 4 6 8 10





Rate

cove

rage

(ℛ)

Φa = Φc2

Φa = Φw3

Φa = Φc2 + Φw3

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

06

05

04

03

02

01

000 250 500 750 1000 1250 1500 1750 2000









Stab

le q

ueue

pro

babi

lity

()

= 15 dB

tw=046

10

08

06

04

02

0000 02 04 06 08 10

Φc -domΦw -dom

Φc -modΦw-mod





120600119908 (120579 120585 119902) = [1 minus exp( log (120585P119905119908)119902P119905



1198772

11990816 )]11199031199003ge1198771199084

+ [exp(minus120587120582119908

1198772

11990816 )minus exp( log (120585P119905


(29)

respectively where

1199032119900V = minus log (120585P119905119908)120587119902P119905





Φc -domΦw -dom

Φc -modΦw-mod

Stab

le q

ueue

pro

babi

lity

()

10

08

06

04

02

00


= 03


000

005

010

015

020

025

030

035

040St

able

que

ue p

roba

bilit

y (

) = 15 dB = 03

tw=029

tw=031


250 500 750 1000 1250 1500 1750 2000

Φc -domΦw -dom

Φc -modΦw-mod





Φcj

Φwj


0 250 500 750 1000 1250 1500 1750 2000

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

Φa = Φc2 + Φw3

00

02

04

06

08

10A

ssoc

iatio

n pr

obab

ility

(ij

)








120582119908119895 = 120582119888119895

119891119908119895119891119888119895

radic 119875119888119895119875119908119895

(31)





200 400 600 800 1000 1200 1400 1600 1800 2000


c2 = w3 = 3 ＞

SIR

cove

rage

()

Exs[14] Φa = Φc1

Exs[14] Φa = Φc2

Exs[14] Φa = Φw3

080

075

070

065

060

055

050

045









SIR

cove

rage

()

070

075

080

085



w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3


Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2







Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20


040

045

050

055

060

065

070




Rate

cove

rage

(ℛ)


Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3


08

07

06

05

04

03

02

01

00




6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



lt





120582119908119899119860119908n)sum119899isinΦ119908120582119908119899119860119908119899

(11)

where

119860119908119899 = intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 (12)







250 500 750 1000 1250 1500 1750 2000



w3 = w

w2 = 02w w3 = 08w

w2 = 05w w3 = 05w

w2 = w

00

02

04

06

08

10

Med

ium

acce

ss p

roba

bilit

y (

t w)



120582119908119899119860119908119899)sum119899isinΦ119908120582119908119899119860119908119899

(13)


and

119877119908119899 = ( 1205821198884120587radic119875119908119899Γ119908119899

)(2120572119899)

(14)


120582119908119899119860119908119899



119908 gt 119875119905119908


(a)

(b)

(c)




P119905119908119895

= intinfin

0


120582119908119899 [119860119908119899 minus 119860119908119899 (119910)] 119891119884119908119895(119910) (15)

where

119860119908119899 (119910)= 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909 (16)



tagged APrandom AP

10

08

06

04

02

00

Med

ium

acce

ss p

roba

bilit

y (

t w)

w3 = w

w2 = 02w w3 = 08w

w2 = 05w w3 = 05w

w2 = w




119908 =sum119895isinΦ119908A119908119895P

119905119908119895



P[N

o A

P clo

ser t

hanR

]


w = 100 ０ＥＧ2

w = 500 ０ＥＧ2

w = 1000 ０ＥＧ2

w = 1500 ０ＥＧ2


10

08

06

04

02

000 20 40 60 80 100 120






4 Coverage




119908)with density120582119905119908 = P119905



119908119899) with density120582119905119908119899 = P119905



LI119908119899(119904119908119895)


) | 119910 lt 1198771199081198734 )+ (Z (119904119908119895119875119908119899 120572119899 119911120572119899

119908119899119887) | 119910 gt 1198771199081198734 )]

(17)


and 119911119908119899119887are defined







2120587120582119888119895 intinfin

0exp(minus1199041198881198951205902

119888 minus 120587 sum119899isinΦ119888

120582119888119899Z (119904119888119895119875119888119899 120572119899 119911120572119899119888119899119887) + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)119910119889119910+ sum

119895isinΦ119908

2120587120582119908119895 intinfin

0exp(minus1199041199081198951205902

119908 minus 120587P119905119908119895


120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119886) | 119910 lt 1198771199081198734 ] + 120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119887

) | 119910 gt 1198771199081198734 ] + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)sdot 119910119889119910(18)



(119888119886)2119887

1198891199061 + 1199061198872 (19)


S119894119895 (120579119894119895) 119886= int119910gt0

E[[expminus

120579119894119895119897 (10038171003817100381710038171199101003817100381710038171003817)119875119894119895

(sum119899isinΦ119894

119875119894119899 sum119909isinΦ119899cap119861119888(01199101015840)

119890119905119909ℎ119909119897 (119909) + 1205902119894 ) | 119890119905119909119900 = 1119891119884119894119895

(119910) 119889119910]] (20)


E[[expminus119904119894119895(sum

119899isinΦ119905119894

I119894119899 + 1205902119894 )119891119884119894119895

(119910) 119889119910]] (21)

119888= int119910gt0

119890minus1199041198941198951205902119894 prod119899isinΦ119905119894

LI119894119899(minus119904119894119895) 119891119884119894119895

(119910) 119889119910 (22)






119908 = 0 with 120572119895 = 4 and P119905119908119895 asymp P119905



11990811989516) 120574119908119895 (3))120574119908119895 (3)+ exp (minus120587120582119908119895 (1198772

11990811989516) 120574119908119895 (1))120574119908119895 (1) (23)

where


V2) (24)




SIR

cove

rage

(w

)w = 1000 ０ＥＧ2

SimNum


10

08

06

04

02

00

w3 = w

w2 = 02w w3 = 08w

w2 = 052 w3 = 05ww2 = w

minus10 minus5 0 5 10 15 20







R asymp sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895 = 120591119894119895 (N119894119895)) (25)


00

02

04

06

08

10

SIR

cove

rage

()

Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

minus5 0 5 10 15 20minus10






120588119888119895N119888119895

P119905119888119895B119888119895

= 120588119888119895B119888119895

+ 128120588119888119895120582119906A119888119895120582119888119895B119888119895

(26)



119908119895 asymp P119905119908119895

we get

120588119908119895N119908119895

P119905119908119895B119908119895


(27)





te co

vera

ge (ℛ

)


Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

10

08

06

04

02

000 2 4 6 8 10





Rate

cove

rage

(ℛ)

Φa = Φc2

Φa = Φw3

Φa = Φc2 + Φw3

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

06

05

04

03

02

01

000 250 500 750 1000 1250 1500 1750 2000









Stab

le q

ueue

pro

babi

lity

()

= 15 dB

tw=046

10

08

06

04

02

0000 02 04 06 08 10

Φc -domΦw -dom

Φc -modΦw-mod





120600119908 (120579 120585 119902) = [1 minus exp( log (120585P119905119908)119902P119905



1198772

11990816 )]11199031199003ge1198771199084

+ [exp(minus120587120582119908

1198772

11990816 )minus exp( log (120585P119905


(29)

respectively where

1199032119900V = minus log (120585P119905119908)120587119902P119905





Φc -domΦw -dom

Φc -modΦw-mod

Stab

le q

ueue

pro

babi

lity

()

10

08

06

04

02

00


= 03


000

005

010

015

020

025

030

035

040St

able

que

ue p

roba

bilit

y (

) = 15 dB = 03

tw=029

tw=031


250 500 750 1000 1250 1500 1750 2000

Φc -domΦw -dom

Φc -modΦw-mod





Φcj

Φwj


0 250 500 750 1000 1250 1500 1750 2000

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

Φa = Φc2 + Φw3

00

02

04

06

08

10A

ssoc

iatio

n pr

obab

ility

(ij

)








120582119908119895 = 120582119888119895

119891119908119895119891119888119895

radic 119875119888119895119875119908119895

(31)





200 400 600 800 1000 1200 1400 1600 1800 2000


c2 = w3 = 3 ＞

SIR

cove

rage

()

Exs[14] Φa = Φc1

Exs[14] Φa = Φc2

Exs[14] Φa = Φw3

080

075

070

065

060

055

050

045









SIR

cove

rage

()

070

075

080

085



w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3


Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2







Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20


040

045

050

055

060

065

070




Rate

cove

rage

(ℛ)


Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3


08

07

06

05

04

03

02

01

00




6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



(a)

(b)

(c)




P119905119908119895

= intinfin

0


120582119908119899 [119860119908119899 minus 119860119908119899 (119910)] 119891119884119908119895(119910) (15)

where

119860119908119899 (119910)= 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909 (16)



tagged APrandom AP

10

08

06

04

02

00

Med

ium

acce

ss p

roba

bilit

y (

t w)

w3 = w

w2 = 02w w3 = 08w

w2 = 05w w3 = 05w

w2 = w




119908 =sum119895isinΦ119908A119908119895P

119905119908119895



P[N

o A

P clo

ser t

hanR

]


w = 100 ０ＥＧ2

w = 500 ０ＥＧ2

w = 1000 ０ＥＧ2

w = 1500 ０ＥＧ2


10

08

06

04

02

000 20 40 60 80 100 120






4 Coverage




119908)with density120582119905119908 = P119905



119908119899) with density120582119905119908119899 = P119905



LI119908119899(119904119908119895)


) | 119910 lt 1198771199081198734 )+ (Z (119904119908119895119875119908119899 120572119899 119911120572119899

119908119899119887) | 119910 gt 1198771199081198734 )]

(17)


and 119911119908119899119887are defined







2120587120582119888119895 intinfin

0exp(minus1199041198881198951205902

119888 minus 120587 sum119899isinΦ119888

120582119888119899Z (119904119888119895119875119888119899 120572119899 119911120572119899119888119899119887) + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)119910119889119910+ sum

119895isinΦ119908

2120587120582119908119895 intinfin

0exp(minus1199041199081198951205902

119908 minus 120587P119905119908119895


120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119886) | 119910 lt 1198771199081198734 ] + 120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119887

) | 119910 gt 1198771199081198734 ] + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)sdot 119910119889119910(18)



(119888119886)2119887

1198891199061 + 1199061198872 (19)


S119894119895 (120579119894119895) 119886= int119910gt0

E[[expminus

120579119894119895119897 (10038171003817100381710038171199101003817100381710038171003817)119875119894119895

(sum119899isinΦ119894

119875119894119899 sum119909isinΦ119899cap119861119888(01199101015840)

119890119905119909ℎ119909119897 (119909) + 1205902119894 ) | 119890119905119909119900 = 1119891119884119894119895

(119910) 119889119910]] (20)


E[[expminus119904119894119895(sum

119899isinΦ119905119894

I119894119899 + 1205902119894 )119891119884119894119895

(119910) 119889119910]] (21)

119888= int119910gt0

119890minus1199041198941198951205902119894 prod119899isinΦ119905119894

LI119894119899(minus119904119894119895) 119891119884119894119895

(119910) 119889119910 (22)






119908 = 0 with 120572119895 = 4 and P119905119908119895 asymp P119905



11990811989516) 120574119908119895 (3))120574119908119895 (3)+ exp (minus120587120582119908119895 (1198772

11990811989516) 120574119908119895 (1))120574119908119895 (1) (23)

where


V2) (24)




SIR

cove

rage

(w

)w = 1000 ０ＥＧ2

SimNum


10

08

06

04

02

00

w3 = w

w2 = 02w w3 = 08w

w2 = 052 w3 = 05ww2 = w

minus10 minus5 0 5 10 15 20







R asymp sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895 = 120591119894119895 (N119894119895)) (25)


00

02

04

06

08

10

SIR

cove

rage

()

Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

minus5 0 5 10 15 20minus10






120588119888119895N119888119895

P119905119888119895B119888119895

= 120588119888119895B119888119895

+ 128120588119888119895120582119906A119888119895120582119888119895B119888119895

(26)



119908119895 asymp P119905119908119895

we get

120588119908119895N119908119895

P119905119908119895B119908119895


(27)





te co

vera

ge (ℛ

)


Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

10

08

06

04

02

000 2 4 6 8 10





Rate

cove

rage

(ℛ)

Φa = Φc2

Φa = Φw3

Φa = Φc2 + Φw3

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

06

05

04

03

02

01

000 250 500 750 1000 1250 1500 1750 2000









Stab

le q

ueue

pro

babi

lity

()

= 15 dB

tw=046

10

08

06

04

02

0000 02 04 06 08 10

Φc -domΦw -dom

Φc -modΦw-mod





120600119908 (120579 120585 119902) = [1 minus exp( log (120585P119905119908)119902P119905



1198772

11990816 )]11199031199003ge1198771199084

+ [exp(minus120587120582119908

1198772

11990816 )minus exp( log (120585P119905


(29)

respectively where

1199032119900V = minus log (120585P119905119908)120587119902P119905





Φc -domΦw -dom

Φc -modΦw-mod

Stab

le q

ueue

pro

babi

lity

()

10

08

06

04

02

00


= 03


000

005

010

015

020

025

030

035

040St

able

que

ue p

roba

bilit

y (

) = 15 dB = 03

tw=029

tw=031


250 500 750 1000 1250 1500 1750 2000

Φc -domΦw -dom

Φc -modΦw-mod





Φcj

Φwj


0 250 500 750 1000 1250 1500 1750 2000

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

Φa = Φc2 + Φw3

00

02

04

06

08

10A

ssoc

iatio

n pr

obab

ility

(ij

)








120582119908119895 = 120582119888119895

119891119908119895119891119888119895

radic 119875119888119895119875119908119895

(31)





200 400 600 800 1000 1200 1400 1600 1800 2000


c2 = w3 = 3 ＞

SIR

cove

rage

()

Exs[14] Φa = Φc1

Exs[14] Φa = Φc2

Exs[14] Φa = Φw3

080

075

070

065

060

055

050

045









SIR

cove

rage

()

070

075

080

085



w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3


Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2







Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20


040

045

050

055

060

065

070




Rate

cove

rage

(ℛ)


Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3


08

07

06

05

04

03

02

01

00




6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



P[N

o A

P clo

ser t

hanR

]


w = 100 ０ＥＧ2

w = 500 ０ＥＧ2

w = 1000 ０ＥＧ2

w = 1500 ０ＥＧ2


10

08

06

04

02

000 20 40 60 80 100 120






4 Coverage




119908)with density120582119905119908 = P119905



119908119899) with density120582119905119908119899 = P119905



LI119908119899(119904119908119895)


) | 119910 lt 1198771199081198734 )+ (Z (119904119908119895119875119908119899 120572119899 119911120572119899

119908119899119887) | 119910 gt 1198771199081198734 )]

(17)


and 119911119908119899119887are defined







2120587120582119888119895 intinfin

0exp(minus1199041198881198951205902

119888 minus 120587 sum119899isinΦ119888

120582119888119899Z (119904119888119895119875119888119899 120572119899 119911120572119899119888119899119887) + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)119910119889119910+ sum

119895isinΦ119908

2120587120582119908119895 intinfin

0exp(minus1199041199081198951205902

119908 minus 120587P119905119908119895


120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119886) | 119910 lt 1198771199081198734 ] + 120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119887

) | 119910 gt 1198771199081198734 ] + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)sdot 119910119889119910(18)



(119888119886)2119887

1198891199061 + 1199061198872 (19)


S119894119895 (120579119894119895) 119886= int119910gt0

E[[expminus

120579119894119895119897 (10038171003817100381710038171199101003817100381710038171003817)119875119894119895

(sum119899isinΦ119894

119875119894119899 sum119909isinΦ119899cap119861119888(01199101015840)

119890119905119909ℎ119909119897 (119909) + 1205902119894 ) | 119890119905119909119900 = 1119891119884119894119895

(119910) 119889119910]] (20)


E[[expminus119904119894119895(sum

119899isinΦ119905119894

I119894119899 + 1205902119894 )119891119884119894119895

(119910) 119889119910]] (21)

119888= int119910gt0

119890minus1199041198941198951205902119894 prod119899isinΦ119905119894

LI119894119899(minus119904119894119895) 119891119884119894119895

(119910) 119889119910 (22)






119908 = 0 with 120572119895 = 4 and P119905119908119895 asymp P119905



11990811989516) 120574119908119895 (3))120574119908119895 (3)+ exp (minus120587120582119908119895 (1198772

11990811989516) 120574119908119895 (1))120574119908119895 (1) (23)

where


V2) (24)




SIR

cove

rage

(w

)w = 1000 ０ＥＧ2

SimNum


10

08

06

04

02

00

w3 = w

w2 = 02w w3 = 08w

w2 = 052 w3 = 05ww2 = w

minus10 minus5 0 5 10 15 20







R asymp sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895 = 120591119894119895 (N119894119895)) (25)


00

02

04

06

08

10

SIR

cove

rage

()

Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

minus5 0 5 10 15 20minus10






120588119888119895N119888119895

P119905119888119895B119888119895

= 120588119888119895B119888119895

+ 128120588119888119895120582119906A119888119895120582119888119895B119888119895

(26)



119908119895 asymp P119905119908119895

we get

120588119908119895N119908119895

P119905119908119895B119908119895


(27)





te co

vera

ge (ℛ

)


Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

10

08

06

04

02

000 2 4 6 8 10





Rate

cove

rage

(ℛ)

Φa = Φc2

Φa = Φw3

Φa = Φc2 + Φw3

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

06

05

04

03

02

01

000 250 500 750 1000 1250 1500 1750 2000









Stab

le q

ueue

pro

babi

lity

()

= 15 dB

tw=046

10

08

06

04

02

0000 02 04 06 08 10

Φc -domΦw -dom

Φc -modΦw-mod





120600119908 (120579 120585 119902) = [1 minus exp( log (120585P119905119908)119902P119905



1198772

11990816 )]11199031199003ge1198771199084

+ [exp(minus120587120582119908

1198772

11990816 )minus exp( log (120585P119905


(29)

respectively where

1199032119900V = minus log (120585P119905119908)120587119902P119905





Φc -domΦw -dom

Φc -modΦw-mod

Stab

le q

ueue

pro

babi

lity

()

10

08

06

04

02

00


= 03


000

005

010

015

020

025

030

035

040St

able

que

ue p

roba

bilit

y (

) = 15 dB = 03

tw=029

tw=031


250 500 750 1000 1250 1500 1750 2000

Φc -domΦw -dom

Φc -modΦw-mod





Φcj

Φwj


0 250 500 750 1000 1250 1500 1750 2000

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

Φa = Φc2 + Φw3

00

02

04

06

08

10A

ssoc

iatio

n pr

obab

ility

(ij

)








120582119908119895 = 120582119888119895

119891119908119895119891119888119895

radic 119875119888119895119875119908119895

(31)





200 400 600 800 1000 1200 1400 1600 1800 2000


c2 = w3 = 3 ＞

SIR

cove

rage

()

Exs[14] Φa = Φc1

Exs[14] Φa = Φc2

Exs[14] Φa = Φw3

080

075

070

065

060

055

050

045









SIR

cove

rage

()

070

075

080

085



w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3


Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2







Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20


040

045

050

055

060

065

070




Rate

cove

rage

(ℛ)


Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3


08

07

06

05

04

03

02

01

00




6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







2120587120582119888119895 intinfin

0exp(minus1199041198881198951205902

119888 minus 120587 sum119899isinΦ119888

120582119888119899Z (119904119888119895119875119888119899 120572119899 119911120572119899119888119899119887) + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)119910119889119910+ sum

119895isinΦ119908

2120587120582119908119895 intinfin

0exp(minus1199041199081198951205902

119908 minus 120587P119905119908119895


120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119886) | 119910 lt 1198771199081198734 ] + 120582119908119899 [Z (119904119908119895119875119908119899 120572119899 119911120572119899119908119899119887

) | 119910 gt 1198771199081198734 ] + sum(119898119899)isinΦ119886

120582119898119899 (119898119899119910120572119895119891119898119899

)2120572119899)sdot 119910119889119910(18)



(119888119886)2119887

1198891199061 + 1199061198872 (19)


S119894119895 (120579119894119895) 119886= int119910gt0

E[[expminus

120579119894119895119897 (10038171003817100381710038171199101003817100381710038171003817)119875119894119895

(sum119899isinΦ119894

119875119894119899 sum119909isinΦ119899cap119861119888(01199101015840)

119890119905119909ℎ119909119897 (119909) + 1205902119894 ) | 119890119905119909119900 = 1119891119884119894119895

(119910) 119889119910]] (20)


E[[expminus119904119894119895(sum

119899isinΦ119905119894

I119894119899 + 1205902119894 )119891119884119894119895

(119910) 119889119910]] (21)

119888= int119910gt0

119890minus1199041198941198951205902119894 prod119899isinΦ119905119894

LI119894119899(minus119904119894119895) 119891119884119894119895

(119910) 119889119910 (22)






119908 = 0 with 120572119895 = 4 and P119905119908119895 asymp P119905



11990811989516) 120574119908119895 (3))120574119908119895 (3)+ exp (minus120587120582119908119895 (1198772

11990811989516) 120574119908119895 (1))120574119908119895 (1) (23)

where


V2) (24)




SIR

cove

rage

(w

)w = 1000 ０ＥＧ2

SimNum


10

08

06

04

02

00

w3 = w

w2 = 02w w3 = 08w

w2 = 052 w3 = 05ww2 = w

minus10 minus5 0 5 10 15 20







R asymp sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895 = 120591119894119895 (N119894119895)) (25)


00

02

04

06

08

10

SIR

cove

rage

()

Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

minus5 0 5 10 15 20minus10






120588119888119895N119888119895

P119905119888119895B119888119895

= 120588119888119895B119888119895

+ 128120588119888119895120582119906A119888119895120582119888119895B119888119895

(26)



119908119895 asymp P119905119908119895

we get

120588119908119895N119908119895

P119905119908119895B119908119895


(27)





te co

vera

ge (ℛ

)


Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

10

08

06

04

02

000 2 4 6 8 10





Rate

cove

rage

(ℛ)

Φa = Φc2

Φa = Φw3

Φa = Φc2 + Φw3

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

06

05

04

03

02

01

000 250 500 750 1000 1250 1500 1750 2000









Stab

le q

ueue

pro

babi

lity

()

= 15 dB

tw=046

10

08

06

04

02

0000 02 04 06 08 10

Φc -domΦw -dom

Φc -modΦw-mod





120600119908 (120579 120585 119902) = [1 minus exp( log (120585P119905119908)119902P119905



1198772

11990816 )]11199031199003ge1198771199084

+ [exp(minus120587120582119908

1198772

11990816 )minus exp( log (120585P119905


(29)

respectively where

1199032119900V = minus log (120585P119905119908)120587119902P119905





Φc -domΦw -dom

Φc -modΦw-mod

Stab

le q

ueue

pro

babi

lity

()

10

08

06

04

02

00


= 03


000

005

010

015

020

025

030

035

040St

able

que

ue p

roba

bilit

y (

) = 15 dB = 03

tw=029

tw=031


250 500 750 1000 1250 1500 1750 2000

Φc -domΦw -dom

Φc -modΦw-mod





Φcj

Φwj


0 250 500 750 1000 1250 1500 1750 2000

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

Φa = Φc2 + Φw3

00

02

04

06

08

10A

ssoc

iatio

n pr

obab

ility

(ij

)








120582119908119895 = 120582119888119895

119891119908119895119891119888119895

radic 119875119888119895119875119908119895

(31)





200 400 600 800 1000 1200 1400 1600 1800 2000


c2 = w3 = 3 ＞

SIR

cove

rage

()

Exs[14] Φa = Φc1

Exs[14] Φa = Φc2

Exs[14] Φa = Φw3

080

075

070

065

060

055

050

045









SIR

cove

rage

()

070

075

080

085



w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3


Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2







Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20


040

045

050

055

060

065

070




Rate

cove

rage

(ℛ)


Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3


08

07

06

05

04

03

02

01

00




6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



SIR

cove

rage

(w

)w = 1000 ０ＥＧ2

SimNum


10

08

06

04

02

00

w3 = w

w2 = 02w w3 = 08w

w2 = 052 w3 = 05ww2 = w

minus10 minus5 0 5 10 15 20







R asymp sum(119894119895)isinΦ119886

A119894119895S119894119895 (120579119894119895 = 120591119894119895 (N119894119895)) (25)


00

02

04

06

08

10

SIR

cove

rage

()

Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

minus5 0 5 10 15 20minus10






120588119888119895N119888119895

P119905119888119895B119888119895

= 120588119888119895B119888119895

+ 128120588119888119895120582119906A119888119895120582119888119895B119888119895

(26)



119908119895 asymp P119905119908119895

we get

120588119908119895N119908119895

P119905119908119895B119908119895


(27)





te co

vera

ge (ℛ

)


Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

10

08

06

04

02

000 2 4 6 8 10





Rate

cove

rage

(ℛ)

Φa = Φc2

Φa = Φw3

Φa = Φc2 + Φw3

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

06

05

04

03

02

01

000 250 500 750 1000 1250 1500 1750 2000









Stab

le q

ueue

pro

babi

lity

()

= 15 dB

tw=046

10

08

06

04

02

0000 02 04 06 08 10

Φc -domΦw -dom

Φc -modΦw-mod





120600119908 (120579 120585 119902) = [1 minus exp( log (120585P119905119908)119902P119905



1198772

11990816 )]11199031199003ge1198771199084

+ [exp(minus120587120582119908

1198772

11990816 )minus exp( log (120585P119905


(29)

respectively where

1199032119900V = minus log (120585P119905119908)120587119902P119905





Φc -domΦw -dom

Φc -modΦw-mod

Stab

le q

ueue

pro

babi

lity

()

10

08

06

04

02

00


= 03


000

005

010

015

020

025

030

035

040St

able

que

ue p

roba

bilit

y (

) = 15 dB = 03

tw=029

tw=031


250 500 750 1000 1250 1500 1750 2000

Φc -domΦw -dom

Φc -modΦw-mod





Φcj

Φwj


0 250 500 750 1000 1250 1500 1750 2000

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

Φa = Φc2 + Φw3

00

02

04

06

08

10A

ssoc

iatio

n pr

obab

ility

(ij

)








120582119908119895 = 120582119888119895

119891119908119895119891119888119895

radic 119875119888119895119875119908119895

(31)





200 400 600 800 1000 1200 1400 1600 1800 2000


c2 = w3 = 3 ＞

SIR

cove

rage

()

Exs[14] Φa = Φc1

Exs[14] Φa = Φc2

Exs[14] Φa = Φw3

080

075

070

065

060

055

050

045









SIR

cove

rage

()

070

075

080

085



w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3


Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2







Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20


040

045

050

055

060

065

070




Rate

cove

rage

(ℛ)


Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3


08

07

06

05

04

03

02

01

00




6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




te co

vera

ge (ℛ

)


Φc = Φc1 + Φc2

Φw = Φw2 + Φw3

Φma = Φc1 + Φw2

Φfa = Φc2 + Φw3

Φw = Φw2

Φw = Φw3

10

08

06

04

02

000 2 4 6 8 10





Rate

cove

rage

(ℛ)

Φa = Φc2

Φa = Φw3

Φa = Φc2 + Φw3

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

06

05

04

03

02

01

000 250 500 750 1000 1250 1500 1750 2000









Stab

le q

ueue

pro

babi

lity

()

= 15 dB

tw=046

10

08

06

04

02

0000 02 04 06 08 10

Φc -domΦw -dom

Φc -modΦw-mod





120600119908 (120579 120585 119902) = [1 minus exp( log (120585P119905119908)119902P119905



1198772

11990816 )]11199031199003ge1198771199084

+ [exp(minus120587120582119908

1198772

11990816 )minus exp( log (120585P119905


(29)

respectively where

1199032119900V = minus log (120585P119905119908)120587119902P119905





Φc -domΦw -dom

Φc -modΦw-mod

Stab

le q

ueue

pro

babi

lity

()

10

08

06

04

02

00


= 03


000

005

010

015

020

025

030

035

040St

able

que

ue p

roba

bilit

y (

) = 15 dB = 03

tw=029

tw=031


250 500 750 1000 1250 1500 1750 2000

Φc -domΦw -dom

Φc -modΦw-mod





Φcj

Φwj


0 250 500 750 1000 1250 1500 1750 2000

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

Φa = Φc2 + Φw3

00

02

04

06

08

10A

ssoc

iatio

n pr

obab

ility

(ij

)








120582119908119895 = 120582119888119895

119891119908119895119891119888119895

radic 119875119888119895119875119908119895

(31)





200 400 600 800 1000 1200 1400 1600 1800 2000


c2 = w3 = 3 ＞

SIR

cove

rage

()

Exs[14] Φa = Φc1

Exs[14] Φa = Φc2

Exs[14] Φa = Φw3

080

075

070

065

060

055

050

045









SIR

cove

rage

()

070

075

080

085



w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3


Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2







Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20


040

045

050

055

060

065

070




Rate

cove

rage

(ℛ)


Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3


08

07

06

05

04

03

02

01

00




6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Stab

le q

ueue

pro

babi

lity

()

= 15 dB

tw=046

10

08

06

04

02

0000 02 04 06 08 10

Φc -domΦw -dom

Φc -modΦw-mod





120600119908 (120579 120585 119902) = [1 minus exp( log (120585P119905119908)119902P119905



1198772

11990816 )]11199031199003ge1198771199084

+ [exp(minus120587120582119908

1198772

11990816 )minus exp( log (120585P119905


(29)

respectively where

1199032119900V = minus log (120585P119905119908)120587119902P119905





Φc -domΦw -dom

Φc -modΦw-mod

Stab

le q

ueue

pro

babi

lity

()

10

08

06

04

02

00


= 03


000

005

010

015

020

025

030

035

040St

able

que

ue p

roba

bilit

y (

) = 15 dB = 03

tw=029

tw=031


250 500 750 1000 1250 1500 1750 2000

Φc -domΦw -dom

Φc -modΦw-mod





Φcj

Φwj


0 250 500 750 1000 1250 1500 1750 2000

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

Φa = Φc2 + Φw3

00

02

04

06

08

10A

ssoc

iatio

n pr

obab

ility

(ij

)








120582119908119895 = 120582119888119895

119891119908119895119891119888119895

radic 119875119888119895119875119908119895

(31)





200 400 600 800 1000 1200 1400 1600 1800 2000


c2 = w3 = 3 ＞

SIR

cove

rage

()

Exs[14] Φa = Φc1

Exs[14] Φa = Φc2

Exs[14] Φa = Φw3

080

075

070

065

060

055

050

045









SIR

cove

rage

()

070

075

080

085



w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3


Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2







Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20


040

045

050

055

060

065

070




Rate

cove

rage

(ℛ)


Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3


08

07

06

05

04

03

02

01

00




6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Φcj

Φwj


0 250 500 750 1000 1250 1500 1750 2000

Φa = Φc1 + Φw3

Φa = Φc2 + Φw2

Φa = Φc2 + Φw3

00

02

04

06

08

10A

ssoc

iatio

n pr

obab

ility

(ij

)








120582119908119895 = 120582119888119895

119891119908119895119891119888119895

radic 119875119888119895119875119908119895

(31)





200 400 600 800 1000 1200 1400 1600 1800 2000


c2 = w3 = 3 ＞

SIR

cove

rage

()

Exs[14] Φa = Φc1

Exs[14] Φa = Φc2

Exs[14] Φa = Φw3

080

075

070

065

060

055

050

045









SIR

cove

rage

()

070

075

080

085



w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3


Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2







Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20


040

045

050

055

060

065

070




Rate

cove

rage

(ℛ)


Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3


08

07

06

05

04

03

02

01

00




6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



SIR

cove

rage

()

070

075

080

085



w3 = 400 APkＧ2

w3 = 600 APkＧ2

w3 = 1200 APkＧ2

Φa = Φc2 + Φw3


Φa = Φc2 + Φw3

02

03

04

05

06

07

08

Rate

cove

rage

(ℛ)


w3 = 600 ０ＥＧ2

w3 = 1200 ０ＥＧ2u = 1000 ＯＭＬＭＥＧ2







Φa = Φc2 + Φw3

6 8 10 12 14 16 18 20


040

045

050

055

060

065

070




Rate

cove

rage

(ℛ)


Φa = Φc2 + Φw3

Rate

cove

rage

(ℛ)


0 250 500 750 1000 1250 1500 1750 2000

u = 4w3

u = 6w3

u = 10w3


08

07

06

05

04

03

02

01

00




6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



6 Conclusion



Appendix

A Proof of Lemma 1


119890119905119895= E

119909119895Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119899119909119895

(1119905119899ge119905119895+ 1119905119899lt119905119895

1ℎ119899119895119897(119909119899minus119909119895)leΓ119908119899119875119908119899)]]

(A1)

119886= E119909119895Φ119908[prod

119899isinΦ119908

prod119909119899

1 minus 119905 exp(minus Γ119908119899119875119908119899

119897 (10038171003817100381710038171003817119909119899 minus 119909119895

10038171003817100381710038171003817))] (A2)


120582119908119899119860119908119899] (A3)



P [119890119905119909119900 = 1 | 119909119900 = (119910 0)] = E119909119900Φ119908[[ prod

119899isinΦ119908

prod119909119899isinΦ119908119909119900

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899) | 119909119900 = (119910 0)]] (B1)

119886= E[[ prod119899isinΦ119908

prod119909119899isinΦ119899cap119861c(01199101015840)

(1119905119899ge119905119900+ 1119905119899lt119905119900

1ℎ119899119900119897(119909119899)leΓ119908119899119875119908119899)]] (B2)

119887= int1

0exp[minus sum

119899isinΦ119908

120582119908119899119905 intR2

exp(minus Γ119908119899119875119908119899

119897 (119909)) 119889119909 minus 2int21199101015840

0arccos ( 11990921199101015840

) exp(minus Γ119908119899119875119908119899

119897 (119909)) 119909119889119909119889119905] (B3)





ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



ΦnltN

Φn=N

TaggedActive



C Proof of Lemma 8


LI119899(119904119895)

= EΦ119905119899ℎ119909[[exp


ℎ119909119897 (119909))]] (C1)

119886asymp EΦ119905119899

prod119909isinΦ119905119899cap119861119888(01199101015840)

Lℎ119909(119904119895119875119899119909minus120572119899) (C2)


119911119899

1 minusLℎ119909(119904119895119875119899119909minus120572119899) 119909119889119909] (C3)


119895120582119899 intinfin

119911119899

21199091 + (119904119895119875119899)minus1 119909120572119899

(C4)



Z (119886 119887 119888) = 1198862119887 intinfin

(119888119886)2119887

1198891199061 + 1199061198872 (C5)

Hence

LI119899(119904119895)


+ (Z (119904119895119875119899 120572119899 119911120572119899119899119887 ) | 119910 gt 1198771198734 )] (C6)

where

119911119899119886 = (4 119877119899119877119873

minus 1)119910 (C7)

and

119911119899119887 = 1120572119899119899 1199101120572119899 (C8)






lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



lt

=

(a)

lty

lt (lt

minus )

(b)






R119894119895 (120588119894119895) = P (C119894119895 gt 120588119894119895) (D1)

= P[B119894119895

N119894119895

log (1 + 119878119868119873119877119894119895) P119905119894119895 gt 120588119894119895] (D2)

= P [119878119868119873119877119894119895 gt 2120588119894119895N119894119895P119905119894119895B119894119895 minus 1] (D3)

= EN119894119895[S119894119895 120591119894119895 (N119894119895)] (D4)







P [N119900119894119895 = 119899]= (35)119889 Γ (119899 + 119889) (120582119906A119894119895120582119894119895)119899Γ (119889) Γ (119899 + 1) (120582119906A119894119895120582119894119895 + 35)119899+119889

(D5)


EN119894119895[S119894119895 120591119894119895 (N119894119895)]= sum

119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D6)

R = sum(119894119895)isinΦ119886

A119894119895sum119899ge0

P [N119900119894119895 = 119899]S119894119895 [120591119894119895 (119899 + 1)] (D7)





120583Φ119894119894 = P (119878119868119877119903119900

gt 120579 | Φ119894) (E1)





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





120583119908





120583119888 ge 120585 (E4)


120583119908P119905119908 ge 120585 (E5)






1199032119900V le minus log (120585P119905119908)120587119902P119905



120600119908 (120579 120585 119902) =

1 minus exp( log (120585P119905119908)119902P119905


1 minus exp(minus120587120582119908

1198772

11990816 ) 1199031199003 ge 1198771199084exp(minus120587120582119908

1198772

11990816 ) minus exp( log (120585P119905119908)119902P119905


(E9)


Data Availability




References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Toward a Unified Framework for Analysis of Multi-RAT...

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