Research Article Connectivity Analysis of Millimeter-Wave...

Research ArticleConnectivity Analysis of Millimeter-Wave Device-to-DeviceNetworks with Blockage

Haejoon Jung1 and In-Ho Lee2

1The Department of Information and Telecommunication Engineering Incheon National University Incheon 22012 Republic of Korea2TheDepartment of Electrical Electronic andControl EngineeringHankyongNationalUniversity Anseong 456-749 Republic of Korea

Correspondence should be addressed to In-Ho Lee ihleehknuackr

Received 3 August 2016 Revised 8 October 2016 Accepted 17 October 2016

Academic Editor Mohammad Abdul Matin

Copyright copy 2016 H Jung and I-H Lee This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We consider device-to-device (D2D) communications in millimeter-wave (mmWave) for the future fifth generation (5G) cellularnetworks While the mm Wave systems can support multiple D2D pairs simultaneously through beamforming with highlydirectional antenna arrays the mmWave channel is significantly more susceptible to blockage compared to microwave mmWavechannel studies indicate that if line-of-sight (LoS) paths are blocked reliable mmWave communications may not be achieved forhigh data-rate applications Therefore assuming that an outage occurs in the absence of the LoS path between two wireless devicesby obstructions we focus on connectivity of the mmWave D2D networks We consider two types of D2D communications directand indirect schemes The connectivity performances of the two schemes are investigated in terms of (i) the probability to achievea fully connected network 119875FC and (ii) the average number of reliably connected devices 120574 Through analysis and simulation we showthat as the network size increases 119875FC and 120574 decrease Also 119875FC and 120574 decrease when the blockage parameter increases Moreoversimulation results indicate that the hybrid direct and indirect scheme can improve both 119875FC and 120574 up to about 35 compared tothe nonhybrid scheme

1 Introduction

With rapidly growing volume of mobile devices the demandfor capacity in mobile broadband communications increasesdramatically As a result wireless industry is required toseek greater capacity and find new wireless spectrum beyondfourth generation (4G) standards For this reason fifth gen-eration (5G) is envisioned to have significantly greater spec-trum allocations at millimeter-wave (mm Wave) frequencybands highly directional antenna arrays larger coveragearea lower infrastructure costs and higher aggregate capac-ity for many simultaneous users [1 2] While microwavecommunication systems suffer from the limited spectrumthe bandwidths of several gigahertz could be available at mmWave frequencies for 5G communication systems

Moreover instead of the complete control at the infras-tructure side 5G systems will exploit intelligence at thedevice side within different layers of the protocol stackby allowing device-to-device (D2D) connectivity [3] The

direct communications between nearby mobile devices canprovide higher spectrum utilization enhanced throughputand better energy efficiency while facilitating new peer-to-peer applications and location-based services such as3GPP Proximity Services (ProSe) [4] and IEEE 80215 Peer-Aware Communications (PAC) [5] Along the lines of theincreasing demand of such high-rate local services local D2Dcommunications have been studied as underlay to Long TermEvolution-Advanced (LTE-A) 4G cellular networks [6] Ashighlighted in [6 7] the main challenge of a D2D-enabledair interface for the cellular networks is how to share wire-less resources between cellular and D2D communicationsFor instance the local D2D communications should useorthogonal channels or opportunistically access the spectrumoccupied by cellular communications

In this context we consider D2D communications in themm Wave frequency bands for the future cellular networkbecause the mmWave can aid a resource sharing scheme be-tween theD2D and cellular communications which supports

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016 Article ID 7939671 9 pageshttpdxdoiorg10115520167939671

2 International Journal of Antennas and Propagation

noninterfering concurrent links To be specific directionalantenna arrays in mm Wave can reduce cochannel interfer-ence and improve spatial reuse of communication systemsthrough large beamforming gain This characteristic of mmWave systems is also desirable to guarantee connectivitybetween a huge number of devices in the future wirelessnetworks [8] In other words thanks to highly directionalbeam inmmWave a user equipment (UE) can communicatewith another UE in proximity over a D2D link whichenables multiple D2D pairs to use the same radio resourcessimultaneously

However compared to microwave an mm Wave com-munication channel experiences higher path-loss and issusceptible to blockage such as walls trees or even humanbodies as revealed in [2 9 10] Hence the mm Wave com-munication channel is a nearly bimodal channel dependingon the existence of line-of-sight (LoS) path [11] Based onthis property of the mm Wave propagation a new channelmodel is introduced in [12] which characterizes large-scaleblockage effects using random shape theory [13] The modelproposed in [12] shows that the probability of a blockage eventbetween two radios increases exponentially as the separationbetween the two increases This propagation model hasbeen applied to analyze various wireless communicationsystems using mm Wave The author of [14] derives theoutage probability improved by macrodiversity with multi-ple base stations (BSs) assuming the outage event occurswhen the LoS path is blocked The authors in [15] analyzethe coverage and rate performance in mm Wave cellularnetworks Moreover [16] presents the outage performanceof the mm Wave wireless backhaul link between a 5Gmacro base station (MBS) and small-cell base stations (SBSs)Assuming Poisson point process (PPP) on the plane thestochastic geometry used in these studies is known to be aneffective tool to evaluate system performances in the cellularnetworks [17]

In this paper based on the framework in [12 14] weconsider D2D connectivity in mm Wave networks ReliableD2D connections are required for D2D data transmissions aswell as cellular data offloading onto D2D connections whichcan provide considerable wireless capacity gains [18] In mmWave 5G cellular networks two kinds of D2D communi-cations can be enabled direct communications between twowireless devices in proximity and indirect communicationswhich connect two devices through base station(s) [8] In thispaper we investigate both types of the D2D communicationsassuming a network consisting of one mm Wave BS andmultiple wireless devices distributed according to a two-dimensional Poisson point process (PPP)

The contributions of this paper are fourfold First wederive the probability distribution mean and variance of theinterdevice distance (ie the distance between two randomlylocated devices) Second we derive the probability that theD2D network is fully connected which means that all thewireless devices have reliable communication links to eachother [19] Third we also quantify the D2D connectivityin terms of the average number of reliable connections (orcommunication links) both for the direct and indirect D2D

communication systems Lastly we consider a hybrid schemein which both the direct and indirect communications can beselectively used and present the simulation results to comparethe performances of the direct and indirect schemes

This paper is organized as follows The system modelis introduced in Section 2 The direct and indirect D2Dcommunication systems are analyzed in Sections 3 and 4respectively Numerical results are presented in Section 5Conclusions are provided in Section 6

2 System Model

We consider a 5G cellular network enabled with D2D com-munications with two tiers the cell tier and the device tierThe conventional cellular communications are supported bythe cell (macro or small-cell) tier while D2D communica-tions are covered by the device tier The BS may have a fullor partial control over the D2D communications dependingon the system architecture to establish and manage the D2Dlinks In this paper we focus on the connectivity of the D2Dlinks which indicates the potential extent of cellular dataoffloading onto D2D connections Thus the analysis in thispaper does not depend on specific design aspects of thenetwork architecture Based on [12] we focus on outdoorenvironments

We assume mm Wave 5G communication systems asshown in Figures 1(a) and 1(b) which illustrate direct andindirect D2D communications respectively In both caseswe assume 119873 devices are uniformly distributed over AreaS (ie the area of the gray circle with radius 119877) withintensity 120582 which means the average number of devicesper unit area (ie devicesm2) according to homogeneousPPP [20] We assume a quasi-stationary scenario where thenetwork topology is stationary while D2D communicationsare performed

For the indirect D2D communications in Figure 1(b)at the center of the circle there exists a single mm WaveBS which can be a small-cell BS with low power and lowcost in heterogeneous networks We note that we do notconsider indirect (multihop) communications via devicesas relays because of the complexity to build a multihoproute with directional antennas in the mobile scenario Inboth types of D2D networks the number of the devicesin the network is assumed to be 119873 which is a randomvariable following Poisson probability distribution as119875119873(119899) =((1205821205871198772)119899119899)119890minus1205821205871198772 where 119875119873(119899) is the probability that thereare 119899 devices in Area S and 119899 is a nonnegative integerTherefore the average number of the devices is given byE119873 = 1205821205871198772 where Esdot is the expectation operator

As in [15] for analytical tractability we assume thesectored antenna model which characterizes key features ofan antenna pattern As noted in [8] the interference issue byconcurrent cochannel links with omnidirectional antennasis of little concern in the mm Wave communications withhighly directional antenna arraysWe assume that directionalbeamforming is performed at devices as well as the mmWave BS We note a half-duplex system is assumed There-fore instead of the interference-limited performance metric

International Journal of Antennas and Propagation 3

Device 1

Device 2 Device 3

DeviceN

Area

OR

Blockage

(a)

Device 1

mm Wave BS

Device 2 Device 3

DeviceN

Area

R

Blockage

(b)

Figure 1 Two types of mmWave D2D communications (a) direct D2D communications and (b) indirect D2D communications

(eg signal-to-noise-plus-interference ratio) the blockageproblem of line-of-sight (LoS) paths may induce an outageevent in mm Wave communications especially for high-speed data transfer in multimedia or interactive applicationsAs in [12 14 16] blockages are assumed to be impenetrableAlso following [12 14 16] we define an outage event as thecase that the LoS path is blocked by obstacles

In case of the direct D2D communications suppose 119889119894119895is the distance between devices 119894 and 119895 As in [12 14ndash16]the probability of no blockage in the LoS path between thetwo devices is 119875LoS119894119895 = 119890minus120573119889119894119895 where 120573 is the parameterthat captures density and size of obstacles which cause anoutage due to blockage The greater 120573 means obstacles withhigher density and larger sizes which results in lower 119875LoS119894119895[12] In this paper we assume 119875LoS119894119895 can be interpretedas the probability that the communication link betweendevices 119894 and 119895 is reliable In other words if the LoS path isblockage-free we declare the corresponding communication(or connection) is reliable

As in [12 14 15] for analytical tractability we assumethe blockage events on different mm Wave links are mutu-ally independent For example if there are three wirelessterminals 1 2 and 3 where they can be either the mmWave BS or a device the three possible communication links(ie 1-2 1-3 and 2-3) have independent outage events Ingeneral depending on the location of wireless terminalsblockage events are not independent For example if theangle between two links with a common end-point or start-point (eg 1-2 and 1-3) is narrow enough the two D2D linksmight experience the same blockage effect Therefore theprobability of a fully connected D2D network derived in thispaper may be an upper bound as the references of systemdesign and analysis Moreover we note that the numericalresults in [12] show that the error caused by the independentlink assumption is minor and acceptable in accuracy

3 Direct D2D Communications

In this section we consider direct D2D communicationswhich do not require infrastructure such as BSs Withdirectional antenna arrays in mm Wave noninterferingconcurrent D2D pairs are able to share radio resourceswhich significantly enhances network capacity Assuming thespatially uniform distribution of the devices we will explorevarious aspects of the direct D2D communications

31 Probability Distribution of 119889119894119895 To investigate the per-formance of the mm Wave communications between twodevices we first consider the probability distribution function(PDF) of 119889119894119895 since 119875LoS119894119895 is a function of 119889119894119895 which isa random variable Using Croftonrsquos fixed point theorem in[21] we derive the PDF of 119889119894119895 Fixed point theorem permitsthe evaluation of some definite integrals without directlyperforming the integrations which is especially useful toderive geometric probability distributions Suppose that thereare 119899 points 1205851 1205852 120585119899 which are randomly distributed ona domain S Let 119876 be an event that depends on the positionof the 119899 points and let 119889119878 be an infinitesimal boundary ofS Then Croftonrsquos fixed point theorem gives the followingformula

119889Pr [119876] = 119899 (Pr [119876 | 1205851 isin 119889S] minus Pr [119876])S

119889S (1)

where Pr[119876 | 1205851 isin 119889S] is the probability that 119876 occurs whenone of the random points 1205851 is on the boundary 119889S of S

To derive the PDF of 119889119894119895 suppose that119876 is the event thattwo points (ie 119899 = 2) in the circle with the radius 119877 (ieS = 1205871198772) are separated by a distance between 119909 and 119909 + 119889119909as shown in Figure 2 Also let 119862 be the event that one pointis on the circumference 119889S The probability of the event 119876 is


R R

dx

x

x2

Figure 2 Illustration to derive the PDF of 119889119894119895

denoted by Pr[119876] and the conditional probability of119876 given119862 is denoted by Pr[119876 | 119862] From (1) we have

119889Pr [119876] = 2 (Pr [119876 | 119862] minus Pr [119876])S

119889S= 2 (Pr [119876 | 119862] minus Pr [119876])1205871198772 2120587119877119889119877= 4 (Pr [119876 | 119862] minus Pr [119876])119877 119889119877

(2)

where Pr[119876 | 119862] is given by

Pr [119876 | 119862] = 2119909119889119909 cosminus1 (1199092119877)1205871198772 (3)

Therefore when plugging (3) into (2) it gives

119877119889Pr [119876] + 4Pr [119876] 119889119877 = 8119909119889119909 cosminus1 (1199092119877)1205871198772 119889119877 (4)

Multiplying both sides by 1198773 we obtain1198774119889 Pr [119876] + 41198773 Pr [119876] 119889119877= 8119877119909119889119909 cosminus1 (1199092119877)120587 119889119877 (5)

Thus if we integrate both sides it gives

1198774 Pr [119876] = 41199092119889119909120587 int 2119877119909 cosminus1 ( 1199092119877)119889119877= 119909119889119909120587 (41198772 cosminus1 ( 1199092119877) minus 2119909119877radic1 minus 119909241198772) + 119870

(6)

where the constant 119870 = 0 because Pr[119876] = 0 when 119877 = 1199092Hence the PDF of 119889119894119895 is expressed as

119891119889119894119895 (119909) = Pr [119876]119889119909= 21199091205871198772 (2 cosminus1 ( 1199092119877) minus 119909119877radic1 minus 119909241198772)

(7)

where 0 lt 119909 lt 2119877The derived PDF in (7) is the same as [22]Themean and variance of119889119894119895 are given byE119889119894119895 = 12811987745120587and VAR119889119894119895 = 1198772 minus (12811987745120587)2 respectively32 Fully Connected Network via Direct D2D Communica-tions In this section we derive the probability that all thewireless devices are interconnected via direct D2D commu-nications in mmWave Suppose a Bernoulli random variable119880119894119895 is defined as

119880119894119895 = 1 wp 119875LoS119894119895 = exp (minus120573119889119894119895) 0 wp 1 minus 119875LoS119894119895 = 1 minus exp (minus120573119889119894119895) (8)

where if 119880119894119895 = 1 it means that the mm Wave linkbetween devices 119894 and 119895 is reliable In other words followingassumption of impenetrable blockage in [12 14] we declare anoutage event of the direct D2D transmission between devices119894 and 119895 when 119880119894119895 = 0 If there are119873 (ge 2) devices there canexist (1198732 ) = 119873(119873 minus 1)2 possible mm Wave links betweenany two devices as in [19] (with 119873 devices or nodes in anetwork the maximum number of concurrent ldquoactiverdquo D2Dlinks (or pairs) is1198732 However we consider all the possiblelink combinations among 119873 devices (1198732 ) = 119873(119873 minus 1)2to check if the network is fully connected) When all of the119873(119873 minus 1)2 direct D2D links are reliable we assume that thenetwork is fully connected [19] which can be regarded as afully meshed network [23]

Therefore the probability to achieve the fully connectednetwork can be represented as

119875FC = suminfin119899=2 Pr [all 119880119894119895rsquos are one | 119873 = 119899]Pr [119873 = 119899]Pr [119873 ge 2] (9)

where 1 le 119894 119895 le 119899 where 119894 = 119895 With the independent linkassumption as in [12 14] the conditional probability in (9)can be simplified as

Pr [all 119880119894119895rsquos are one | 119873 = 119899] = (E 119890minus120573119889119894119895)119899(119899minus1)2 (10)

where the distance 119889119894119895 between devices 119894 and 119895 follows thePDF in (7) Suppose A = E119890minus120573119889119894119895 which can be calculatedby Taylor expansion as follows

A = int21198770119890minus120573119909119891119889119894119895 (119909) 119889119909

asymp [1 + 12057322 (1198772 minus (12811987745120587 )2)] exp(minus12057312811987745120587 ) (11)


Thus 119875FC in (9) can be simplified as

119875FC = E A119873(119873minus1)2 | 119873 ge 2= suminfin119899=2A119899(119899minus1)2 sdot Pr [119873 = 119899]

Pr [119873 ge 2]= suminfin119899=2A119899(119899minus1)2 sdot ((1205821205871198772)119899 119899) 119890minus12058212058711987721 minus exp (minus1205821205871198772) (1 + 1205821205871198772)

(12)

33 Ratio of Reliable Direct D2D Connections If the fullyconnected network cannot be achieved we can estimate theaverage number of reliably connected devices through directD2D communications If we define a set 119879dir = 119880119894119895 | 119880119894119895 =1 its cardinality |119879dir| is the number of the reliable directD2D links Thus for 119873 ge 2 the ratio of the reliable D2Dconnections to all of the possible D2D pairs in the networkcan be derived as

120574dir = 1Pr [119873 ge 2]

infinsum119899=2

E 1003816100381610038161003816119879dir1003816100381610038161003816 | 119873 = 119899119899 (119899 minus 1) 2 Pr [119873 = 119899]= A

(13)

where E|119879dir| | 119873 = 119899 is in fact the mean of binomialdistribution 119861(119899(119899 minus 1)2 119901 = A) which gives E|119879dir| | 119873 =119899 = A119899(119899minus1)2 As a result 120574dir = A and it is not a functionof the device density 12058234 Impacts of System Parameters In this section we con-sider the impacts of the network size 119877 and the blockageparameter 120573 on 119875FC and 120574dir The analysis in this sectionwill be verified by comparing with the simulation results inSection 5

341 Radius 119877 To analyze the impact of the network sizeindicated by 119877 we first need to investigate the derivative ofA in (11) in terms of 119877 as below

119889A119889119877 = minus120573119890minus12812057311987745120587911251205873 [(8120573radic20251205872 minus 16384119877+ 737280120587 minus 91125120587316radic20251205872 minus 16384)

2 2592001205872

minus ( 737280120587 minus 91125120587316radic20251205872 minus 16384)2]

(14)

which satisfies 119889A119889119877 lt 0 for any 120573 and 119877 Therefore 120574dir =A is a decreasing function of 119877 Moreover we can find that119875FC rarr 0 as 119877 rarr infin

342 Blockage Parameter 120573 As in the previous sectionregarding 119877 we find the derivative of A in terms of 120573 toinvestigate the impact of 120573 as

119889A119889120573 = minus119877119890minus12812057311987745120587911251205873 [(8119877radic20251205872 minus 16384120573

+ 737280120587 minus 91125120587316radic20251205872 minus 16384)2 2592001205872


(15)

which satisfies 119889A119889120573 lt 0 for any 120573 and 119877 Therefore 120574dir =A is a decreasing function of 120573 Also when 120573 rarr 0 A rarr 1Consequently 119875FC rarr 1 as 120573 rarr 0 On the other hand as120573 rarr infinArarr 0 which results in 119875FC rarr 04 Indirect D2D Communications via

mm Wave Base Station

In this section we investigate indirect D2D communicationshopping via the mm Wave BS where it is assumed thatdirect D2D links are unavailable If we have an indirect D2Dcommunication pair two consecutive LoS links ldquofrom onedevice to the mm Wave BSrdquo and ldquofrom the mm Wave BS tothe other devicerdquo are required to be reliable

41 Fully Connected Network via Indirect D2D Communica-tions For the indirect D2D communication from devices 119894to 119895 via the BS both ldquodevice 119894 to BSrdquo and ldquoBS to device 119895rdquolinks should have blockage-free LoS pathsTherefore to havea fully connected network with 119873 ge 2 hopping over the BSall the LoS links between the mm Wave BS and 119873 devicesshould not have blockage As in the previous section supposea Bernoulli random variable119882119894 is defined as


where if119882119894 = 1 the mm Wave communication between themmWave BS and device 119894 is successful

Also 119903119894 is the distance between the mm Wave BS anddevice 119894 which follows the PDF of 119891119903119894(119909) = 21199091198772 for 0 le119909 le 119877 Therefore the probability to create a fully connectednetwork is given by

119875FC = suminfin119899=2 Pr [all 119882119894rsquos are one | 119873 = 119899]Pr [119873 = 119899]Pr [119873 ge 2]

= suminfin119899=2 (E 119890minus120573119903119894)119899 Pr [119873 = 119899]Pr [119873 ge 2]

(17)

where 1 le 119894 le 119899 IfB = E119890minus120573119903119894 it givesB = int119877

0119890minus120573119909119891119903119894 (119909) 119889119909 = 21205731198772 [ 1120573 minus (119877 + 1120573) 119890minus120573119877] (18)


Hence 119875FC is obtained as

119875FC = suminfin119899=2B119899Pr [119873 = 119899]Pr [119873 ge 2]

= infinsum119899=2

B119899 (1205821205871198772)119899 sdot 119890minus1205821205871198772Pr [119873 ge 2] sdot 119899

= 119890B1205821205871198772 minusB1205821205871198772 minus 11198901205821205871198772 minus 1205821205871198772 minus 1 (19)

42 Ratio of Reliable Indirect D2D Connections If we definea set 119879ind = 119882119894 | 119882119894 = 1 its cardinality |119879ind| is thenumber of reliable ldquodevice to themmWaveBSrdquo links Becauseany devices connected to the mm Wave BS can indirectlycommunicate with each other for |119879ind| ge 2 there are|119879ind|(|119879ind| minus 1)2 reliable D2D connections For a given119873 =119899 ge 2 |119879ind| follows a binomial distribution 119861(119899 119901 = B)whereB is derived in (18) Therefore the ratio of the reliableindirect D2D connections is given by

120574ind = 1Pr [119873 ge 2]

sdot infinsum119899=2

E 1003816100381610038161003816119879ind1003816100381610038161003816 (1003816100381610038161003816119879ind1003816100381610038161003816 minus 1) 2 | 119873 = 119899 sdot Pr [119873 = 119899]119899 (119899 minus 1) 2= 1Pr [119873 ge 2]


(E 1003816100381610038161003816119879ind10038161003816100381610038162 | 119873 = 119899 minus E 1003816100381610038161003816119879ind1003816100381610038161003816 | 119873 = 119899) 2119899 (119899 minus 1) 2

sdot Pr [119873 = 119899] = 1Pr [119873 ge 2]

infinsum119899=2

1198992B2 minus 119899B2119899 (119899 minus 1)sdot Pr [119873 = 119899] =B

2

(20)

As 120574dir in (13) 120574ind is not a function of the device density 120582either

43 Impacts of System Parameters As in the previous sectionabout the direct D2D communications we analyze theimpacts of 119877 and 120573 on 119875FC and 120574dir431 Radius 119877 We first consider howB changes as 119877 variesby considering two extreme cases For119877 rarr 0 andinfin we have

lim119877rarr0

B = lim119877rarr0

1120573 minus 119890minus120573119877 (1120573 + 119877)120573119877 = 120573120573 = 1lim119877rarrinfin

B = lim119877rarr0

212057321198772 = 0(21)

Thus 120574ind = B2 rarr 1 as 119877 rarr 0 On the other hand 120574ind =B2 rarr 0 as119877 rarr infin Furthermore lim119877rarr0119875FC = lim119877rarr0B

2 =1 In contrast 119875FC rarr 0 as 119877 rarr infin

20 40 60 80 100

Radius R (m)0

01

02

03

04

05

06

07

08

09

1

P

Numericalresults results

Simulation( )(5 times 10minus4 10minus3)

(10minus3 10minus3)(10minus3 5 times 10minus3)

Figure 3 Direct D2D 119875FC versus 119877 with 120582 = 0001 00005 and120573 = 0001 0005

432 Blockage Parameter 120573 We can readily find thatBrarr 1and 0 as 120573 rarr 0 and infin respectively Therefore 120574ind = B2

has the same limiting values Also it gives 119875FC rarr 1 and 0 as120573 rarr 0 andinfin respectively

5 Numerical Results

In this section we present numerical results of the mmWaveD2D communication systems via both the direct and indirectcommunications

51 Direct D2D Communications Figure 3 shows the prob-ability to achieve a fully connected D2D network using thedirect D2D communications The horizontal axis 119877 repre-sents the radius of Area S in Figure 1(a) The solid dashedand dotted lines represent 119875FC in (12) obtained numericallywith different combinations of 120582 and 120573 Moreover thesymbols indicate the corresponding Monte Carlo simulationresults which are consistent with the numerical results Allthe three graphs decrease as the radius 119877 increases Alsocomparing the solid and dashed lines (120582 = 00005 and 0001with the same 120573) the higher 120582 gives the lower likelihood ofthe fully connected network Similarly as 120573 increases (thedashed and dotted lines) 119875FC decreases In other words theprobability 119875FC is a decreasing function of both the devicedensity 120582 and the blockage parameter 120573

Figure 4 shows the ratio of the reliable direct D2Dconnections 120574dir in (13)The three curves denoted by the soliddashed and dotted lines correspond to 120573 = 0001 001 01respectively Also the simulation results indicated by thesymbols are in line with the numerical results As119877 increases120574dir decreases Furthermore the ratio of the reliable D2D con-nections decreases as 120573 increases because of higher blockage


20 70 9030 40 50 60 80 100

Radius R (m)10


Simulation

01

0

02

03

04

05

06

07

08

09

1

0001

001

01

＞ＣＬ

Figure 4 Direct D2D 120574dir = A versus 119877 with 120573 = 0001 001 01

20 40 60 80 100

Radius R (m)0




01

02

03

04

05

06

07

08

09

1

P

Figure 5 Indirect D2D 119875FC versus 119877 with 120582 = 0001 00005 and120573 = 0001 0005

effects These observations about the system parameters 119877and 120573 are consistent with our analysis in Section 34

52 Indirect D2D Communications Figure 5 shows theresults of 119875FC for the indirect D2D communications in(19) The three (solid dashed and dotted) lines indicatedifferent 120582 and 120573 combinations while the symbols representthe corresponding simulation results As in the direct D2Dcommunication case the probability 119875FC decreases as 119877

20 40 60 80 100

Radius R (m)00


Simulation

0001

001

01

01

02

03

04

05

06

07

08

09

1

ＣＨ＞

Figure 6 Indirect D2D 120574ind = B2 versus 119877 with 120573 = 0001 00101

increases We can observe that 119875FC is a decreasing functionof 120582 Also 119875FC decreases as 120573 increases The same trends canbe observed in Figure 6 which is in line with our analysis inSection 43

Compared to the direct communication case for the sameparameters 120582 120573 and 119877 119875FC is significantly higher with theD2D communication hopping via the mmWave BS becausethe possible distance of the indirect D2D links that is 0 le119903119894 le 119877 is shorter than that of the direct D2D links thatis 0 le 119889119894119895 le 2119877 Also for the average separation betweentwo terminals (which are one device and the mm Wave BSfor the indirect D2D but two devices in the direct D2Dtransmissions) E119889119894119895 asymp 091119877 gt E119903119894 asymp 067119877 whichimplies a higher probability of blockage in the direct D2Dcompared to the indirect D2D communications

In contrast when it comes to the average ratios of thereliable D2D connections 120574ind is smaller than 120574dir This canbe explained by comparing the exponential terms whichdominate the polynomial terms for large enough 120573 and 119877 ofthe two ratios as

120574dir = A prop exp(minus 12845120587120573119877) asymp exp (minus09120573119877) 120574ind =B

2 prop [exp (minus120573119877)]2 asymp exp (minus2120573119877) (22)

Therefore for large enough 120573 and 119877 120574dir gt 120574ind53 HybridD2DCommunications In this section we presentthe D2D connectivity with both the direct and indirect com-munications jointly exploited In other words we considera hybrid scheme where all the devices are permitted to useboth direct and indirect links to communicate with eachother Therefore for a D2D pair we assume their D2D link


Table 1 119875FC for direct indirect and hybrid (direct + indirect) D2Dcommunications

Parameters 119875FC Change ()(120582 120573) 119877 Direct Indirect Hybrid

(00005 0001) 10 09992 09863 10000 00850 07417 08687 09722 119100 00063 03646 04828 324

(0001 0001) 10 09983 09859 10000 01750 03479 07715 09241 198100 00000 01329 01563 176

(0001 0005) 10 09913 09315 09918 00550 00595 02990 04026 346100 00000 00002 00002 mdash

00

20 70 9030 40 50 60 80 100

Radius R (m)10

resultsSimulation( )

(5 times 10minus4 10minus3)(10minus3 10minus3)

(10minus3 5 times 10minus3)

01

02

03

04

05

06

07

08

09

1

P

Figure 7 Hybrid D2D 119875FC versus 119877 with 120582 = 0001 00005 and120573 = 0001 0005

is reliable as long as one of the two (direct or indirect links) isreliable

Table 1 and Figure 7 show 119875FC with the hybrid (direct +indirect) D2D communications Also Table 2 and Figure 8present the ratio of the reliable D2D connections using thehybrid scheme In the two tables the last column indicatesthe performance improvements in percentage by using thehybrid scheme compared to the nonhybrid case assumingthat we pick the one with the better performance between thedirect and indirect connections As shown in the tables andfigures when the two communication links are jointly usedboth 119875FC and 120574 could be improved significantly For examplein Table 1 119875FC becomes almost one with the small 119877 for thegiven 120582 and 120573 Also Table 2 shows the increase in 120574 up to355

Table 2 120574 for direct indirect and hybrid (direct + indirect) D2Dcommunications

Parameters 120574 Change ()120573 119877 Direct Indirect Hybrid

000110 09909 09866 09999 00950 09559 09355 09972 43100 09143 08757 09893 82

00110 09134 08745 09891 8350 06497 05201 08319 280100 04408 02793 05970 354

0110 04376 02759 05928 35550 00349 00059 00406 163100 00012 00004 00016 333

20 70 9030 40 50 60 80 100

Radius R (m)10

resultsSimulation

0001

001

01

0

01

02

03

04

05

06

07

08

09

1

Figure 8 Hybrid D2D 120574 versus 119877 with 120573 = 0001 001 01

6 Conclusion

In this paper we consider D2D communications in mmWave where a device can communicate with another devicedirectly or via the mmWave BS Assuming uniform distribu-tion of devices according to PPP and the blockage model wederive the probability of the fully connected network for bothdirect and indirect communications Moreover we analyzethe ratio of the reliable D2D connections with the two typesof communications Through analysis and numerical resultswe observe that both connectivity performance metricsdecrease as the network size and the blockage parameterincrease Also the simulation results show that if the hybriddirect and indirect schemes are used both connectivityperformances can be enhanced up to about 35 comparedto the nonhybrid case


Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the Incheon National UniversityResearch Grant in 2016

References

[1] C-X Wang F Haider X Gao et al ldquoCellular architecture andkey technologies for 5G wireless communication networksrdquoIEEE Communications Magazine vol 52 no 2 pp 122ndash1302014

[2] T S Rappaport S Sun R Mayzus et al ldquoMillimeter wavemobile communications for 5G cellular it will workrdquo IEEEAccess vol 1 pp 335ndash349 2013

[3] 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

[4] 3rd Generation Partnership Project (3GPP) ldquoTechnical specifi-cation group SA Feasibility study for proximity services (ProSe)(Release 12)rdquo Tech Rep TR 22803 V100 2012

[5] IEEE 802158 ldquoPeer-Aware Communications (PAC)rdquo httpwwwieee802org15pubTG8html

[6] K Doppler M Rinne C Wijting C B Ribeiro and K HugldquoDevice-to-device communication as an underlay to LTE-advanced networksrdquo IEEE Communications Magazine vol 47no 12 pp 42ndash49 2009

[7] X Lin J G Andrews A Ghosh and R Ratasuk ldquoAn overviewof 3GPP device-to-device proximity servicesrdquo IEEE Communi-cations Magazine vol 52 no 4 pp 40ndash48 2014

[8] J Qiao X S Shen J W Mark Q Shen Y He and LLei ldquoEnabling device-to-device communications inmillimeter-wave 5G cellular networksrdquo IEEE Communications Magazinevol 53 no 1 pp 209ndash215 2015

[9] T S Rappaport F Gutierrez E Ben-Dor J N Murdock YQiao and J I Tamir ldquoBroadband millimeter-wave propagationmeasurements and models using adaptive-beam antennas foroutdoor urban cellular communicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 4 pp 1850ndash1859 2013

[10] S Rajagopal S Abu-Surra and M Malmirchegini ldquoChan-nel feasibility for outdoor non-line-of-sight mmwave mobilecommunicationrdquo in Proceedings of the 76th IEEE VehicularTechnology Conference (VTC rsquo12) pp 1ndash6 September 2012

[11] 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

[12] T Bai R Vaze and R W Heath ldquoAnalysis of blockage effectson urban cellular networksrdquo IEEE Transactions on WirelessCommunications vol 13 no 9 pp 5070ndash5083 2014

[13] D Stoyan W Kendall and J Mecke Geometry and Its Appli-cations John Wiley amp Sons New York NY USA 2nd edition1995

[14] J Choi ldquoOn the macro diversity with multiple BSs to mitigateblockage in millimeter-wave communicationsrdquo IEEE Commu-nications Letters vol 18 no 9 pp 1623ndash1656 2014

[15] T Bai and R W Heath ldquoCoverage and rate analysis formillimeter-wave cellular networksrdquo IEEE Transactions onWire-less Communications vol 14 no 2 pp 1100ndash1114 2015

[16] H Jung and I-H Lee ldquoOutage analysis of millimeter-wavewireless backhaul in the presence of blockagerdquo IEEE Commu-nications Letters vol 20 no 11 pp 2268ndash2271 2016

[17] 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

[18] S AndreevOGalinina A Pyattaev K Johnsson andYKouch-eryavy ldquoAnalyzing assisted offloading of cellular user sessionsonto D2D links in unlicensed bandsrdquo IEEE Journal on SelectedAreas in Communications vol 33 no 1 pp 67ndash80 2015

[19] J E Wieselthier G D Nguyen and A Ephremides ldquoOn theconstruction of energy-efficient broadcast and multicast treesin wireless networksrdquo in Proceedings of the 19th Annual JointConference of the IEEE Computer and Communications Societies(IEEE INFOCOM rsquo00) vol 2 pp 585ndash594 Tel Aviv IsraelMarch 2000

[20] P Billingsley Probability andMeasure JohnWiley amp Sons 2012[21] M Crofton ldquoProbabilityrdquo in Encyclopedia Britannica Britan-

nica 9th edition 1885[22] D E Barton F N David and E Fix ldquoRandom points in a circle

and the analysis of chromosome patternsrdquo Biometrika vol 50no 1-2 pp 23ndash29 1963

[23] Y Wang Z Wang and L Zhang ldquoInternet traffic engineeringwithout full mesh overlayingrdquo in Proceedings of the 20th AnnualJoint Conference on the IEEE Computer and CommunicationsSocieties (IEEE INFOCOM rsquo01) vol 1 pp 565ndash571 AnchorageAlaska USA April 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



noninterfering concurrent links To be specific directionalantenna arrays in mm Wave can reduce cochannel interfer-ence and improve spatial reuse of communication systemsthrough large beamforming gain This characteristic of mmWave systems is also desirable to guarantee connectivitybetween a huge number of devices in the future wirelessnetworks [8] In other words thanks to highly directionalbeam inmmWave a user equipment (UE) can communicatewith another UE in proximity over a D2D link whichenables multiple D2D pairs to use the same radio resourcessimultaneously

However compared to microwave an mm Wave com-munication channel experiences higher path-loss and issusceptible to blockage such as walls trees or even humanbodies as revealed in [2 9 10] Hence the mm Wave com-munication channel is a nearly bimodal channel dependingon the existence of line-of-sight (LoS) path [11] Based onthis property of the mm Wave propagation a new channelmodel is introduced in [12] which characterizes large-scaleblockage effects using random shape theory [13] The modelproposed in [12] shows that the probability of a blockage eventbetween two radios increases exponentially as the separationbetween the two increases This propagation model hasbeen applied to analyze various wireless communicationsystems using mm Wave The author of [14] derives theoutage probability improved by macrodiversity with multi-ple base stations (BSs) assuming the outage event occurswhen the LoS path is blocked The authors in [15] analyzethe coverage and rate performance in mm Wave cellularnetworks Moreover [16] presents the outage performanceof the mm Wave wireless backhaul link between a 5Gmacro base station (MBS) and small-cell base stations (SBSs)Assuming Poisson point process (PPP) on the plane thestochastic geometry used in these studies is known to be aneffective tool to evaluate system performances in the cellularnetworks [17]

In this paper based on the framework in [12 14] weconsider D2D connectivity in mm Wave networks ReliableD2D connections are required for D2D data transmissions aswell as cellular data offloading onto D2D connections whichcan provide considerable wireless capacity gains [18] In mmWave 5G cellular networks two kinds of D2D communi-cations can be enabled direct communications between twowireless devices in proximity and indirect communicationswhich connect two devices through base station(s) [8] In thispaper we investigate both types of the D2D communicationsassuming a network consisting of one mm Wave BS andmultiple wireless devices distributed according to a two-dimensional Poisson point process (PPP)

The contributions of this paper are fourfold First wederive the probability distribution mean and variance of theinterdevice distance (ie the distance between two randomlylocated devices) Second we derive the probability that theD2D network is fully connected which means that all thewireless devices have reliable communication links to eachother [19] Third we also quantify the D2D connectivityin terms of the average number of reliable connections (orcommunication links) both for the direct and indirect D2D

communication systems Lastly we consider a hybrid schemein which both the direct and indirect communications can beselectively used and present the simulation results to comparethe performances of the direct and indirect schemes

This paper is organized as follows The system modelis introduced in Section 2 The direct and indirect D2Dcommunication systems are analyzed in Sections 3 and 4respectively Numerical results are presented in Section 5Conclusions are provided in Section 6

2 System Model

We consider a 5G cellular network enabled with D2D com-munications with two tiers the cell tier and the device tierThe conventional cellular communications are supported bythe cell (macro or small-cell) tier while D2D communica-tions are covered by the device tier The BS may have a fullor partial control over the D2D communications dependingon the system architecture to establish and manage the D2Dlinks In this paper we focus on the connectivity of the D2Dlinks which indicates the potential extent of cellular dataoffloading onto D2D connections Thus the analysis in thispaper does not depend on specific design aspects of thenetwork architecture Based on [12] we focus on outdoorenvironments

We assume mm Wave 5G communication systems asshown in Figures 1(a) and 1(b) which illustrate direct andindirect D2D communications respectively In both caseswe assume 119873 devices are uniformly distributed over AreaS (ie the area of the gray circle with radius 119877) withintensity 120582 which means the average number of devicesper unit area (ie devicesm2) according to homogeneousPPP [20] We assume a quasi-stationary scenario where thenetwork topology is stationary while D2D communicationsare performed

For the indirect D2D communications in Figure 1(b)at the center of the circle there exists a single mm WaveBS which can be a small-cell BS with low power and lowcost in heterogeneous networks We note that we do notconsider indirect (multihop) communications via devicesas relays because of the complexity to build a multihoproute with directional antennas in the mobile scenario Inboth types of D2D networks the number of the devicesin the network is assumed to be 119873 which is a randomvariable following Poisson probability distribution as119875119873(119899) =((1205821205871198772)119899119899)119890minus1205821205871198772 where 119875119873(119899) is the probability that thereare 119899 devices in Area S and 119899 is a nonnegative integerTherefore the average number of the devices is given byE119873 = 1205821205871198772 where Esdot is the expectation operator

As in [15] for analytical tractability we assume thesectored antenna model which characterizes key features ofan antenna pattern As noted in [8] the interference issue byconcurrent cochannel links with omnidirectional antennasis of little concern in the mm Wave communications withhighly directional antenna arraysWe assume that directionalbeamforming is performed at devices as well as the mmWave BS We note a half-duplex system is assumed There-fore instead of the interference-limited performance metric


Device 1

Device 2 Device 3

DeviceN

Area

OR

Blockage

(a)

Device 1

mm Wave BS

Device 2 Device 3

DeviceN

Area

R

Blockage

(b)









119889S (1)




R R

dx

x

x2



119889Pr [119876] = 2 (Pr [119876 | 119862] minus Pr [119876])S


(2)


Pr [119876 | 119862] = 2119909119889119909 cosminus1 (1199092119877)1205871198772 (3)


119877119889Pr [119876] + 4Pr [119876] 119889119877 = 8119909119889119909 cosminus1 (1199092119877)1205871198772 119889119877 (4)




(6)



(7)









A = int21198770119890minus120573119909119891119889119894119895 (119909) 119889119909






(12)


120574dir = 1Pr [119873 ge 2]

infinsum119899=2

E 1003816100381610038161003816119879dir1003816100381610038161003816 | 119873 = 119899119899 (119899 minus 1) 2 Pr [119873 = 119899]= A

(13)




2 2592001205872


(14)






(15)










(17)






= infinsum119899=2




120574ind = 1Pr [119873 ge 2]





sdot Pr [119873 = 119899] = 1Pr [119873 ge 2]

infinsum119899=2


2

(20)



lim119877rarr0

B = lim119877rarr0


B = lim119877rarr0

212057321198772 = 0(21)



20 40 60 80 100

Radius R (m)0

01

02

03

04

05

06

07

08

09

1

P







5 Numerical Results





20 70 9030 40 50 60 80 100

Radius R (m)10


Simulation

01

0

02

03

04

05

06

07

08

09

1

0001

001

01

＞ＣＬ


20 40 60 80 100

Radius R (m)0




01

02

03

04

05

06

07

08

09

1

P




20 40 60 80 100

Radius R (m)00


Simulation

0001

001

01

01

02

03

04

05

06

07

08

09

1

ＣＨ＞











(00005 0001) 10 09992 09863 10000 00850 07417 08687 09722 119100 00063 03646 04828 324

(0001 0001) 10 09983 09859 10000 01750 03479 07715 09241 198100 00000 01329 01563 176

(0001 0005) 10 09913 09315 09918 00550 00595 02990 04026 346100 00000 00002 00002 mdash

00

20 70 9030 40 50 60 80 100

Radius R (m)10




01

02

03

04

05

06

07

08

09

1

P






000110 09909 09866 09999 00950 09559 09355 09972 43100 09143 08757 09893 82

00110 09134 08745 09891 8350 06497 05201 08319 280100 04408 02793 05970 354

0110 04376 02759 05928 35550 00349 00059 00406 163100 00012 00004 00016 333

20 70 9030 40 50 60 80 100

Radius R (m)10

resultsSimulation

0001

001

01

0

01

02

03

04

05

06

07

08

09

1


6 Conclusion



Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Device 1

Device 2 Device 3

DeviceN

Area

OR

Blockage

(a)

Device 1

mm Wave BS

Device 2 Device 3

DeviceN

Area

R

Blockage

(b)









119889S (1)




R R

dx

x

x2



119889Pr [119876] = 2 (Pr [119876 | 119862] minus Pr [119876])S


(2)


Pr [119876 | 119862] = 2119909119889119909 cosminus1 (1199092119877)1205871198772 (3)


119877119889Pr [119876] + 4Pr [119876] 119889119877 = 8119909119889119909 cosminus1 (1199092119877)1205871198772 119889119877 (4)




(6)



(7)









A = int21198770119890minus120573119909119891119889119894119895 (119909) 119889119909






(12)


120574dir = 1Pr [119873 ge 2]

infinsum119899=2

E 1003816100381610038161003816119879dir1003816100381610038161003816 | 119873 = 119899119899 (119899 minus 1) 2 Pr [119873 = 119899]= A

(13)




2 2592001205872


(14)






(15)










(17)






= infinsum119899=2




120574ind = 1Pr [119873 ge 2]





sdot Pr [119873 = 119899] = 1Pr [119873 ge 2]

infinsum119899=2


2

(20)



lim119877rarr0

B = lim119877rarr0


B = lim119877rarr0

212057321198772 = 0(21)



20 40 60 80 100

Radius R (m)0

01

02

03

04

05

06

07

08

09

1

P







5 Numerical Results





20 70 9030 40 50 60 80 100

Radius R (m)10


Simulation

01

0

02

03

04

05

06

07

08

09

1

0001

001

01

＞ＣＬ


20 40 60 80 100

Radius R (m)0




01

02

03

04

05

06

07

08

09

1

P




20 40 60 80 100

Radius R (m)00


Simulation

0001

001

01

01

02

03

04

05

06

07

08

09

1

ＣＨ＞











(00005 0001) 10 09992 09863 10000 00850 07417 08687 09722 119100 00063 03646 04828 324

(0001 0001) 10 09983 09859 10000 01750 03479 07715 09241 198100 00000 01329 01563 176

(0001 0005) 10 09913 09315 09918 00550 00595 02990 04026 346100 00000 00002 00002 mdash

00

20 70 9030 40 50 60 80 100

Radius R (m)10




01

02

03

04

05

06

07

08

09

1

P






000110 09909 09866 09999 00950 09559 09355 09972 43100 09143 08757 09893 82

00110 09134 08745 09891 8350 06497 05201 08319 280100 04408 02793 05970 354

0110 04376 02759 05928 35550 00349 00059 00406 163100 00012 00004 00016 333

20 70 9030 40 50 60 80 100

Radius R (m)10

resultsSimulation

0001

001

01

0

01

02

03

04

05

06

07

08

09

1


6 Conclusion



Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










R R

dx

x

x2



119889Pr [119876] = 2 (Pr [119876 | 119862] minus Pr [119876])S


(2)


Pr [119876 | 119862] = 2119909119889119909 cosminus1 (1199092119877)1205871198772 (3)


119877119889Pr [119876] + 4Pr [119876] 119889119877 = 8119909119889119909 cosminus1 (1199092119877)1205871198772 119889119877 (4)




(6)



(7)









A = int21198770119890minus120573119909119891119889119894119895 (119909) 119889119909






(12)


120574dir = 1Pr [119873 ge 2]

infinsum119899=2

E 1003816100381610038161003816119879dir1003816100381610038161003816 | 119873 = 119899119899 (119899 minus 1) 2 Pr [119873 = 119899]= A

(13)




2 2592001205872


(14)






(15)










(17)






= infinsum119899=2




120574ind = 1Pr [119873 ge 2]





sdot Pr [119873 = 119899] = 1Pr [119873 ge 2]

infinsum119899=2


2

(20)



lim119877rarr0

B = lim119877rarr0


B = lim119877rarr0

212057321198772 = 0(21)



20 40 60 80 100

Radius R (m)0

01

02

03

04

05

06

07

08

09

1

P







5 Numerical Results





20 70 9030 40 50 60 80 100

Radius R (m)10


Simulation

01

0

02

03

04

05

06

07

08

09

1

0001

001

01

＞ＣＬ


20 40 60 80 100

Radius R (m)0




01

02

03

04

05

06

07

08

09

1

P




20 40 60 80 100

Radius R (m)00


Simulation

0001

001

01

01

02

03

04

05

06

07

08

09

1

ＣＨ＞











(00005 0001) 10 09992 09863 10000 00850 07417 08687 09722 119100 00063 03646 04828 324

(0001 0001) 10 09983 09859 10000 01750 03479 07715 09241 198100 00000 01329 01563 176

(0001 0005) 10 09913 09315 09918 00550 00595 02990 04026 346100 00000 00002 00002 mdash

00

20 70 9030 40 50 60 80 100

Radius R (m)10




01

02

03

04

05

06

07

08

09

1

P






000110 09909 09866 09999 00950 09559 09355 09972 43100 09143 08757 09893 82

00110 09134 08745 09891 8350 06497 05201 08319 280100 04408 02793 05970 354

0110 04376 02759 05928 35550 00349 00059 00406 163100 00012 00004 00016 333

20 70 9030 40 50 60 80 100

Radius R (m)10

resultsSimulation

0001

001

01

0

01

02

03

04

05

06

07

08

09

1


6 Conclusion



Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













(12)


120574dir = 1Pr [119873 ge 2]

infinsum119899=2

E 1003816100381610038161003816119879dir1003816100381610038161003816 | 119873 = 119899119899 (119899 minus 1) 2 Pr [119873 = 119899]= A

(13)




2 2592001205872


(14)






(15)










(17)






= infinsum119899=2




120574ind = 1Pr [119873 ge 2]





sdot Pr [119873 = 119899] = 1Pr [119873 ge 2]

infinsum119899=2


2

(20)



lim119877rarr0

B = lim119877rarr0


B = lim119877rarr0

212057321198772 = 0(21)



20 40 60 80 100

Radius R (m)0

01

02

03

04

05

06

07

08

09

1

P







5 Numerical Results





20 70 9030 40 50 60 80 100

Radius R (m)10


Simulation

01

0

02

03

04

05

06

07

08

09

1

0001

001

01

＞ＣＬ


20 40 60 80 100

Radius R (m)0




01

02

03

04

05

06

07

08

09

1

P




20 40 60 80 100

Radius R (m)00


Simulation

0001

001

01

01

02

03

04

05

06

07

08

09

1

ＣＨ＞











(00005 0001) 10 09992 09863 10000 00850 07417 08687 09722 119100 00063 03646 04828 324

(0001 0001) 10 09983 09859 10000 01750 03479 07715 09241 198100 00000 01329 01563 176

(0001 0005) 10 09913 09315 09918 00550 00595 02990 04026 346100 00000 00002 00002 mdash

00

20 70 9030 40 50 60 80 100

Radius R (m)10




01

02

03

04

05

06

07

08

09

1

P






000110 09909 09866 09999 00950 09559 09355 09972 43100 09143 08757 09893 82

00110 09134 08745 09891 8350 06497 05201 08319 280100 04408 02793 05970 354

0110 04376 02759 05928 35550 00349 00059 00406 163100 00012 00004 00016 333

20 70 9030 40 50 60 80 100

Radius R (m)10

resultsSimulation

0001

001

01

0

01

02

03

04

05

06

07

08

09

1


6 Conclusion



Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












= infinsum119899=2




120574ind = 1Pr [119873 ge 2]





sdot Pr [119873 = 119899] = 1Pr [119873 ge 2]

infinsum119899=2


2

(20)



lim119877rarr0

B = lim119877rarr0


B = lim119877rarr0

212057321198772 = 0(21)



20 40 60 80 100

Radius R (m)0

01

02

03

04

05

06

07

08

09

1

P







5 Numerical Results





20 70 9030 40 50 60 80 100

Radius R (m)10


Simulation

01

0

02

03

04

05

06

07

08

09

1

0001

001

01

＞ＣＬ


20 40 60 80 100

Radius R (m)0




01

02

03

04

05

06

07

08

09

1

P




20 40 60 80 100

Radius R (m)00


Simulation

0001

001

01

01

02

03

04

05

06

07

08

09

1

ＣＨ＞











(00005 0001) 10 09992 09863 10000 00850 07417 08687 09722 119100 00063 03646 04828 324

(0001 0001) 10 09983 09859 10000 01750 03479 07715 09241 198100 00000 01329 01563 176

(0001 0005) 10 09913 09315 09918 00550 00595 02990 04026 346100 00000 00002 00002 mdash

00

20 70 9030 40 50 60 80 100

Radius R (m)10




01

02

03

04

05

06

07

08

09

1

P






000110 09909 09866 09999 00950 09559 09355 09972 43100 09143 08757 09893 82

00110 09134 08745 09891 8350 06497 05201 08319 280100 04408 02793 05970 354

0110 04376 02759 05928 35550 00349 00059 00406 163100 00012 00004 00016 333

20 70 9030 40 50 60 80 100

Radius R (m)10

resultsSimulation

0001

001

01

0

01

02

03

04

05

06

07

08

09

1


6 Conclusion



Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










20 70 9030 40 50 60 80 100

Radius R (m)10


Simulation

01

0

02

03

04

05

06

07

08

09

1

0001

001

01

＞ＣＬ


20 40 60 80 100

Radius R (m)0




01

02

03

04

05

06

07

08

09

1

P




20 40 60 80 100

Radius R (m)00


Simulation

0001

001

01

01

02

03

04

05

06

07

08

09

1

ＣＨ＞











(00005 0001) 10 09992 09863 10000 00850 07417 08687 09722 119100 00063 03646 04828 324

(0001 0001) 10 09983 09859 10000 01750 03479 07715 09241 198100 00000 01329 01563 176

(0001 0005) 10 09913 09315 09918 00550 00595 02990 04026 346100 00000 00002 00002 mdash

00

20 70 9030 40 50 60 80 100

Radius R (m)10




01

02

03

04

05

06

07

08

09

1

P






000110 09909 09866 09999 00950 09559 09355 09972 43100 09143 08757 09893 82

00110 09134 08745 09891 8350 06497 05201 08319 280100 04408 02793 05970 354

0110 04376 02759 05928 35550 00349 00059 00406 163100 00012 00004 00016 333

20 70 9030 40 50 60 80 100

Radius R (m)10

resultsSimulation

0001

001

01

0

01

02

03

04

05

06

07

08

09

1


6 Conclusion



Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












(00005 0001) 10 09992 09863 10000 00850 07417 08687 09722 119100 00063 03646 04828 324

(0001 0001) 10 09983 09859 10000 01750 03479 07715 09241 198100 00000 01329 01563 176

(0001 0005) 10 09913 09315 09918 00550 00595 02990 04026 346100 00000 00002 00002 mdash

00

20 70 9030 40 50 60 80 100

Radius R (m)10




01

02

03

04

05

06

07

08

09

1

P






000110 09909 09866 09999 00950 09559 09355 09972 43100 09143 08757 09893 82

00110 09134 08745 09891 8350 06497 05201 08319 280100 04408 02793 05970 354

0110 04376 02759 05928 35550 00349 00059 00406 163100 00012 00004 00016 333

20 70 9030 40 50 60 80 100

Radius R (m)10

resultsSimulation

0001

001

01

0

01

02

03

04

05

06

07

08

09

1


6 Conclusion



Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Connectivity Analysis of Millimeter-Wave...

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