Research Article Multicast Capacity Analysis for High...

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2013 Article ID 234728 9 pageshttpdxdoiorg1011552013234728

Research ArticleMulticast Capacity Analysis for High Mobility SocialProximity Machine-to-Machine Networks

Xin Guan

School of Information Science and Technology Heilongjiang University Harbin 150080 China

Correspondence should be addressed to Xin Guan guanxinhljugmailcom

Received 16 August 2013 Revised 25 October 2013 Accepted 28 October 2013

Academic Editor Yan Zhang

Copyright copy 2013 Xin Guan This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Wireless machine-to-machine (M2M) networks enable ubiquitous sensing and controlling via sensors vehicles and other types ofwireless nodes Capacity scaling law is one of the fundamental properties for high mobility M2M networks As for high mobilityM2Mnetworks vehicular ad hoc networks (VANETs) are a typical case Since vehicles have social property their moving trajectoryis according to the fixed community With the purpose of transmitting packets to different communities of VANETs and furtherimproving the network capacity we study the multicast capacity of bus-assisted VANETs in two scenarios forwarding scenario androuting scenario All the ordinary vehicles obey the restricted mobility model Thus the spatial stationary distribution decays aspower law with the distance from the center spot of a restrict region of each vehicle In forwarding scenario all the buses deployedin all roads as intermediate nodes are used to forward packets for ordinary vehicles In routing scenario buses and ordinary carsconstruct a highway path supported by percolation theory to transmit urgent packets Each ordinary vehicle randomly chooses119896 minus 1 vehicles from the other ordinary vehicles as receivers For the two kinds of scenarios we derived the upper bound and lowerbound respectively

1 Introduction

In recent years M2M networks have been a subject ofintense interests due to the rapid evolution of wirelesscommunication systems M2M networks can be used toenable a wide variety of automated complex operationswhich include advanced sensing remote control and mon-itoring technologies M2M networks have many differencescomparedwith conventional wireless networks becauseM2Mis a machine-oriented networking technology First of allM2Mnetworks commonly need to support a large number ofdevices Second power consumption of M2M devices is veryimportant due to the limited battery capacity Third variousdata transfer delay requirements may exist depending on theM2M application being executed [1 2] Examples of M2Mcommunications networks include vehicular ad hoc networks(VANETs) [3] underwater sensor networks (UWSNs) [4]wireless body area networks (WBANs) [5] and wirelessmobile social networks [6ndash8]

Capacity scaling laws of wireless networks have attracteda lot of attention Since Gupta and Kumar initiated the study

of capacity scaling laws of ad hoc networks [9] the capacityof different types of wireless networks has been widelystudied such as sensor networks ad hoc networks cellularnetworks vehicular networks and so forth Grossglauserand Tse showed how the mobility increases the capacity ofad hoc wireless network in 2002 [10] Li proposed a newmethod to calculate the multicast capacity of wireless ad hocnetworks in 2009 [11] Garetto and Leonardi proved restrictedmobility improves delay-throughput tradeoffs in mobile adhoc networks [12] and Mao et al calculated the multicastcapacity for hybrid wireless networks [13] in 2012 Thereare also some people who work on the capacity of energy-constrained networks [14] some people study the capacity ofarbitrary networks [15] and inhomogeneous networks [16]some people let nodes cooperate with a hierarchical MIMOnetwork [17] and some people use network coding [18] toimprove the network capacity

The capacity of vehicular ad hoc networks was first stud-ied in 2007 by Pishro-Nik et al [19]They proposed a grid-likeframework as shown in Figure 1 to demonstrate the streets ofurban region We have abstracted one real construction from

2 International Journal of Distributed Sensor Networks

Tier (1)

Tier (2)

Tier (3)

Road segment

Road line

middot middot middot

middot middot middot middot middot middot


Figure 1 The real map of urban area

the real urban map However we realized we also have to usestatistic methods to calculate the capacity of high mobilitysocial M2M networks If the statistic methods were appliedin the calculation the detail of the real construction willbe neglected in the calculation Thus there is no differencebetween real construction and grid-like construction Fur-thermore grid-like construction is easy to understand andconvenient to be used in calculation As a result we choosethe grid-like construction to be the geometrical constructionin our calculation Lu et al extended their work and obtainedthe per-vehicle throughputΩ(1 log(119899)) and delay O(log2(119899))[20] However Lu et al only consider the scenario thatsource vehicles can only transmit packets to the vehicle whichbelongs to the same community as the source vehicle and theyonly consider the unicast transmissionThere are rare studiesthat work on the calculation of capacity for different VANETsscenarios The capacity of different typical ad hoc scenarioscannot be adopted for theoretical analysis for VANETs andcannot provide guidelines in designing VANETs Thereforewe calculate the multicast capacity for bus-assisted VANETsin this paper

Assume that 119899 ordinary vehicles and 119899119887

buses aredeployed in a grid-like road framework This construction iscomposed of119898 parallel lines with other intersected119898 parallellines Since the larger urban region means more numberof vehicles 119898 increases linearly with 119899 Vehicles that havesocial proximity mobility always move in a localized areacentered at the driverrsquos home or work space and seldommove out Thus we let all the ordinary vehicles obey therestricted mobility model As a result the spatial stationarydistribution of vehicles decays as the form of power law withthe distance between the home point and each vehicle Eachordinary vehicle randomly chooses 1198961 vehicles from the otherordinary vehicles as receiversThe network can be consideredas unicast when 119896 is 2 and as broadcast when 119896 is 1198991

In the forwarding scenario all the buses deployed in allroads as intermediate nodes are used to forward packetsfor ordinary vehicles The buses of each road were consid-ered as one bus trajectory The bus forward scheme wasproposed in [21] Each transmission flow transmits packetsvia two-hop relay scheme proposed in [22] All the forwardprocesses of buses are considered as one-hop The packetscould be transmitted directly from source to destination orbe transmitted to an intermediate vehicle or bus then beforwarded to the destination In the routing scenario we usebuses and ordinary cars to construct the highway systemWe use percolation theory to prove that there are highwaypath clusters that cross through the network vertically andhorizontally Thus based on a simple routing protocol thehighway system can ensure the packets can be transmitted todestination located in any place of VANET Simultaneouslythe urgent TTL requirement can be satisfied

To schedule the interference in MAC layer of wirelesstransmission we use the protocol model [6] as our inter-ference model in this paper We assume the bandwidthof wireless transmission is 119882

119886bits per time slot and the

bandwidth of bus system is 119882119887bits per time slot For

simplicity we also assume that there is only one wirelesschannel in the bus-assisted forwarding and routing and allthe vehicles and buses have enough memory to buffer all thepacketsOur Main Contributions With the purpose of transmittingpackets to different communities of VANETs and furtherimproving the network capacity in this paper we derive theupper and lower bound of the multicast capacity for bus-assisted VANET in two kinds of scenarios

In the forwarding scenario the per vehicle capacity ofbus-assisted VANET is as follows

119874max [119882119886

119896min(

119882119887radic119888

119899radic119896119882119886119888

119896119899)] if 119896 = 119874 (119888)

119874 max [119882119886

119896min(

119882119887

119899119882119886119888

119896119899)] if 119896 = Ω (119888)

(1)

and 119908ℎ119901 scales of at leastΩ((119896 minus 1) log(119899))In the routing scenario we construct the highway system

to transmit the urgent packets We also derive the capacityof per vehicle via busminusassisted mode it can achieve at most119874(1119896119889) and cannot be lower thanΩ(1radic119896119889)

The rest of the paper is organized as follows In Section 2we represent the network model in detail The upper andlower bounds of multicast capacity by ad hoc forwardingare derived in Section 3 In Section 4 we calculate the upperand lower bounds of multicast capacity by bus forwardingWe combine the results of Sections 3 and 4 and obtain themulticast capacity for bus-assisted forwarding in Section 5Section 6 constructs the highway path and derives the capac-ity of bus-assisted VANET in routing scenario Section 7discusses the results and the remaining challenges Section 8concludes this paper and reviews the results on multicastcapacity for bus-assisted VANETs

International Journal of Distributed Sensor Networks 3

2 Network Model

21 Network Geometry We use the grid-like construction(as shown in Figure 1) to represent the urban region Theconstruction comprises 119898 parallel streets intersected withother 119898 parallel streets Each street considers a bus line Let119904 denote the total number of road segments (street sectionbetween any two street lines) and let 119888 denote the numberof squares in the grid-like construction Thus 119904 = 2119888 =

2(119898 minus 1)2 The density of the network is 119889 = 119899119904 = 1198992(119898 minus

1)2 One has the experience that the center spots (such as

workplaces markets and malls) always have high vehiculardensity than other locations If we use amathematicalmethodto analyse this phenomenon we can find out that the densityof one location is steady Furthermore the density of thelocation would decrease as the distance from the center spotincreasesTherefore we can derive the density of one locationaccording to its distance from the center spot From Figure 1we assume that Tier (1) is one of the center spots so it hasthe highest vehicular density According to the formulationcalculated in [17] we can get the density of Tier (2) and Tier(3) based on their distance from Tier (1) Obviously Tier (3)has lower vehicular density than Tier (2) due to its longerdistance from Tier (1) than Tier (2) Larger urban regionmeans the more number of vehicles so we assume 119889 is aconstant to make the number of street segments 119904 increaselinearly with the number of vehicles 119899

22 Mobility Model Vehicles always move in a localized areacentered at the driverrsquos home or work space and seldommoveout Thus we let all the ordinary vehicles obey the restrictedmobility model Each vehicle randomly selects one square asits home-point (as Tier (1) which is showed in Figure 1) Tier(2) was formed by the adjacent squares surrounding Tier (1)Tier (3)was formed so forthThe restrictedmobility region isTier (120572) 120572 isin 1 2 120596 where Tier (120596) is the outermost tierof themobility region of a vehicleThe stationary distributionof vehicles on the road segments of Tier (120572) is denoted by120587120572 Thus the steady-state location of vehicles is modeled by

power law function 120587120572= 120572minus1205741205871 where exponent 120574 gt 0

Moreover 1205871= 1sum

120596

120572=1(16120572 minus 12)120572

minus120574 is calculated in [17]All the buses are deployed in all roads as intermediate nodesThey were used to forward packets for ordinary vehicles inthe forwarding scenario and construct highway system withordinary vehicles in the routing scenario The buses of eachroad construct one bus line Thus there are 2119898 bus lines inthe network

23 Communication and Interference Model For simplicitywe assume that there is only one wireless channel in thebus-assisted forwarding and during a time slot 119905 the wirelesschannel can only transmit 119882

119886bits The bandwidth of the

bus system is 119882119887bits per time slot Due to the interference

of wireless transmission one vehicle cannot transmit withtwo other vehicles at the same time slot Let 119903 denote thetransmission rangeThe length of one segment is equal to 119903 toensure the transmission range covers the entire road segmentThus 119903 = (1119898 minus 1) To schedule the transmission flows we

adopt the protocol interference model Protocol interferencemodel schedule is defined as follows

At each time slot a transmission from vehicle 119894 to vehicle119895 is successful only if the following inequality stands

10038171003817100381710038171003817119883119894(119905) minus 119883

119895(119905)10038171003817100381710038171003817le 119903 (2)

and for any other vehicle 119897 that transmits at 11990510038171003817100381710038171003817119883119897(119905) minus 119883

119895(119905)10038171003817100381710038171003817le (1 + Δ) 119903 (3)

where Δ is a guard factor defining a protection zone aroundthe receivers and 119909

119894(119905) denotes the location of vehicle 119894 at the

time slot 119905 we will add the missing definition in the paper

24 TrafficModel and Relay Scheme We consider 119899multicastflows existing in the network concurrently Each packet istransmitted to 119896minus1 destinations Each vehicle is the source ofonemulticast flowandone of the destinations of anothermul-ticast flow The network can be considered as unicast when 119896is 2 and as broadcast when 119896 is 1198991 There are three differentforwarding strategies in the bus-assisted VANETs ad hocforwarding bus forwarding and bus-assisted forwardingWith the first forwarding method multicast flows transmitpackets via two-hop relay scheme With the application oftwo-hop relay scheme a vehicle can transmit packets toa destination directly or relayed through one intermediatevehicle that has more contact opportunities with the desti-nation vehicle and all the intermediate vehicles are ordinaryvehicles Since each vehicle has the restrictedmobility regionthe source and destination have the same home-point and 119896is less than or equal to the number of vehicles that have thesame home-point The second forwarding method uses busas intermediate vehicle The forwarding scheme of buses isdiscussed in [18] In this forwarding method the bus is usedin intercommunity communication Thus the source anddestination have different home-pointsThe third forwardingmethod combines the above two forwarding methods

25 Definitions of Capacity In this paper capacity denotesthe feasible throughput The capacity of VANETs is definedas follows

Definition 1 (feasible throughput) A throughput of 120582(119899) bitsper second for each vehicle is feasible if there is a spatialand temporal scheme for scheduling transmissions and everyvehicle can send 120582(119899) bits per second on average to its chosendestination

Definition 2 (capacity of vehicle network) The average capac-ity of vehicular network is of order 120579(119892(119899)) (we use Knuthrsquosnotation given two functions119892(119899) ge 0where119891(119899) = 119874(119892(119899))means lim sup

119899rarrinfin119891(119899)119892(119899) = 119888 lt infin 119891(119899) = Ω(119892(119899))

is equivalent to 119891(119899) = 119874(119892(119899)) 119891(119899) = Θ(119892(119899)) means119891(119899) = 119874(119892(119899)) and 119891(119899) = Ω(119892(119899))) bits per second if thereare deterministic constants 119888 gt 0 and 119888 lt 1198881015840 lt +infin such that

lim119899rarrinfin

Pr (120582 (119899) = 119888 (119892 (119899)) is feasible) = 1

lim119899rarrinfin

inf Pr (120582 (119899) = 119888 (119892 (119899)) is feasible) lt 1(4)


Definition 3 (throughput capacity) Let 119866(119879) denote theamount of data received by all the vehicles during time 119879 Acapacity throughput 120582(119899) is feasible if there is a schedulingscheme for which the following properties hold

lim119879rarrinfin

Pr(119866 (119879)119879

ge 120582) = 1 (5)

3 Bounds in Multicast Capacity byAd Hoc Forwarding

Forwarding scenario has two parts ad hoc forwarding partsand bus forwarding parts In this section we discuss thebounds of capacity for pure ad hoc forwarding for theforwarding scenario All the packets are transmitted todestination directly or via an ordinary vehicle Since sourcevehicle and destination vehicle have the same home-point 119896is less than or equal to the number of vehicles that have thesame home-point Broadcast cannot be achieved only withthis forwarding method

31 Upper Bound in Multicast Throughput Capacity by AdHoc Forwarding We first calculate the upper bound ofmulticast throughput capacity by ad hoc forwarding Thefollowing lemma is used in the calculation of upper bound ofthroughput capacity of ad hoc forwarding This lemma wasproved in [17]

Lemma 4 Let 119901119886denote the probability of a road segment

being active as follows

119901119886=

1

(2 lceil1 + Δrceil lceil2 + Δrceil) (6)

where lceil119909rceil represents the smallest integer number greater thanor equal to 119909

According to Lemma 4 we first derive the upper boundin multicast throughput capacity by ad hoc forwarding

Theorem 5 For the social proximity VANETs with two-hopscheme the average per-multicast throughput capacity by adhoc forwarding cannot be better than 1(2119896119889lceil1 + Δrceillceil2 + Δrceil)

Proof Let 119866119889(119879) denote the amount of data transmitted

through direct transmission from source to destination dur-ing the time interval [0 119879] and let119866

119903(119879)denote the amount of

data transmitted through relay transmission during the timeinterval [0 119879] According to Definition 3 throughput 120582(119899)satisfies the following inequality

119866119889(119879) + 119866

119903(119879)

119879ge 119896119899120582 (119899) minus 120576 (7)

where 120576 gt 0 is an arbitrary and fixed number and 120576 rarr 0 as119879 rarr infin Let 119870(119879) denote the total transmit opportunitiesduring [0 119879] The total number of transmitted packets mustbe less than the total number of transmit opportunities duringa long time interval Since the relay transmission needs the

transmitting opportunities twice to transmit one packet wehave the following

1

119879119870 (119879)119882

119886ge1

119879119866119889(119879) +

2

119879119866119903(119879) (8)

Substituting (7) into (8) we have the following

1

119879119870 (119879)119882

119886ge1

119879119866119889(119879) + 2 (119896119899120582 (119899) minus 120576 minus

1

119879119866119889(119879)) (9)

Sorting (9) we have the following

120582 (119899) le(1119879)119870 (119879)119882

119886+ (1119879)119866

119889(119879) + 2120576

2119896119899 (10)

When 120576 rarr 0 as 119879 rarr infin

120582 (119899) le(1119879)119870 (119879)119882

119886+ (1119879)119866

119889(119879)

2119896119899 (11)

Due to the interference of wireless transmission the totaltransmission must be less than the concurrent transmissionsduring time [0 119879] According to the law of large numbers wehave the following

lim119909rarr119879

1

119879119870 (119879)119882

119886le 119904119901119886119882119886 (12)

Similarly we have the following

lim119909rarr119879

1

119879119866119889(119879) le 119904119901

119886119882119886 (13)

The two equalities hold when there is always a transmissionflow exits on each unit of a concurrent transmissions groupduring each time slot According to Lemma 4 by substituting(12) and (13) into (11) we have the following

120582 (119899V) le119904119901119886

119896119899=119901119886

119896119889=

1

2119896119889 lceil1 + Δrceil lceil2 + Δrceil (14)

Thus the theorem then follows

32 Lower Bound in Multicast Throughput Capacity by AdHoc Forwarding To obtain the lower bound of averagemulticast throughput capacity we introduce the followinglemma which was proved in [17]

Lemma 6 The number of vehicles whose mobility regioncontains road segment 119894 is denoted by 119873 Thus 119873 scales as119874(log(119899)) when 119899 rarr infin the probability approaches 1

With the two-hop relay scheme a packet can be suc-cessfully transmitted only if there exists at least one source-destination pair or source-intermediate pair when the roadsegment is active Since sources and destinations have thesame home-point there is one source-destination pair orsource-intermediate pair with probability (119896minus1)2119873 Accord-ing to Lemma 6 the probability of success transmission is atleast Ω((119896 minus 1) log(119899)) Therefore we derive the followingtheorem

Theorem 7 The throughput capacity of average per-multicastflow can be scaled at least Ω((119896 minus 1) log(119899)) 119908ℎ119901


4 Bounds in Multicast Capacity byBus Forwarding

In this section we calculate the bounds of multicast capacityby bus forwarding for the forwarding scenario All the packetsare relayed by bus and all the ordinary vehicles can be thedestinations of any source vehicle With the bus forwarding119896 can be equal to 119899 to achieve broadcast in the network

41 Upper Bound in Multicast Throughput Capacity by BusForwarding We first calculate the upper bound in multicastthroughput capacity by bus-to-bus transmissions We intro-duce the Euclidean tree to demonstrate the bus transmissionprocess of multicast flows and each segment of the bustransmission process used is one edge of a multicast treeLet 119879119894denote the 119894th multicast tree and let 119878(119879

119894) denote the

number of segments the tree 119879119894will use The total number

of used segments of total transmissions in the network isdenoted by 119871 = sum

119899

119894=1119878(119879119894) To obtain the value of 119871 we

introduce the following lemma which was proved in [20]

Lemma 8 Given 119899 nodes randomly and uniformly distributedin a 2-dimensional cube divide the cube into 119888 cells as Voronoidiagrams with the same side length Each node transmitspackets to 119896 destination concurrent via base-station Thebase-station forwarding of a transmission is considered as aEuclidean tree When 119896 = O(119898) with probability of at least1 minus 2119890

minus119899120579232 the total edge number of all Euclidean tree is

119871 ge 119899120579radic11989611989816 When 119896 = Ω(119898) with probability at least1 minus 2119890

minus1198998 then 119871 ge 1198991198984

Different from the base-station connected by fiber theones connected by bus carry packets and move to thedestination along roads represented by segments in the gridconstruction However if we consider the intersections ofroad segments as the vertex of Euclidean tree and eachintersection belonging to a unique square we can obtain thesame conditions with Lemma 8 Thus the results are suitablefor the Euclidean tree of bus-to-bus transmissions Accordingto the above analysis we derive the following corollary

Corollary 9 In the grid-like construction if one usesEuclidean tree which represents the bus-to-bus transmissionsone can have the following results When 119896 = 119874(119898) withprobability at least 1 minus 2119890minus119899120579

232 the total edge number of all

Euclidean tree is 119871 ge 119899120579radic11989611989816 When 119896 = Ω(119898) withprobability at least 1 minus 2119890minus1198998 then 119871 ge 1198991198984

Recall that there are 119904 segments in the constructionThenaccording to Pigeonhole principle when 119896 = 119874(119888) there is atleast one cell that will be used by at least (119899120579radic11989611988816)119888 flowswith probability at least 1minus2119890minus119899120579

232 andwhen 119896 = Ω(119888) there

is at least one cell that will be used by at least 1198991199044 flows with

probability at least 1minus2119890minus1198998 Let119882119887denote the packets num-

ber of transmission by bus during one time slot then we havethe following theorem

Theorem 10 When 119896 = O(119888) the per-multicast flow through-put capacity of bus-to-bus transmission is at most 119882

119887radic119888

119899120579radic119896 119908ℎ119901 When 119896 = Ω(119888) the per-multicast flow through-put capacity of bus-to-bus transmissions is at most 4119882

119887

119899 119908ℎ119901

Theorem 10 is the capacity of bus-to-bus transmissionsbuses have to transmit packets to ordinary vehicles at lastThen we calculate the up bound of the transmission betweenbuses and ordinary vehicles for bus forwarding methodRecall that the active probability of one segment is 119901

119886 The

total transmission opportunities are at most 119904119901119886 Each packet

firstly is transmitted to bus then is transmitted to ordinaryvehicles by bus One multicast flow totally has 119896 links Thusaccording to Pigeonhole principle at least one link of amulticast has atmost119901

119886119896119899 transmission opportunitiesThen

we derived the upper bound of the transmission betweenbuses and ordinary vehicles for bus forwarding method

Theorem 11 The upper bound of the throughput capacitybetween bus and ordinary vehicles for bus forwarding is119882119886119904119901119886119896119899 = 119882

119886119901119886119896119889

Obviously the minimum throughput of bus forwardingprocess and bus-to-vehicle process determines the through-put of whole bus forwarding By summarizing Theorems 10and 11 we derive the upper bound of multicast throughputcapacity for bus forwarding

Theorem12 Theupper bound of themulticast capacity for busforwarding is as follows

119874[min(119882119887radic119888

119899radic119896119882119886119888

119896119899)] if 119896 = 119874 (119888)

119874 [min(119882119887

119899119882119886119888

119896119899)] if 119896 = Ω (119888)

(15)

42 Lower Bound in Multicast Throughput Capacity by BusForwarding By using the bus forwarding method a packetcan be successfully transmitted by bus only if there exists atleast one bus-destination vehicle when the road segment isactive Reference to the proof of Lemma 6 we can easily provethe following lemma

Lemma 13 The number of vehicles whose mobility regioncontains bus line 119894 is denoted by 119873

119887 Thus 119873

119887scales as

119874(log(119899)) 119908ℎ119901

According to Lemma 13 we know that in any segmentthere is one bus-destination vehicle pair with probability (119896minus1)119873119887and the probability of success transmission is at least

Ω(119896 log(119899)) Therefore we derive the following theorem

Theorem14 The throughput capacity of average per-multicastflow can be scales at least Ω((119896 minus 1) log(119899)) 119908ℎ119901

5 Capacity Bounds for Bus-AssistedForwarding

In this section we will analyze the throughput capacity forbus-assisted forwarding The analysis is based on the results


of ad hoc forwarding and bus forwarding derived aboveWiththe purpose of transmitting packets to other communitiesthat have different home-point and further improving thenetwork capacity both ad hoc forwarding and bus forward-ing are used in bus-assisted forwarding In particular busforwarding is the only way to transmit packets to othercommunities According to the destination of packets bus-assisted forwarding adaptively selects a better forwardingmethod Therefore the throughput capacity of bus-assistedforwarding cannot surpass the maximum capacity of ad hocforwarding and bus forwarding The maximum capacity ofad hoc forwarding and bus forwarding is optimum for bus-assisted forwarding Similarly bus-assisted forwarding hasthe same lower bound with ad hoc forwarding and busforwarding According to the analysis above we can obtainthe bounds of multicast capacity for bus-assisted forwardingas the following theorem

Theorem 15 The upper bound of the multicast capacity forbus-assisted VANET is as follows

119874max [119882119886

119896min(

119882119887radic119888

119899radic119896119882119886119888

119896119899)] if 119896 = 119874 (119888)

119874 max [119882119886

119896min(

119882119887

119899119882119886119888

119896119899)] if 119896 = Ω (119888)

(16)

and 119908ℎ119901 scales of at least Ω((119896 minus 1) log(119899))

6 Capacity of Highway Routing

The forwarding protocol can transmit packets with tinydelivery cost However the delay cannot be diminishedby the protocol because it mainly depends on the velocityof the bus For the urgent packets the forward scenariocannot match the requirement Thus the capacity of forwardscenario cannot satisfy the transmission of urgent packetsFor the purpose of calculating the capacity of urgent packetstransmissions in the bus-assistedVANETwe designed a basicrouting protocol for the bus-assisted VANET We call it thehighway routing Then we calculate the capacity with thehighway routing

61 Introduction of PercolationTheory Thehighway system isconstructed based on the percolationmodel on square latticeBefore we construct the highway we first introduce the bondpercolation of percolation theory

Assume that some packets are generated on top of thenetwork region Will the packets be able to make their wayfrom edge to edge and reach the bottom This question ismodeledmathematically as a two-dimensional network of 119899times119899 vertices inwhich the edge between each two neighborsmaybe open (allowing the liquid through) with probability 119901 orclosed with probability 1minus119901 For the purpose of transmittingpackets all over the network we have to derive the probabilitythat an open path exists from the top to the bottom For theregular square lattice if there is an open path from the topto the bottom the open path from left to right of the networkalso exists

Figure 2 Construction of highway path

The open probability 119901 of the edge is independent Theprobability of an open path existing is determined by 119901 Thesingle open path is not enough to ensure the all over networktransmissions As the number of vehicles increases to infinitythe network construction increases to infinity Thus theremust be infinite open path clusters to ensure all over networktransmissions

By Kolmogorovrsquos zero-one law in the regular squarelattice for any given 119901 the probability that an infinite openpath cluster exists is either zero or one The probabilityof 119901 is an increasing function that was proved in [19] Itincreases sharply from approach zero to one in a short spanof 119875 Therefore there must be a critical 119901 (denoted by 119901

119888)

determining the probability 119875 that an infinite open pathcluster exists in the regular square latticeWhen119901 is below119901

119888

the probability 119875 is zero When 119901 is above 119901119888 the probability

119875 is one and an infinite open path cluster exists in the regularsquare lattice

62 The Construction of Highway As in Figure 2 we assumethat the intersection is the center of a virtual square Theroad between two intersections is one edgeThe virtual squarelattice is denoted by red dotted lines The black lines denotethe open paths To calculate the 119901

119888in the square lattice of

Figure 2 we introduce the definition of the coordinationnumber which is denoted by 119911 in this paper It means the totalnumber of neighbors of center intersection Obviously in theregular square lattice in Figure 1 the coordination numberis four According to the result of Harry Kesten [20] theprobability threshold 119901

119888is 1(119911minus1) Therefore the open edge

cluster exits only if the probability that the edge between twosquares is open is larger than 13

Packets are forwarded by intermediate vehicles In thenetwork each edge between two squares is open if the roadbetween the two squares has at least one vehicle Thus theprobability of each road having at least one vehicle mustbe larger than 13 to make sure the open edge clusterexits in the network For the regular road segment without


Road a

S D


buses in any time slot the probability of not finding anyvehicles is prod120596

120572=1(1 minus 120587

120572)119873119894

120572 where 119873119894

120572 denotes the roadsegment number of layer 120572 Thus we have the probabilitythat road segment 119894 has at least one vehicle It is also the openprobability 119901

119894of edge 119894 as follows

119901119894= 1 minus

120596

prod

120572=1

(1 minus 120587120572)119873120572

119894 (17)

For any road segment 119901119894ge 1205872

1 Obviously 1205872

1is less

than 13 Thus only ordinary vehicle cannot ensure theopening edge cluster in the network The highway cannotbe constructed by ordinary vehicles However the regularbuses can significantly increase the probability that at anytime slot the road segment has at least one vehicle Accordingto the real mobility trace collected by [21] we can know theprobability 119901

119887that in any time slot the road segment has a

bus larger than 15 Adding 119901119886and 119901

119887 we can derive the

probability that the edge between two squares has at least onevehicle is bigger than 13 Therefore with the bus-assistedroads we can construct the highway path to transmit theurgent packets in the VANET

63 Highway Routing Protocol Based on the open edgecluster derivation above we use basic routing protocol totransmit packets in VANETs Following the protocol allsource vehicles upload packets to the highway path andthen packets are transmitted through the highway path untilthey approach the destinations Destination vehicles willdownload packets from highway path Time slots will be wellarranged to ensure the highway path has priority to occupytransmit opportunity

Upload Source vehicles upload the packets to highway pathwhen there are some transmission opportunities Otherwisethe source vehicle can add the packets to highway transmis-sion flow when it is chosen as the intermediate vehicle of thehighway path

Routing Packets are forwarded along the shortest highwaypath to the destination or the intermediate vehicle besides the

destination If the shortest bus path has a closed edge thenhighway path will detour to avoid the closed path

Download Packets are forwarded to the destination vehiclethrough the highway pathThedestination vehicle downloadsthe packet from the highway path when it has opportunity toaccess the link Otherwise the destination vehicle can get thepackets from highway transmission flow when it is chosen asthe intermediate vehicle of the highway path

Figure 3 is used to show a simple routing process Sourcevehicle 119878 can be assumed as the highway path and transmitpackets to destination vehicle119863 When road 119886 has no vehiclethe highway path detours to avoid road 119886 Then packets willachieve the destination vehicle or the neighbor of destination119863 Destination vehicle will get the packets or download themfrom its neighbor which is in the highway path

64 Calculation of Capacity When vehicle upload or down-load packet from the highway path one packet occupiesone transmission opportunity That process is equivalent totransmit one packet through one edge Thus the upload anddownload process can be assumed as the first edge and thelast edge of the highway pathThe transmission opportunitiesare arranged by protocol and thus an interference groupcan fully use the transmission opportunities Each packetwill cost one transmit opportunity to pass through one edgeThe network has total 119904119901

119886transmission opportunities When

packets can be transmitted to each destination they onlypass through one edge The capacity of each vehicle canachieve at most (119904119901

119886119896119899)119882

119886 The upper capacity bound of

VANET with highway is identical with the forward scenarioHowever the lower capacity bound of VANET with highwayis different from forward scenario To calculate the uppercapacity bound we introduce a lemma proved by Neely andModiano in [22]

Lemma 16 Given a square 119887 in the lattice the probability thata random highway path will be routed via the square 119904 is atmost 119888radic119896 sdot (119903119886) where 119888 is a constant number

Therefore a square 119887 can be used by at most 119899119888radic119896 sdot (119903119886)packets We consider the protocol can give the opportunityto the busier square Thus the busiest square will have moretransmission opportunities than other squares in the sameinterference group Thus the opportunities of the busiestsquare 119887 must be more than the average Thus we can easilyderive the lower capacity bound in each vehicle as follows

120582 (119899) ge119901119886119882119886

1198992119888radic119896 sdot (119903119886)=

119882119886

21198881198992radic119896 lceil1 + Δrceil lceil2 + Δrceil (18)

The above analysis is summarized by the following theorem

Theorem 17 With the application of highway protocol theper-vehicle capacity of VANET can be achieved at mostO(1119896119889) and cannot be lower than Ω(1radic119896119889)


7 Discussion

Notice that 119888 = Θ(1198982) and we did not use Θ(1198982) in the

capacity results of bus routing method The reason is thatwe use 119898 to denote both road and bus lines in the grid-likeconstruction in this paper and the roads number is equal tothe number of bus lines However the cell used in bus routingis constructed by bus lines not roads If we use a differentnumber of bus lines the cell numberwill be different from thenumber of squares in the grid-like constructionThus for thecapacity of scenarios with different bus lines we only need toreplace the value of 119888

To calculate the bounds of multicast for the bus-assistedVANET we assume that the TTL (time to live) of packetsis infinite However in the realistic VANET TTL is oneof the most important characteristics of the packets in anykind of ad hoc network Therefore if we can tolerate thedelay of transmission the study of achievable capacity isalso essential for the bus-assisted VANET We will focuson the tradeoff between capacity and delay in the futurework Similarly applying more real interference model is alsoessential such as physical interference model and Gaussianinterference model All of our results are derived under grid-like construction A more realistic framework may close thegap of capacity between theoretic results and real value Wewill consider all the remaining challenges in the future work

We calculate the achievement per-vehicle capacity forthe forwarding scenario and routing scenario of bus-assistedVANETsThe forwarding scenario can save lots of energy andtransmission coasts to diminish the overhead of bus-assistedVANETs The routing scenario can transmit packets withinvery little time to satisfy the urgent packets by sacrificing theoverhead of the network We just derive the performanceThe selection of transmission scenario needs an additionalprotocol

8 Conclusion

The capacity scaling law of high mobility M2M networks hasbeen considered as one of the most fundamental issues Inthis paper we derive the upper and lower bounds ofmulticastcapacity for high mobility social proximity M2M networksvia bus-assisted forwarding method In the routing scenariowe use buses and ordinary cars to construct the highwaysystem for VANETs which is a typical case of high mobilityM2M networks We use percolation theory to prove thatthere is a highway path cluster cross through the networkvertically and horizontally Therefore the highway systemcan ensure the packets can be transmitted to destinationlocated anywhere in VANETs The per-vehicle capacity ofrouting scenario is also derived At last we discussed howthe different forwarding processes influence the results ofcapacity scaling law for high mobility M2M networks Ourwork provides new insights for the design of bus-assistedVANETs as intermediate vehicle to relay packets

Acknowledgment

This work is supported by Heilongjiang Province EducationDepartment Foundation 12531Z007

References

[1] Y Zhang R Yu S Xie W Yao Y Xiao andM Guizani ldquoHomeM2M networks architectures standards and QoS improve-mentrdquo IEEE Communications Magazine vol 49 no 4 pp 44ndash52 2011

[2] Y Zhang R Yu M Nekovee Y Liu S Xie and S GjessingldquoCognitive machine-to-machine communications visions andpotentials for the smart gridrdquo IEEE Network Magazine vol 26no 3 pp 6ndash13 2012

[3] H Hartenstein and K P Laberteaux ldquoA tutorial survey on veh-icular ad hoc networksrdquo IEEE Communications Magazine vol46 no 6 pp 164ndash171 2008

[4] I F Akyildiz D Pompili and TMelodia ldquoUnderwater acousticsensor networks research challengesrdquo Ad Hoc Networks vol 3no 3 pp 257ndash279 2005

[5] M AHansonH C Powell Jr A T Barth et al ldquoBody area sen-sor networks challenges and opportunitiesrdquo Computer vol 42no 1 zpp 58ndash65 2009

[6] G Nan Z Mao M Li et al ldquoDistributed resource allocation incloud-based wireless multimedia social networksrdquo IEEE Net-work Magazine In press

[7] G Nan Z Mao M Yu M Li H Wang and Y Zhang ldquoStacke-lberg game for bandwidth allocation in cloud-based wirelesslive-streaming social networksrdquo IEEE Systems Journal no 992013

[8] E Palomar A Alcaide E Molina and Y Zhang ldquoCoalitionalgames for the management of anonymous access in online soc-ial networksrdquo in Proceedings of the 11th International Conferenceon Privacy Security and Trust (PST rsquo13) pp 1ndash10 TarragonaSpain July 2013

[9] P Gupta and P R Kumar ldquoThe capacity of wireless networksrdquoIEEETransactions on InformationTheory vol 46 no 2 pp 388ndash404 2000

[10] M Grossglauser and D N C Tse ldquoMobility increases the cap-acity of ad hoc wireless networksrdquo IEEEACM Transactions onNetworking vol 10 no 4 pp 477ndash486 2002

[11] X-Y Li ldquoMulticast capacity of wireless ad hoc networksrdquo IEEEACM Transactions on Networking vol 17 no 3 pp 950ndash9612009

[12] M Garetto and E Leonardi ldquoRestricted mobility improvesdelay-throughput tradeoffs in mobile ad hoc networksrdquo IEEETransactions on Information Theory vol 56 no 10 pp 5016ndash5029 2010

[13] X Mao X-Y Li and S Tang ldquoMulticast capacity for hybridwireless networksrdquo in Proceedings of the 9th ACM InternationalSymposium on Mobile Ad Hoc Networking and Computing(MobiHoc rsquo08) pp 189ndash198 Hong Kong May 2008

[14] R Zheng ldquoAsymptotic bounds of information dissemination inpower-constrained wireless networksrdquo IEEE Transactions onWireless Communications vol 7 no 1 pp 251ndash259 2008

[15] O Goussevskaia R Wattenhofer M M HalldorssoN and EWelzl ldquoCapacity of arbitrary wireless networksrdquo in Proceedingsof the 28th Annual IEEE International Conference on Com-puter Communications (INFOCOM rsquo09) pp 1872ndash1880 Rio deJaneiro Brazil April 2009

[16] G Alfano M Garetto and E Leonardi ldquoCapacity scaling ofwireless networks with inhomogeneous node density lower bo-undsrdquo in Proceedings of the 28th Conference on Computer Com-munications (INFOCOM rsquo09) pp 1890ndash1898 Rio de JaneiroBrazil April 2009


[17] C Jiang Y Shi Y T Hou and S Kompella ldquoOn the asymptoticcapacity of multi-hop MIMO ad hoc networksrdquo IEEE Transac-tions on Wireless Communications vol 10 no 4 pp 1032ndash10372011

[18] Z Li and L C Lau ldquoA constant bound on throughput improve-ment of multicast network coding in undirected networksrdquoIEEE Transactions on Information Theory vol 55 no 3 pp1016ndash1026 2009

[19] H Pishro-Nik A Ganz andDNi ldquoThe capacity of vehicular adhoc networksrdquo in Proceedings of the Annual Allerton Conferenceseptember 2007

[20] N Lu T H Luan M Wang X Shen and F Bai ldquoCapacity anddelay analysis for social-proximity urban vehicular networksrdquoin Proceedings of the 31st Annual IEEE International Conferenceon Computer Communications (INFOCOM rsquo12) pp 1476ndash1484Orlando Fla USA 25-30 March 2012

[21] W K Lai K-T Yang andM-C Li ldquoBus assisted connectionlessrouting protocol for metropolitan VANETrdquo in Proceedings ofthe 5th International Conference on Genetic and EvolutionaryComputing (ICGEC rsquo11) pp 57ndash60 Xiamen Cihna September2011

[22] M J Neely and EModiano ldquoCapacity and delay tradeoffs for adhoc mobile networksrdquo IEEE Transactions on Information The-ory vol 51 no 6 pp 1917ndash1937 2005

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



Tier (1)

Tier (2)

Tier (3)

Road segment

Road line


middot middot middot middot middot middot


Figure 1 The real map of urban area

the real urban map However we realized we also have to usestatistic methods to calculate the capacity of high mobilitysocial M2M networks If the statistic methods were appliedin the calculation the detail of the real construction willbe neglected in the calculation Thus there is no differencebetween real construction and grid-like construction Fur-thermore grid-like construction is easy to understand andconvenient to be used in calculation As a result we choosethe grid-like construction to be the geometrical constructionin our calculation Lu et al extended their work and obtainedthe per-vehicle throughputΩ(1 log(119899)) and delay O(log2(119899))[20] However Lu et al only consider the scenario thatsource vehicles can only transmit packets to the vehicle whichbelongs to the same community as the source vehicle and theyonly consider the unicast transmissionThere are rare studiesthat work on the calculation of capacity for different VANETsscenarios The capacity of different typical ad hoc scenarioscannot be adopted for theoretical analysis for VANETs andcannot provide guidelines in designing VANETs Thereforewe calculate the multicast capacity for bus-assisted VANETsin this paper

Assume that 119899 ordinary vehicles and 119899119887

buses aredeployed in a grid-like road framework This construction iscomposed of119898 parallel lines with other intersected119898 parallellines Since the larger urban region means more numberof vehicles 119898 increases linearly with 119899 Vehicles that havesocial proximity mobility always move in a localized areacentered at the driverrsquos home or work space and seldommove out Thus we let all the ordinary vehicles obey therestricted mobility model As a result the spatial stationarydistribution of vehicles decays as the form of power law withthe distance between the home point and each vehicle Eachordinary vehicle randomly chooses 1198961 vehicles from the otherordinary vehicles as receiversThe network can be consideredas unicast when 119896 is 2 and as broadcast when 119896 is 1198991

In the forwarding scenario all the buses deployed in allroads as intermediate nodes are used to forward packetsfor ordinary vehicles The buses of each road were consid-ered as one bus trajectory The bus forward scheme wasproposed in [21] Each transmission flow transmits packetsvia two-hop relay scheme proposed in [22] All the forwardprocesses of buses are considered as one-hop The packetscould be transmitted directly from source to destination orbe transmitted to an intermediate vehicle or bus then beforwarded to the destination In the routing scenario we usebuses and ordinary cars to construct the highway systemWe use percolation theory to prove that there are highwaypath clusters that cross through the network vertically andhorizontally Thus based on a simple routing protocol thehighway system can ensure the packets can be transmitted todestination located in any place of VANET Simultaneouslythe urgent TTL requirement can be satisfied

To schedule the interference in MAC layer of wirelesstransmission we use the protocol model [6] as our inter-ference model in this paper We assume the bandwidthof wireless transmission is 119882

119886bits per time slot and the

bandwidth of bus system is 119882119887bits per time slot For

simplicity we also assume that there is only one wirelesschannel in the bus-assisted forwarding and routing and allthe vehicles and buses have enough memory to buffer all thepacketsOur Main Contributions With the purpose of transmittingpackets to different communities of VANETs and furtherimproving the network capacity in this paper we derive theupper and lower bound of the multicast capacity for bus-assisted VANET in two kinds of scenarios

In the forwarding scenario the per vehicle capacity ofbus-assisted VANET is as follows

119874max [119882119886

119896min(

119882119887radic119888

119899radic119896119882119886119888

119896119899)] if 119896 = 119874 (119888)

119874 max [119882119886

119896min(

119882119887

119899119882119886119888

119896119899)] if 119896 = Ω (119888)

(1)

and 119908ℎ119901 scales of at leastΩ((119896 minus 1) log(119899))In the routing scenario we construct the highway system

to transmit the urgent packets We also derive the capacityof per vehicle via busminusassisted mode it can achieve at most119874(1119896119889) and cannot be lower thanΩ(1radic119896119889)

The rest of the paper is organized as follows In Section 2we represent the network model in detail The upper andlower bounds of multicast capacity by ad hoc forwardingare derived in Section 3 In Section 4 we calculate the upperand lower bounds of multicast capacity by bus forwardingWe combine the results of Sections 3 and 4 and obtain themulticast capacity for bus-assisted forwarding in Section 5Section 6 constructs the highway path and derives the capac-ity of bus-assisted VANET in routing scenario Section 7discusses the results and the remaining challenges Section 8concludes this paper and reviews the results on multicastcapacity for bus-assisted VANETs


2 Network Model








120596

120572=1(16120572 minus 12)120572








10038171003817100381710038171003817119883119894(119905) minus 119883

119895(119905)10038171003817100381710038171003817le 119903 (2)


119895(119905)10038171003817100381710038171003817le (1 + Δ) 119903 (3)










lim119899rarrinfin

Pr (120582 (119899) = 119888 (119892 (119899)) is feasible) = 1

lim119899rarrinfin




lim119879rarrinfin

Pr(119866 (119879)119879

ge 120582) = 1 (5)






119901119886=

1









119866119889(119879) + 119866

119903(119879)

119879ge 119896119899120582 (119899) minus 120576 (7)



1

119879119870 (119879)119882

119886ge1

119879119866119889(119879) +

2

119879119866119903(119879) (8)


1

119879119870 (119879)119882

119886ge1

119879119866119889(119879) + 2 (119896119899120582 (119899) minus 120576 minus

1

119879119866119889(119879)) (9)


120582 (119899) le(1119879)119870 (119879)119882

119886+ (1119879)119866

119889(119879) + 2120576

2119896119899 (10)


120582 (119899) le(1119879)119870 (119879)119882

119886+ (1119879)119866

119889(119879)

2119896119899 (11)


lim119909rarr119879

1

119879119870 (119879)119882

119886le 119904119901119886119882119886 (12)


lim119909rarr119879

1

119879119866119889(119879) le 119904119901

119886119882119886 (13)


120582 (119899V) le119904119901119886

119896119899=119901119886

119896119889=

1











119894) denote the



119899






minus1198998 then 119871 ge 1198991198984






232 andwhen 119896 = Ω(119888) there





119887radic119888


119887

119899 119908ℎ119901


119886 The






119886119901119886119896119889



119874[min(119882119887radic119888

119899radic119896119882119886119888

119896119899)] if 119896 = 119874 (119888)

119874 [min(119882119887

119899119882119886119888

119896119899)] if 119896 = Ω (119888)

(15)



119887 Thus 119873

119887scales as

119874(log(119899)) 119908ℎ119901









119874max [119882119886

119896min(

119882119887radic119888

119899radic119896119882119886119888

119896119899)] if 119896 = 119874 (119888)

119874 max [119882119886

119896min(

119882119887

119899119882119886119888

119896119899)] if 119896 = Ω (119888)

(16)









119888)


119888










Road a

S D



120572=1(1 minus 120587

120572)119873119894

120572 where 119873119894



119901119894= 1 minus

120596

prod

120572=1

(1 minus 120587120572)119873120572

119894 (17)


1 Obviously 1205872

1is less















119886119896119899)119882





120582 (119899) ge119901119886119882119886

1198992119888radic119896 sdot (119903119886)=

119882119886





7 Discussion





8 Conclusion


Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2 Network Model








120596

120572=1(16120572 minus 12)120572








10038171003817100381710038171003817119883119894(119905) minus 119883

119895(119905)10038171003817100381710038171003817le 119903 (2)


119895(119905)10038171003817100381710038171003817le (1 + Δ) 119903 (3)










lim119899rarrinfin

Pr (120582 (119899) = 119888 (119892 (119899)) is feasible) = 1

lim119899rarrinfin




lim119879rarrinfin

Pr(119866 (119879)119879

ge 120582) = 1 (5)






119901119886=

1









119866119889(119879) + 119866

119903(119879)

119879ge 119896119899120582 (119899) minus 120576 (7)



1

119879119870 (119879)119882

119886ge1

119879119866119889(119879) +

2

119879119866119903(119879) (8)


1

119879119870 (119879)119882

119886ge1

119879119866119889(119879) + 2 (119896119899120582 (119899) minus 120576 minus

1

119879119866119889(119879)) (9)


120582 (119899) le(1119879)119870 (119879)119882

119886+ (1119879)119866

119889(119879) + 2120576

2119896119899 (10)


120582 (119899) le(1119879)119870 (119879)119882

119886+ (1119879)119866

119889(119879)

2119896119899 (11)


lim119909rarr119879

1

119879119870 (119879)119882

119886le 119904119901119886119882119886 (12)


lim119909rarr119879

1

119879119866119889(119879) le 119904119901

119886119882119886 (13)


120582 (119899V) le119904119901119886

119896119899=119901119886

119896119889=

1











119894) denote the



119899






minus1198998 then 119871 ge 1198991198984






232 andwhen 119896 = Ω(119888) there





119887radic119888


119887

119899 119908ℎ119901


119886 The






119886119901119886119896119889



119874[min(119882119887radic119888

119899radic119896119882119886119888

119896119899)] if 119896 = 119874 (119888)

119874 [min(119882119887

119899119882119886119888

119896119899)] if 119896 = Ω (119888)

(15)



119887 Thus 119873

119887scales as

119874(log(119899)) 119908ℎ119901









119874max [119882119886

119896min(

119882119887radic119888

119899radic119896119882119886119888

119896119899)] if 119896 = 119874 (119888)

119874 max [119882119886

119896min(

119882119887

119899119882119886119888

119896119899)] if 119896 = Ω (119888)

(16)









119888)


119888










Road a

S D



120572=1(1 minus 120587

120572)119873119894

120572 where 119873119894



119901119894= 1 minus

120596

prod

120572=1

(1 minus 120587120572)119873120572

119894 (17)


1 Obviously 1205872

1is less















119886119896119899)119882





120582 (119899) ge119901119886119882119886

1198992119888radic119896 sdot (119903119886)=

119882119886





7 Discussion





8 Conclusion


Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











lim119879rarrinfin

Pr(119866 (119879)119879

ge 120582) = 1 (5)






119901119886=

1









119866119889(119879) + 119866

119903(119879)

119879ge 119896119899120582 (119899) minus 120576 (7)



1

119879119870 (119879)119882

119886ge1

119879119866119889(119879) +

2

119879119866119903(119879) (8)


1

119879119870 (119879)119882

119886ge1

119879119866119889(119879) + 2 (119896119899120582 (119899) minus 120576 minus

1

119879119866119889(119879)) (9)


120582 (119899) le(1119879)119870 (119879)119882

119886+ (1119879)119866

119889(119879) + 2120576

2119896119899 (10)


120582 (119899) le(1119879)119870 (119879)119882

119886+ (1119879)119866

119889(119879)

2119896119899 (11)


lim119909rarr119879

1

119879119870 (119879)119882

119886le 119904119901119886119882119886 (12)


lim119909rarr119879

1

119879119866119889(119879) le 119904119901

119886119882119886 (13)


120582 (119899V) le119904119901119886

119896119899=119901119886

119896119889=

1











119894) denote the



119899






minus1198998 then 119871 ge 1198991198984






232 andwhen 119896 = Ω(119888) there





119887radic119888


119887

119899 119908ℎ119901


119886 The






119886119901119886119896119889



119874[min(119882119887radic119888

119899radic119896119882119886119888

119896119899)] if 119896 = 119874 (119888)

119874 [min(119882119887

119899119882119886119888

119896119899)] if 119896 = Ω (119888)

(15)



119887 Thus 119873

119887scales as

119874(log(119899)) 119908ℎ119901









119874max [119882119886

119896min(

119882119887radic119888

119899radic119896119882119886119888

119896119899)] if 119896 = 119874 (119888)

119874 max [119882119886

119896min(

119882119887

119899119882119886119888

119896119899)] if 119896 = Ω (119888)

(16)









119888)


119888










Road a

S D



120572=1(1 minus 120587

120572)119873119894

120572 where 119873119894



119901119894= 1 minus

120596

prod

120572=1

(1 minus 120587120572)119873120572

119894 (17)


1 Obviously 1205872

1is less















119886119896119899)119882





120582 (119899) ge119901119886119882119886

1198992119888radic119896 sdot (119903119886)=

119882119886





7 Discussion





8 Conclusion


Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119894) denote the



119899






minus1198998 then 119871 ge 1198991198984






232 andwhen 119896 = Ω(119888) there





119887radic119888


119887

119899 119908ℎ119901


119886 The






119886119901119886119896119889



119874[min(119882119887radic119888

119899radic119896119882119886119888

119896119899)] if 119896 = 119874 (119888)

119874 [min(119882119887

119899119882119886119888

119896119899)] if 119896 = Ω (119888)

(15)



119887 Thus 119873

119887scales as

119874(log(119899)) 119908ℎ119901









119874max [119882119886

119896min(

119882119887radic119888

119899radic119896119882119886119888

119896119899)] if 119896 = 119874 (119888)

119874 max [119882119886

119896min(

119882119887

119899119882119886119888

119896119899)] if 119896 = Ω (119888)

(16)









119888)


119888










Road a

S D



120572=1(1 minus 120587

120572)119873119894

120572 where 119873119894



119901119894= 1 minus

120596

prod

120572=1

(1 minus 120587120572)119873120572

119894 (17)


1 Obviously 1205872

1is less















119886119896119899)119882





120582 (119899) ge119901119886119882119886

1198992119888radic119896 sdot (119903119886)=

119882119886





7 Discussion





8 Conclusion


Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119874max [119882119886

119896min(

119882119887radic119888

119899radic119896119882119886119888

119896119899)] if 119896 = 119874 (119888)

119874 max [119882119886

119896min(

119882119887

119899119882119886119888

119896119899)] if 119896 = Ω (119888)

(16)









119888)


119888










Road a

S D



120572=1(1 minus 120587

120572)119873119894

120572 where 119873119894



119901119894= 1 minus

120596

prod

120572=1

(1 minus 120587120572)119873120572

119894 (17)


1 Obviously 1205872

1is less















119886119896119899)119882





120582 (119899) ge119901119886119882119886

1198992119888radic119896 sdot (119903119886)=

119882119886





7 Discussion





8 Conclusion


Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Road a

S D



120572=1(1 minus 120587

120572)119873119894

120572 where 119873119894



119901119894= 1 minus

120596

prod

120572=1

(1 minus 120587120572)119873120572

119894 (17)


1 Obviously 1205872

1is less















119886119896119899)119882





120582 (119899) ge119901119886119882119886

1198992119888radic119896 sdot (119903119886)=

119882119886





7 Discussion





8 Conclusion


Acknowledgment


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










7 Discussion





8 Conclusion


Acknowledgment


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









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