1

Mobility Weakens the Distinction between Multicastand Unicast

Yi Qin, Xiaohua Tian, Weijie Wu, Xinbing WangDept. of Electronic Engineering Shanghai Jiao Tong University, China

Email: qinyi 33, xtian, wwj, xwang8@sjtu.edu.cn

Abstract—Comparing with the unicast technology, multipleflows from the same source in multicast scenario can be aggregat-ed even if their destinations are different. This paper evaluatessuch distinction by the multicast gain on per-node capacityand delay, which are defined as the per-node capacity anddelay ratios between multi-unicast and multicast (m destinationsfor each multicast session). Particularly, the restricted mobilitymodel [1] is proposed, which is a representative mobility modelcharacterizing a class of mobility models with different averagemoving speeds. The theoretical analysis of this model indicatesthat the mobility significantly decreases the multicast gain onper-node capacity and delay, though the per-node capacity ofboth unicast and multicast can be enhanced by mobility. Thisfinding suggests that mobility weakens the distinction betweenmulticast and unicast. Finally, a general framework of multicaststudy is constituted by analyzing the upper-bound (Θ(m)), thelower-bound (Θ(1)) and the main determinants of the multicastgain on both per-node capacity and delay regardless of mobilitymodel.

Index Terms—Restricted mobility, Multicast, Unicast, Capac-ity, Delay

I. INTRODUCTION

Network scaling laws for large-scale wireless ad hoc net-works have been extensively studied since the seminal workby P. Gupta and P. R. Kumar [2]. They focus on the unicastscenario of random wireless networks with n static nodes.This work shows that the per-node throughput upper-boundΘ(

1√n

)1 can be achieved by their scheme. However, there are

a large number of wireless mobile devices in the real world.Thus, many researchers focus on the network scaling lawsof mobile networks [3]-[10]. With the help of mobility, longdistance transmission can be realized, which is not allowed instatic networks because of the limitation of transmission powerand interference. Hence, in contrast with static networks,node mobility can improve the per-node throughput and delayperformance. In [3], D. Shah, et al. show that Θ(1) per-nodethroughput is obtained in random i.i.d. mobility model forunicast scenario.

Nevertheless, [2] and [3] only analyze two extreme casesof mobility, i.e. lowest node speed (static model) and highest

1We use standard asymptotic notations in our paper. Consider t-wo nonnegative function f(·) and g(·): (1) f(n) = o(g(n))means limn→∞ f(n)/g(n) = 0. (2) f(n) = O(g(n)) mean-s limn→∞ f(n)/g(n) < ∞. (3) f(n) = ω(g(n)) mean-s limn→∞ f(n)/g(n) = ∞. (4) f(n) = Ω(g(n)) meanslimn→∞ f(n)/g(n) > 0. (5) f(n) = Θ(g(n)) means f(n) = O(g(n))and g(n) = O(f(n)). (6) f(n) = Θ(g(n)) means f(n) = Θ(g(n)) whenignoring poly-logarithmic factor.

node speed (random i.i.d. mobility model). There are alsosome studies focusing on other mobility models with differentnode speeds, e.g., random walk models [4], restricted mobilitymodels [1][5][6], heterogeneous mobility models [7]. In [5],the network is divided into squares, where the nodes obey i.i.d.mobility model in each square and follow random walk modelwhen moving among different squares. Moreover, [6] studiesthe case that each node’s home-point can only be located inone of m clusters. This model indicates the fact that people aremore likely to be within the region where they live. However,the event of long distance movement happens with low proba-bility in our real world, which is demonstrated in the mobilitymodel in [1]. In this model, each node has its own home-point,and the distance between it and its home-point follows power-law distribution. Although this model cannot well describethe continuity of node’s mobility, it is suitable for the fastmobile networks. M. Garetto, et al. in [1] adopt a multi-hopscheme for the unicast scenario, which can enhance the per-node throughput and delay by optimizing the configuration oftheir scheme. For example, when the exponent of the power-law distribution parameter α equals to 2, constant per-nodethroughput and delay can be obtained simultaneously exceptfor poly-logarithmic factors. Furthermore, some researchersstudy the heterogeneous mobility model which is feasible forthe nodes with different mobility models [7].

The studies above drill down into the impact of mobility onunicast per-node capacity, and their results indicate that themobility enhances the unicast per-node capacity performance.However, unicast is only a special case of multicast standingfor a more general traffic pattern. Therefore, the impact ofmobility on multicast becomes a hot topic [8]-[10]. In [8], J.J.Garcia-Luna-Aceves et al. focus on the multicast scenario forthe static networks. In their study, there are n static nodes inthe network, and each node has m(m < n) destinations. Theresults demonstrate that the achievable per-node throughputupper-bound is Θ

(1√

mn logn

)when m = o

(n

logn

), and

the upper-bound is Θ(

1n

)for the case m = Ω

(n

logn

). By

introducing the mobility into multicast networks, [9] analyzesthe multicast per-node capacity under random i.i.d. mobilitymodel, which proves that mobility can also increase the mul-ticast per-node capacity. Another mobility model with limitednode speed is given in [10]. In this paper, the node speedis limited by R, and the per-node capacity of the network isa non-decreasing function of R. The theoretical results alsodemonstrate that mobility can enhance the multicast per-node

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capacity.In order to further understand the above phenomenon,

the distinction between the performance enhancements bymobility is analyzed for unicast and multicast. In particular,this paper focuses on a class of mobility models proposedby M. Garetto, et al. [1], i.e. restricted mobility model,which includes mobility models such as static model, randomi.i.d. mobility model, etc. Furthermore, the node speed is adecreasing function of α, which is the exponent of the power-law distribution in this restricted mobility model. Therefore,the impact of mobility on the network per-node capacitycan be investigated by adjusting α. In this model, the per-node capacity and delay are analyzed for both unicast andmulticast, respectively. The results demonstrate that mobilitycan increase the per-node capacity for both of them. However,it weakens the distinction between multicast and unicast sincethe opportunity of flow aggregation is reduced by mobility.Moreover, the delay of unicast and multicast are the same inorder sense in the restricted mobility model, and the mobilityonly reduces the multicast delay gain in constant order.

To support the above conclusion, the main contributions ofthis paper are summarized as follows:• Contribution on capacity: This paper focuses on the

multicast capacity gain2 for the restricted mobility mod-el in two traffic patterns, i.e., unicast and multicas-t. The bottleneck of network per-node throughput isconsidered, and the per-node capacity is derived forthe two traffic patterns, which shows the enhancementof per-node capacity by increasing the moving speed.Denoting the number of destinations as m and thetotal number of nodes as n, and the multicast gainon per-node capacity under different circumstances sat-isfies

Θ(1),Θ

(lognlog n

m

),Θ(m

α2−1

),Θ (√m)

for thecases 0 ≤ α < 2, α = 2, 2 < α ≤ 3, α > 3, respective-ly, where α is the exponential coefficient of restrictedmobility model. These results demonstrate that mobilitysignificantly decreases the multicast capacity gain. More-over, the results also indicate that m enhances the dis-tinction as the secondary determinant. Additionally, thispaper further studies the upper-bound and lower-bound ofmulticast capacity gain regardless of the mobility model,which are Θ(m) and Θ(1), respectively. Based on theseresults, the factors determining the multicast capacity gainare summarized, which form a general framework of themulticast capacity.

• Contribution on delay: The impact of mobility onmulticast delay gain is studied by adopting the floodingscheme, which is capable of achieving the delay lower-bound of the two traffic patterns. By considering theinformation expansion speed, it can be derived that themulticast gain of optimal delay with limited transmissionrange is Θ(m), and the mobility strength reduces it inconstant order. In the further study, this paper investigatesthe upper-bound (Θ(m)) and lower-bound (Θ(m)) ofmulticast delay gain in a more general network. Accord-ing to the theoretical results, the multicast delay gain is

2The ratio of the capacity of multicast and multi-unicast. See Definition 3.

mainly determined by the difference of delay for differentnodes.

• Differences from previous work: This work is the firststudy on the optimal capacity and delay performancefor the restricted mobility model in both unicast andmulticast scenarios. Moreover, different from previouswork, this paper mainly focuses on the performance ratiosbetween the two scenarios in order to investigate theimpact of mobility on multicast gain. Additionally, someother essential factors determining the multicast gain arealso analyzed.

The rest of this paper is organized as follows. In SectionII, the network model and definitions are introduced. Theper-node capacity is analyzed for both unicast and multicastscenarios in Section III. In Section IV, the delay performanceis studied for the two scenarios. In Section V, the multicastgain on per-node capacity and delay are investigated. Finally,conclusion is summarized in Section VI.

II. SYSTEM MODEL AND DEFINITIONS

A. Network Model

There are n nodes in the networks. In order to simplify theperformance analysis, the shape of network is assumed to be atorus defined as O with size

√n×√n, and all of the n nodes

(users) are moving on its surface. Hence, the node density is1, and edge effect is ignored in this model.

B. Mobility Model

In our system, time is divided into slots with equal duration.The fast mobility case is considered, in which the nodemobility is of the same time-scale as packet transmission.Under such time division, one node can only transmit onepacket to another node in one hop when they are connected.

The position of node i(i = 1, · · · , n) at time slot t(t =0, 1, · · · ) is denoted as Xi(t). In random i.i.d. mobility model,the Xi(t) is randomly, uniformly and independently selectedin the network in each time slot. Therefore, the moving speedof nodes in the network is Θ(

√n) per-timeslot, which cannot

be adjusted. In this paper, the restricted mobility model in [1]is adopted instead. In this model, for any i and t, the Xi(t)is independently distributed in O. This assumption is widelyintroduced in network performance research as in [11]-[14].However, the mobility of nodes in the network is restricted,i.e., the Xi(t) is not uniformly distributed in the networks,which is different from the random i.i.d. mobility model.

In this model, for each node i, there is a correspondinghome-point located at Hi which is the mobility center of i.Moreover, the home-point is static and uniformly, independent-ly distributed in the network. Although the uniform densitymay appear to be idealized, it can be a good scenario for theinitial study of the impact of mobility due to its mathematicaltractability. In this model, three kinds of distance are defined.The distance between nodes (users) i and j at time slot tis defined as ρi,j(t) = ‖Xi(t) − Xj(t)‖, where ‖ · ‖ is theoperation of Euclidean distance. The distance between home-points Hi and Hj is represented as ρHi,j = ‖Hi − Hj‖.

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Furthermore, the distance between node (user) i and its home-point Hi at time t is ρi(t) = ‖Xi(t)−Hi‖. In this model, thedistribution of ρi(t) is modeled as a non-increasing functionf(ρ) to ensure that the node is more likely to be close to itshome-point. Many papers focus on the distribution functionf(ρ) [1] [6] [15]-[17]. In our paper, according to [1], the f(ρ)is expressed as follow

f(ρ) =s(ρ)∫ ∫O s(ρ)

, (1)

where s(ρ) = min1, ρ−α and α ≥ 0. Consequently, thef(ρ) can be further calculated as

f(ρ) =

Θ(s(ρ)n

α−22

)0 ≤ α < 2,

Θ(s(ρ)logn

)α = 2,

Θ(s(ρ)) α > 2.

(2)

Although this mobility model is not accurate enough com-paring with the practical user movement, it is proved to beappropriate for modeling human and vehicular mobility [1],which is supported by many measurements papers [16] [17].Moreover, it can be found in (2) that the averaged movingdistance in each time slot is determined by α. Therefore, themoving speed of nodes in the network can be adjusted by α,and hence this is an appropriate model for the study of theimpact of mobility on network performance.

C. Communication Model

In this paper, the protocol model is adopted, which is asimplified version of physical model since it ignores the longdistance interference and transmission. Moreover, it is shownin [2] that the physical model can be treated as the protocolmodel on scaling law when the transmission is allowed forthe case that the Signal to Interference Noise Ratio (SINR) isgreater than a given threshold. In this model, a transmissionbetween node i and j is successful if the following inequalityis satisfied

‖Xi −Xj‖ ≤ r(n), (3)

where r(n) is the maximum transmission range of each node.Moreover, any other transmitting node k must satisfy theinequality as

‖Xk −Xj‖ ≥ (1 + ∆)r(n), (4)

where ∆ > 0 is a constant factor that depends on theacceptable SINR of the network. Furthermore, the bandwidthof the network is finite and constant. In this model, thetransmission range r(n) is assumed to be r(n) = Θ(1) [2],and therefore each node can meet another node within itstransmission range with constant probability.

Furthermore, the TDMA scheme is employed to guaran-tee that each transmission is successful. In the K2-TDMAscheme, the network is divided into n

r2(n) cells with equalsize r2(n), and only one of the K2 cells nearby is allowed tobe active, i.e., K2-TDMA scheme allows each K2 adjacentcells to be active with a round-robin fashion. K is a constantand satisfies (K − 1)r(n)/

√2 ≥ (1 + ∆)r(n). Hence, K is

defined as K = d1 + (1 + ∆)√

2e. Without loss of generality,

one of the cells is denoted as cell (1, 1), i.e. C1,1. Then, all thecells in the network can be denoted as Ca,b(1 ≤ a, b ≤

√n),

respectively.

D. Traffic Models

Two traffic patterns are studied in this paper, which areUnicast and Multicast. Firstly, in the unicast scenario, eachnode randomly selects another node as its destination. Thetransmission from one source to its destination (maybe throughmulti-hop) is denoted as one unicast session.

For the multicast scenario, each node i randomly selects mdifferent nodes as its destinations. Node i needs to transmitthe same packet to all of its destinations. The multicast sessionis defined as the transmission from the source to all of itsdestinations (maybe through multi-hop).

E. Network Performance Metrics and Some Notations

Some definitions of the performance metrics are listed asfollows.

Definition 1: (Per-node Throughput) For a given scheme,the per-node throughput is defined as the maximum achievabletransmission rate as in [2]. In t time slots, there are M(i, t)packets transmitted from node i to its destination(s). Firstly,the long term per-node throughput is defined by λi(n) as

λi(n) = lim inft→∞

1

tM(i, t). (5)

Afterwards, the per-node throughput of this model for a givenscheme is defined by the maximum T (n) that satisfies

limn→∞

P(λi(n) ≥ T (n) for all i) = 1. (6)

This paper studies the random networks instead of arbitraryones. The transport throughput, which is defined as the ratetiming the distance, is not appropriate in our networks sinceit is just defined for arbitrary networks [2].

Definition 2: (Per-node Capacity) For a given network, theper-node capacity of it is defined as

C(n) = maxσ∈Σ

Tσ(n), (7)

where σ is a scheme for the network, Σ is the set of all possibleschemes, and Tσ(n) is the per-node throughput of scheme σ.

Definition 3: (Multicast Capacity Gain) For a given net-work, the per-node capacity of multicast is assumed to beCmulti(n). Moreover, if each node has m destinations, eachmulticast session can be treated as m unicast sessions (multi-unicast), and the corresponding sum per-node capacity isdenoted as Cm uni(n). Comparing the capacity of multicastand multi-unicast, the multicast capacity gain is defined as

ξ(n) =Cmulti(n)

Cm uni(n). (8)

The multicast capacity gain ξ(n) indicates the enhancementof per-node capacity by multicast transmission.

Definition 4: (Network Delay) For a given scheme, assumingthat the source sends the packet to the network at time slotts and the destination receives the packet at time slot td, thedelay is defined as the average value of td−ts, i.e., Etd−ts.

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TABLE INOTATIONS AND DEFINITIONS

Notation Definitionn The total number of nodes in the network.m The number of destinations for each source

in multicast scenario.ξ(n) The multicast capacity gain.ζ(n) The multicast delay gain.α The parameter of restricted mobility model.

C(n) The per-node capacity performance.D(n) The delay performance.DSDT The delay for the case that data is only

delivered by short distance transmission.DLDT The delay for the case that data is only

delivered by long distance transmission.

It should be noted that the queuing delay at source is notconsidered here, which is the same as in many important work[1] [4]. Moreover, for wireless networks, the operation timespent in coding/decoding is assumed to be negligible comparedto the transmission time.

Definition 5: (Multicast Delay Gain) For a given network,the network delay of multicast is assumed to be Dmulti(n).Moreover, if each node has m destinations, the sum delayof the transmissions from the source to them by unicast isdenoted as Dm uni(n). Comparing the delay of multicast andmulti-unicast, the multicast delay gain is defined as

ζ(n) =Dm uni(n)

Dmulti(n). (9)

The multicast delay gain ζ(n) indicates the enhancement ofdelay performance by multicast transmission.

Finally, some essential notations and definitions are listedin Table I3.

III. CAPACITY ANALYSIS FOR UNICAST AND MULTICAST

In this section, the capacity performance is analyzed forboth unicast and multicast scenarios. Moreover, the multicastcapacity gain based on the results of this section will bediscussed in Section V.

A. Per-node Capacity of Unicast Scenario

Different from the random i.i.d. mobility model, the prob-ability that two nodes meet each other is related with thedistance between them. Considering two arbitrary nodes i andk, the probability that i meets k can be expressed as

pi,k

=∑Ca,b

∫Xi∈Ca,b

f(‖Xi −Hi‖)∫Xk∈Ca,b

f(‖Xk −Hk‖)dXkdXi.

(10)

Without loss of generality, we assume that node i’s home-point is in C1,1 and node k’s home-point is in Cak,bk . Since

3The detailed discussion of DSDT and DLDT can be found in SectionIV.

this paper focuses on the scaling law of the performance, theorder of pi,k satisfies

pi,k = Θ

S20

∑Xi,Xk∈Ca,b,1≤a,b≤

√n

f(‖Xi −Hi‖)f(‖Xk −Hk‖)

,

(11)

where S0 is the size of one cell and S0 = Θ (1). The pi,kcan be calculated in the similar way to that in [1], and it isexpressed as

p(ρ) =

Θ(n−1

)0 ≤ α ≤ 1,

Θ(ρ2−2αnα−2

)1 < α < 2,

Θ(ρ−2 log ρlog2 n

)α = 2,

Θ(ρ−α

)α > 2,

(12)

where ρ =√a2k + b2k is the distance between two home-

points. It can be found from (12) that the probability is similarto random i.i.d. mobility model when 0 ≤ α ≤ 1. However,for other cases, the probability becomes different. Based onthis mobility model, the per-node capacity of the network isderived in the following theorem.

Theorem 1: The per-node capacity for unicast scenariounder restricted mobility model is

C(n) =

Θ (1) 0 ≤ α < 2,

Θ(

1logn

)α = 2,

Θ(n1−α2

)2 < α ≤ 3,

Θ(

1√n

)α > 3.

(13)

Proof: The proof of this theorem can be divided intotwo parts. Firstly, (13) is proved to be the upper-bound ofthe unicast capacity. Afterwards, the corresponding upper-bound achieving scheme is proposed to demonstrate that (13)is achievable. Finally, according to the definition of capacity,the per-node capacity for unicast scenario under restrictedmobility model satisfies (13). The detailed proof can be foundin Appendix A, and the capacity achieving scheme is proposedas follows.

Scheme I: In this scheme, the transmission range satisfiesr(n) = Θ(1), and there are three kinds of transmissions:source to relay (S-R) transmission, relay to relay transmission(R-R) as well as relay to destination (R-D) transmission. Ineach time slot, when one cell is active according to TDMAscheme, each kind of transmission will be selected with thesame probability. Furthermore, one transmission pair belongsto the selected transmission kind will be chosen to be activeequiprobably.• If 0 ≤ α ≤ 3, R-R transmission is not allowed. For

any source-destination pair, denoting their home-pointsdistance as ρ and the middle point of their home-pointsas C, the source is only permitted to transmit packet toone relay with home-point located in the circle centeredat C with radius 1

3ρ. Afterwards, the relay will send thepacket to the destination when they are in the same cell.

• If α > 3, three kinds of transmissions are allowed.Considering the straight line which connects the home-points of source and destination, the set of the cells it

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lines across can be defined as Path Set of this source-destination pair. The cells in the path set are denoted asCi which is numbered according to the distance betweenCi and source’s home-point. In addition, C0 denotes thesource’s home-point cell. Based on such definition, if thepacket is hold by the node with home-point in Ci, it willbe transmitted to node whose home-point is in cell Ci+1

and deleted from the previous relay.

B. Per-node Capacity of Multicast Scenario

In multicast scenario, there are n sources and m destinationsfor each source, which is different from the unicast scenario.The following lemma is proposed to show the number ofcorresponding sources for each destination.

Lemma 1: In multicast scenario, any node, when acting asa destination, has m sources with probability 1 when n goesto infinity.

Proof: The proof can be found in Appendix B.Based on Lemma 1, the per-node capacity for multicast

scenario is derived in the following theorem.Theorem 2: The per-node capacity for multicast scenario

under restricted mobility model is

C(n) =

Θ(

1m

)0 ≤ α < 2,

Θ(

1m log n

m

)α = 2,

Θ(n1−α2 m

α2−2

)2 < α ≤ 3,

Θ(

1√nm

)α > 3.

(14)

Proof: Similar to Theorem 1, the proof can also bedivided into two parts: 1) C(n) in (14) is proved to be theupper-bound of multicast per-node capacity. 2) The followingScheme II is proposed to achieve the capacity upper-bound.The detailed proof can be found in Appendix C.

Scheme II: In this scheme, the transmission range satisfiesr(n) = Θ(1), and there are four kinds of transmissions: S-R,R-D, R-R and destination to relay (D-R) transmission. In eachtime slot, one kind of transmission is selected with the sameprobability. Afterwards, one transmission pair belonging to itis chosen equiprobably.

• If 0 ≤ α < 2, R-R and D-R transmissions are not al-lowed. Traditional 2-hop relay scheme without redun-dancy [9] is employed here. In particular, the sourcetransmits the packet to each relay within its transmissionrange. Afterwards, the relay will send the packet to thedestinations when it is within the transmission range ofthem.

• If α ≥ 2, Four kinds of transmissions are allowed. Firstly,a Euclidean Minimum Spanning Tree (EMST) of themulticast session is generated among the home-points ofsource and destinations as in [19]. The packet can betransmitted to all the destinations based on the EMST,and it is sent through each edge of the EMST as unicastby employing Scheme I.

IV. DELAY ANALYSIS FOR UNICAST AND MULTICAST

In this section, the optimal delay performance is analyzedfor both unicast and multicast scenarios, and the correspondingmulticast delay gain will be discussed in Section V.

A. Optimal Delay of Unicast Scenario

To optimize the delay, it is necessary to utilize all thepossible hops for one unicast. Therefore, a multi-hop schemenamed flooding scheme is employed, which is shown asfollows.

Scheme III: (Flooding Scheme) Four kinds of transmissions(S-R, R-R, R-D, D-R) are allowed in this scheme and will beselected equiprobably in any active cell. When S-R, R-R orD-R transmission is permitted, randomly choose a source (ora relay) and broadcast the packet to all the nodes in the cellsimultaneously. Besides, if R-D transmission is selected, anR-D pair will be randomly chosen to be active.

In [1], the authors propose a multi-hop scheme withoutredundancy. However, the redundancy is introduced in ourpaper for flooding scheme which is different from theirs. Itis well studied in [4] [18] that redundancy can decrease thedelay of networks because packet can be transmitted to thedestination through all of the possible paths, and the delay isdetermined by the shortest one. Moreover, it is obvious that ifthe transmission range is large enough (e.g. r(n) =

√n), the

delay could be Θ(1). However, it is not reasonable to assumethe transmission range to be related with network scale n, andtherefore the transmission range is assumed as r(n) = Θ(1).

As introduced in [11], in order to achieve optimal delay, thispaper considers the situation that a single packet is deliveredover an empty network, which is analyzed in many work aboutthe optimal delay [4]. Therefore, only one source is allowed totransmit its packet to the network until this packet is receivedby the destination.

However, the flooding scheme in the restricted networks isquite different from the random i.i.d. mobile networks. In therestricted mobility model, the PDF of packet-holding nodes inthe next time slot is determined by the packet-holding nodesin current time slot. Therefore, the transmission process inrestricted mobility model can be treated as a Markov chainwith 2n−1 states. Moreover, there are 2n−2 target states and 1initial state. Hence, it is too complex to obtain the exact orderof delay.

Instead, this paper analyzes the upper-bound of the opti-mal delay. In particular, the transmissions are divided intotwo groups, i.e., long distance transmission (LDT) and shortdistance transmission (SDT). The distance between two nodes’home-points is Θ(

√n) for each LDT, and the distance is

o(√n) for each SDT. In this network, if α is small, the

probability of LDT is large, and the probability of SDTis small, and vice versa. In order to reduce the analysiscomplexity, the packet is assumed to be transmitted to thedestination through only LDT or only SDT. Therefore, thecooperation between LDT and SDT is ignored, which causesthe derived delay greater than the actual delay, and thus it isthe delay upper-bound of flooding scheme.

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Firstly, the delay of LDT is calculated in the followinglemma.

Lemma 2: Considering the transmission from source i toits destination j by flooding scheme, the average delay fromsource to destination by LDT can be expressed as

DLDT =

Θ (log n) 0 ≤ α < 2,

Θ(log2 n

)α = 2,

Θ(nα−22 log n

)α > 2.

(15)

Proof: The proof can be found in Appendix D.In Lemma 2, it shows that delay is great when α is large.

The reason is that there are very few LDTs when α is large,and therefore SDT becomes the main issue. The delay ofpacket transmitted by SDT for the case α ≥ 2 is shownin the following lemma. The DSDT for 0 ≤ α < 2 is notshown here because pSDT = o(1) for this case, and thereforeDSDT = ω(log n).

Lemma 3: Considering the transmission from source i toits destination j by flooding scheme, the delay from source todestination by SDT can be expressed as

DSDT =

O(log3 n

)α = 2,

O(nα−22 log n

)2 < α < 3,

O (√n log n) α ≥ 3.

(16)

Proof: The proof can be found in Appendix E.The delay of the flooding scheme is determined by DLDT

and DSDT . Since the cooperation between LDT and SDT isnot considered, the minimum value of DLDT and DSDT isthe upper-bound of optimal delay, which is indicated in thefollowing theorem.

Theorem 3: The upper-bound of optimal delay for unicastscenario follows

D(n) =

Θ (log n) 0 ≤ α < 2,

O(log2 n

)α = 2,

O(nα−22 log n

)2 < α < 3,

O (√n log n) α ≥ 3.

(17)

B. Optimal Delay of Multicast Scenario

To obtain the optimal delay for multicast scenario, the flood-ing scheme is also adopted under the same assumption thatthe transmission range is constant. In Section IV.A, the upper-bound of optimal delay is derived for the unicast scenario.However, by the Scheme III (flooding scheme), each nodein the network will receive a replica of the packet from thesource within the time scale of (17). Therefore, for multicastscenario, the optimal delay is of the same scale of unicastscenario, which is also proved in random i.i.d. mobility modelin [11].

V. MULTICAST GAIN

The multicast can be treated as m-multi-unicast, and thepackets can also be transmitted in the network in the sameway as unicast. However, with the help of flow aggregation,multicast may perform better than multi-unicast in some con-ditions, which is considered to be the advantage of multicast.

Fig. 1 shows an example of flow aggregation. In this figure,the source needs to transmit a packet to all of its destinations.It can be found that 11 hops are required for multi-unicastcase, and only 5 hops are needed in multicast scenario instead.The flows are aggregated in multicast scenario, which is thedifference between multicast and unicast. In order to studysuch distinction, this paper focuses on the per-node capacityand delay ratio of multicast and m-multi-unicast, i.e. multicastgain on capacity and delay, respectively.

multi-unicast multicast

Number of flows

Source Source

Fig. 1. An example of flow aggregation

A. The Multicast Capacity Gain

From the theoretical results in Section III, it can be foundthat α and m jointly enhance the multicast gain on per-nodecapacity, which shows the essential role of mobility and m inmulticast. Furthermore, more general upper-bound and lower-bound of multicast gain on per-node capacity will be derivedregardless of mobility model in this subsection. Finally, basedon above work, it can be concluded that the multicast capacitygain is mainly determined by three factors.

If each multicast session is treated as m independent unicastsessions, there are mn unicast sessions in the network. Thus,the per-node capacity of this multi-unicast case is 1

m of (13)for the restricted mobility model. This per-node capacity iscompared with multicast per-node capacity in (14) in Fig. 24.It can be found that both of the unicast and multicast per-nodecapacity are non-increasing functions of α, which shows theimpact of mobility on per-node capacity.

In Fig.2, it is indicated that the impact of mobility onper-node capacity for unicast and multicast are different.Particularly, since a small α means strong mobility, the per-node capacity enhancement caused by mobility for unicast isgreater than multicast. Therefore, the multicast gain on per-node capacity decreases with mobility strength (increases withα)5. The per-node capacity gain of multicast is as follow

ξ(n) =

Θ (1) 0 ≤ α < 2,

Θ(

lognlog n

m

)α = 2,

Θ(m

α2−1

)2 < α ≤ 3,

Θ (√m) α > 3.

(18)

4There is capacity hopping when α = 2 because the convergence of seriesis changed in this case. The same phenomenon can be found in Fig. 3.

5The mobility strength in this figure can be evaluated by the average movingdistance per-timeslot in the restricted mobility model

7

( )C n

1 m

1 1logm n m

(1 )nm

1 2 2 2( )n m

Multicast

Multi-unicast

(1 )m n

1

logm n1 2 1( )n m

Mobility strength increases

Fig. 2. Per-node capacity comparison between multi-unicast and multicastscenarios

The impacts of α and m on ξ(n) are illustrated in Fig. 3 andFig. 4, respectively.

( )n

(1)

log( )log

n

n m

m

2 1( )m

Mobility strength increases

Fig. 3. The impact of α on multicast capacity gain

Restricted mobility

m

( )n

(1)

0

0 2

2 3

3

Upper-bound

Lower-bound

m

2( )n l n

Multicast capacity gain

range of restricted mobility

Random i.i.d. mobility

Random walk

(step length l)

One-dimensional staticm

( )n

Fig. 4. The impact of m on multicast capacity gain

In Fig. 3, for a given m, the multicast capacity gainincreases with α when 2 ≤ α ≤ 3. Otherwise, the multicastcapacity gain is not related with α for the case 0 ≤ α < 2and α > 3. In Fig. 4, it is shown that ξ(n) is a non-decreasingfunction of m. In particular, there is no multicast capacity gainas long as 0 ≤ α < 2, and the gain increases with m whenα ≥ 2. Furthermore, the increasing speed is determined byα. The upper-bound, lower-bound and other mobility cases inthis figure will be analyzed later.

In fact, the multicast brings about per-node capacity gaindue to the cooperation among destinations and relays, i.e. flowaggregation. However, the effect of cooperation is determinedby the certainty of node location, namely, nodes will cooperateeffectively if their relative locations are certain with highprobability in each time slot. For this mobility model, if α < 2,the unpredictability of nodes’ locations is so strong that thenetwork can be treated as i.i.d. mobility model, and thereforethe opportunity of flow aggregation is very low. Furthermore,for the case 2 ≤ α ≤ 3, the stability of nodes’ locationsbecomes stronger with α, and the cooperation becomes moreeffective. Moreover, the number of destinations also impactsthe cooperation since it determines the distinction betweenmulticast and unicast. Thus, the per-node capacity gain ofmulticast is a non-decreasing function of α and m. At last,when α > 3, the destinations’ locations are approximatelyfixed. Therefore, the multicast capacity gain is independentfrom α, and the cooperation among nodes has a strongimpact on the per-node capacity gain. Consequently, this paperconsiders the mobility to be the first determinant since itdetermines the form of multicast capacity gain growth, butm only determines the increasing speed.

The above results are specialized for the restricted mobilitymodel. However, the further study of multicast capacity gainshould include more mobility models. Thus, this paper will an-alyze the upper-bound and lower-bound of multicast capacitygain regardless of mobility model.

For the upper-bound, since each destination is randomlyselected, at least one unicast is contained in each multicast.Thus, if the multicast session can be finished during theunicast transmission, the per-node capacity of unicast andmulticast will be the same, and therefore the multicast gainon per-node capacity is Θ(m), which is the upper-bound. Anetwork achieving the upper-bound is proposed as follows.The network size is

√n ×√n. All of the nodes are static,

and their locations are (√n

2 cos 2πin ,√n

2 sin 2πin ), where i is

the label of each node. This network is illustrated in Fig.5, the unicast per-node capacity is derived by analyzing themaximum per-node throughput of the cut in this figure. Ifthe transmission range satisfies r(n) > 2π√

n, it can be easily

obtained that only Θ(1) packets can be transmitted across thecut due to the interference. Therefore, the per-node throughputupper-bound is Θ( 1

n ), which can be achieved by followingscheme: setting r(n) = 3π√

n, for each source destination pair i,

j (assuming i < j), the packet is relayed by i+1, i+2, · · · , j.For the multicast scenario, if the packet is transmitted tothe two farthest destinations of both directions, all the otherdestinations must have already received the packets for thereason that they are relays or within the transmission rangeof relays. Hence, the multicast can be treated as two unicastbetween source and the two farthest destinations of bothdirections, which means the per-node capacity of multicastis also Θ( 1

n ). Consequently, the multicast capacity gain forthis network is m, and it is the upper-bound.

For the lower-bound, one multicast session can be treatedas m unicast sessions no matter what the mobility model is.Therefore, the per-node capacity of multicast is no less than

8

The cut

n

n

Center

Fig. 5. The network with upper-bound of multicast capacity gain

m-multi-unicast, which means the lower-bound of multicastcapacity gain is Θ(1). To achieve the lower-bound, there mustbe no cooperation among nodes, and the multicast session isequivalent to m unicast sessions. The case 0 ≤ α < 2 for therestricted mobility model in this paper is an example of thiscondition.

The upper-bound (Θ(m)) and lower-bound (Θ(1)) areshown in Fig. 4, and the multicast capacity gain of restrictedmobility model is within this range. Moreover, we illustratethe multicast capacity gain of random walk mobility model[3] with step length l, one-dimensional static model [20]6 andthe random i.i.d. mobility model [9] in Fig. 4.

From Fig. 4, it can be found that mobility and m joint-ly impact the multicast gain on per-node capacity becausethe opportunity of flow aggregation increases with m butdecreases with mobility strength. However, there is a gapbetween static networks (restricted mobility model α → ∞)and the upper-bound. Hence, there is at least one other factorwhich impacts the multicast capacity gain. Since the upper-bound is achieved by one-dimensional static model and themulticast capacity gain Θ(

√m) can be achieved by two-

dimensional static model, it is obvious that the distributionof nodes also impacts the multicast capacity gain. To explainthis phenomenon, considering one destination of a multicastsession, Fig. 6 shows the transmission path from the sourceto it, and the corresponding coverage of this transmission (therange of dotted circles). If the other destinations belongingto this multicast session are distributed within the coveragewith high probability, the flows are aggregated effectively,and therefore the multicast capacity gain is very high. Thisexample indicated that the distribution of nodes can increasethe multicast capacity gain. Consequently, to form a generalframework of the multicast capacity gain study, the factorsdetermining the multicast capacity gain are listed as follows:

• The mobility• The number of destinations• The node distribution

6The multicast capacity gain for one-dimensional static model is Θ(m),and it can be obtained in the similar as the network in Fig. 5

Source Destination

Fig. 6. The transmission path from source to a destination

B. The Multicast Delay Gain

In order to investigate the multicast delay gain, a generalnetwork is considered, and the flooding scheme is employed.In fact, the flooding scheme utilizes all of the possible trans-missions to deliver one packet, and therefore the delay ofit is optimal for a given transmission range. Defining theflooding scheme delay of the multicast with m destinationsas Dmulti(n) and the flooding scheme delay of each unicastof its corresponding m-multi-unicast as Dm uni,i(n), (i =1, · · · ,m), the relation between them satisfies

Dmulti(n) = maxi=1,··· ,m

Dm uni,i(n). (19)

According to the results in Section IV, the multicast gainon delay for restricted mobility model is Θ(m) for any α,which means the mobility does not affect the multicast delaygain in order sense. However, when α = 0, the node’smovement satisfies the random i.i.d. mobility model, andtherefore the expectations of Dm uni,i(n) are the same. Onthe other hand, when α is large, the node is approximatelystatic, and hence the packet arrives at the nodes with thecloser home-point earlier, i.e., EDm uni,i(n) < Dmulti(n)for some destinations i with home-point close to the source’shome-point. Thus, the mobility reduces the multicast delaygain in constant order.

Furthermore, the upper-bound and lower-bound of multicastdelay gain is analyzed regardless of mobility model. Accordingto the Definition 5 and (19), the multicast delay gain is upper-bounded by Θ(m) and lower-bounded by Θ(1). It is obviousthat the upper-bound of multicast delay gain can be achievedby the restricted model in this paper.

Moreover, in order to achieve the lower-bound, the networkmust satisfy

Dm uni,imax(n) = Ω

( ∑i=1,··· ,imax−1,imax+1,··· ,m

Dm uni,i(n)

),

(20)where imax is the corresponding index of the maximumDm uni,i(n). For example, considering the network dividedinto two parts O1 and O2 with equal size, there are n − cn

mnodes in O1 satisfying the random i.i.d. mobility model, andthe rest of the nodes are moving in O2 satisfying the randomi.i.d. mobility model, where c > e is constant. Additionally,the distance between two parts is greater than the transmissionrange. Moreover, with probability 1

n3 , one node in one partcan move to another part in one time slot, and then itmoves back to its initial part. Therefore, Dm uni,imax(n) =Θ(n3) where poly-logarithmic factors are ignored. Howev-er,∑i=1,··· ,imax−1,imax+1,··· ,mDm uni,i(n) = O(n3), which

9

means (20) is satisfied, and therefore the multicast delay gainis Θ(1) in this network.

Consequently, the achievable bounds of multicast delay gainare proved, which form a general framework of multicastdelay gain. However, the multicast delay gain is not mainlydetermined by the mobility strength in order sense but by thedifference of delay for different nodes in flooding scheme, i.e.,the difference between Dm uni,i(n).

VI. CONCLUSION

This paper mainly focuses on the impact of mobility onmulticast gain. The analysis of per-node capacity shows thatmobility weakens the per-node capacity distinction betweenmulticast and unicast. In particular, the per-node capacityfor both unicast and multicast are derived, and two schemesbased on restricted relay selection are proposed to achieve theper-node capacity. Afterwards, the multicast capacity gain iscalculated, which is utilized to evaluate the distinction betweenthe per-node capacity enhancements for unicast and multicast.Moreover, the essential role of mobility and m in multicast isanalyzed. Additionally, the upper-bound and lower-bound ofmulticast capacity gain are also studied regardless of mobilitymodel, which may guide the multicast study in the future.Three factors which determine the multicast capacity gain arelisted in order to form a general framework of multicast capac-ity. Furthermore, the multicast delay gain is also investigated.The upper-bound and lower-bound of it are derived and provedto be achievable. Moreover, it can be found that the multicastdelay gain is not mainly determined by the mobility strength inorder sense but by the difference of delay for different nodesin flooding scheme.

APPENDIX A: THE PROOF OF THEOREM 1

In order to derive the per-node capacity, a contact graphis considered, in which the nodes are allocated at their home-points respectively. Moreover, an edge can be put between anytwo-nodes, whose weight is defined as the probability that theyhappen to be within distance Θ(1) of each other. Moreover,since this paper focuses on the fast mobility model, the nodecan only transmit one packet to any other node within distanceΘ(1) of it. Due to the unit density of the network, the contactgraph can be treated as a virtual capacitated graph.

Considering a cut dividing the contact graph into two partswith the same size, there are Θ(n) nodes in each part inaverage. The two parts are defined as O1 and O2, whichare illustrated in Fig. 7. Furthermore, the length of the cut isΘ(√n). If the source and destination is within different parts,

the packet must be transmitted through the cut. Therefore thesum per-node throughput of these pairs is bounded by thesum weight of the edges across the cut. Since the numberof such kind of source-destination pairs is Θ(n), the per-node throughput upper-bound of the network can also bederived from this bound according to the definition of per-node throughput.

In order to derive the sum probability of the edges across thecut, a node i belonging to O1 is considered, and its distancefrom the cut is ρi. Denoting the sum weight of the edges across

1

2

The cut

Fig. 7. The packet is transmitted across the cut

the cut with endpoint i as Wi, the sum weight of the edgesacross the cut can be expressed as

W =∑

i is in O1

Wi. (21)

According to the probability that two nodes happen to bewithin distance Θ(1) in Eq. (11), the Wi can be furtherexpressed as

Wi =∑

j is in O2

p(ρHi,j). (22)

Since the home-points in O2 are uniformly distributed and thesize of the network is

√n×√n, the order of Wi in (22) can

be computed as

Wi = Θ

(2

∫ √nρi

∫ arccos(ρi/ρ)

0

ρp(ρ)dθdρ

)

= Θ

(2

∫ √nρi

ρ arccos(ρi/ρ)p(ρ)dρ

)

= Θ

(2

∫ √nρi

√ρ2 − ρ2

i p(ρ)dρ

)

= Θ

(∫ 2ρi

ρi

√ρi√ρ− ρip(ρi)dρ+

∫ √n2ρi

ρp(ρ)dρ

)

=

Θ (1) 0 ≤ α ≤ 2,Θ(ρ2−αi

)α > 2.

(23)

Moreover, since the home-points in O1 are also uniformlydistributed, the order of W in (21) can be computed as

W =∑

i is in O1

Wi

= Θ

(√n

∫ √n0

Widρi

)

=

Θ (n) 0 ≤ α ≤ 2,Θ(n2−α2

)2 < α ≤ 3,

Θ (√n) α > 3,

(24)

10

where the poly-logarithmic factors are ignored for brevitywhen α = 3. Considering that there are Θ(n) source-destination pairs, the per-node throughput of these pairs canbe limited by

Θ

(W

n

)=

Θ (1) 0 ≤ α ≤ 2,Θ(n1−α2

)2 < α ≤ 3,

Θ(

1√n

)α > 3.

(25)

According to the definition of per-node throughput (i.e., Def-inition 1), (25) is also the per-node throughput bound of thenetwork. Based on Definition 2, in order to prove that the per-node capacity of unicast scenario equals to (25), it is necessaryto testify that the per-node throughput in (25) is achievable.Thus, the Scheme I will be proved to be the capacity achievingscheme in the following part.

Firstly, if α ≤ 3, for the source i and its destination j, theprobability that source meets a relay node whose home-pointis in the circle O 1

3ρHi,j

centered at C with radius 13ρHi,j is

Pri meets a relay =

∫ ∫O 1

3ρHi,j

p(ρ)dO

= Θ

p(√n)

∫ ∫O 1

3ρHi,j

dO

=

Θ (1) 0 ≤ α < 2,

Θ(

1logn

)α = 2,

Θ(n1−α2

)2 < α ≤ 3.

(26)

To calculate the per-node throughput of the network, weassume that M(i, t) packets are transmitted from sourcei to destination j at time slot t. The average delay ofthe first hop is Pr−1i meets a relay. Moreover, if t =ω( n

Pri meets a relay ), there are Θ(n) relays hold packets inthe circle. Therefore, the destination will meet a relaywith probability Prj meets a relay = Pri meets a relay.Moreover, since there are Θ(n) relays in the circle andthe number of source-destination pairs is Θ(n), the prob-ability that one source-relay is selected to be active isΘ(1) when they are in the same cell. Thus, when t =ω( n

Pri meets a relay ), the source will send a packet to relay inPr−1i meets a relay time slots, and the relay will receive apacket in Pr−1j meets a relay time slots. Hence, the longterm per-node throughput for this source-destination pair is

λi(n) = lim inft→∞

1

tM(i, t) = Θ(Pr−1j meets a relay). (27)

Consequently, the per-node throughput of the network for thisscheme is

T (n) =

Θ (1) 0 ≤ α < 2,

Θ(

1logn

)α = 2,

Θ(n1−α2

)2 < α ≤ 3,

(28)

which achieves the upper-bound of per-node throughput in (13)when 0 ≤ α ≤ 3, and there is gap of Θ(log n) when α = 2.

For the case α > 3, the scheme is different. From (12), itis obvious that each node will meet the node whose home-point is ρ = Θ(1) from its home-point with probability Θ(1).

Therefore, there are Θ(n) active transmission pairs in eachtime slot. Furthermore, it takes Θ(

√n) hops to transmit one

packet from source to destination. Hence, the total throughputof the network is Θ(n/

√n) = Θ(

√n) and the per-node

throughput Θ( 1√n

). Thus, the upper-bound is achieved whenα > 3.

Consequently, the upper-bound is achievable, which meansthat this per-node throughput upper-bound is the per-nodecapacity of the unicast scenario.

APPENDIX B: THE PROOF OF LEMMA 1

Since each source randomly selects m destinations, theprobability that a given destination i is selected by the givensource is m

n . Thus, defining the number of node i’s sources asd, the probability that d = d0 can be expressed as

Prd = d0 =n!

(n− d0)!d0!

(mn

)d0 (1− m

n

)m−d0. (29)

Therefore, for two constants 0 < c1 < 1/e and c2 > e (e isthe base of the natural logarithm), the probability that c1m <d < c2m satisfies

Pr c1m < d < c2m

= 1−c1m∑i=1

n!

(n− i)!i!

(mn

)i (1− m

n

)n−i−

n∑i=c2m

n!

(n− i)!i!

(mn

)i (1− m

n

)n−i.

(30)

Denote g(i) = n!i!(n−i)! (

mn )i(1− m

n )n−i, therefore,

g(i)

g(i+ 1)=

n!i!(n−i)! (

mn )i(1− m

n )n−i

n!(i+1)!(n−i−1)! (

mn )i+1(1− m

n )n−i−1

=(i+ 1)(1− m

n )

(n− i)mn>i+ 1

m.

(31)

According to (31), denoting I = c2m, the last item of (30)can be bounded as follow

n∑i=c2m

n!

(n− i)!i!

(mn

)i (1− m

n

)n−i<

n∑i=I

mi−II!

i!g(I)

(a)= Θ

(n∑i=I

mi−I ( Ie

)I √I(

ie

)i√i

g(I)

)

= Θ

(n∑i=I

(I

i

)i+ 12(e

c2

)i−Ig(I)

)

= O

(g(I)

n∑i=I

(e

c2

)i−I)= O(g(I)),

(32)

11

(a) hold due to the fact that limn→∞

(ne

)n √2πnn! = 1. The g(I)

can be further calculated as follow

g(I)

=n!

(n− I)!I!

(mn

)I (1− m

n

)n−I= Θ

( (ne

)n√n(

Ie

)I √I(n−Ie

)n−I √n− I

(mn

)I (1− m

n

)n−I)

= Θ

1

cI2

[(1 +

(c2 − 1)m

n− I

) n−I(c2−1)m

](c2−1)m√1

I+

1

n− I

= O

((e

c2

)I).

(33)

Therefore, if m = ω(1), the last item of (30) equals too(1). Similar results can be found after calculating the seconditem of (30), i.e., the second item of (30) equals to o(1) whenm = o(n). Therefore, Pr c1m < d < c2m = 1 when n goesto infinity.

APPENDIX C: THE PROOF OF THEOREM 2

In multicast scenario, each node needs to send packets tom nodes which are randomly selected among all the nodes.When m = Θ(1), the structure of the multicast session issimilar to unicast scenario, and therefore this paper mainlyfocuses on the condition that m = ω(1). The correspondingper-node capacity can be derived based on the contact graphin the similar way to that in unicast scenario. In particular,considering a circle with radius

√n

8m in the contact graph,there are Θ

(nm

)home-points in this circle with probability 1

due to the uniform distribution of home-points. Since the nodeswith home-points in the circle need to receive packets fromthe corresponding sources, it is necessary to find the numberof sources of these packets, which is denoted as Ncircle.According to Lemma 1, there are Θ

(nm8m

)= Θ(n) sources

send packets into the circle, i.e., Ncircle = Θ(n).The circle region is denoted as O1, the region out of the

circle is denoted as O2, which are illustrated in Fig. 8.The edge of O1 can be treated as a cut of the contact graph.

If node i’s home-point is in O2 and the distance from the edgeof O1 is ρi, the sum weight of the edges across the cut canalso be expressed as in (21), where Wi represents the sumweight of edges across the cut with endpoint i. Therefore, theWi here can be further computed as follow

Wi =∑

j is in O2

p(ρHi,j)

= Θ

(2

∫ √nρi

∫ θρ

0

ρp(ρ)dθdρ

),

(34)

where θρ ∈ [0, π] is illustrated in Fig. 8. After some geometricmanipulations, the value of θρ can be computed accordingto Herons formula as in (35) in the next page, where s =√

n2m+ρ−ρi

2 . Afterwards, it is necessary to discuss the order of(35). If ρ ≥

√n

8m , it is obvious that θρ > 2π3 , and therefore

8

n

m

i

8

n

m

Node i

1

2

The cut

Fig. 8. The relation among θρ, ρ, ρi

θρ = Θ(1). Hence, we only needs to focus on the conditionthat ρ <

√n

8m . In this case, after some manipulations, theorder of θρ can be expressed as

θρ =

Θ

√1−( √

n8m−ρ√n

8m−ρi

)2 ρi ≥ 1

2

√n

8m ,

Θ

(√1−

(ρiρ

)2)

ρi <12

√n

8m .

(36)

Thus, the Wi in (34) can be further derived as

Wi = Θ

(∫ √ n8m

ρi

∫ θρ

0

ρp(ρ)dθdρ+

∫ √n√

n8m

∫ θρ

0

ρp(ρ)dθdρ

)

=

Θ (1) 0 ≤ α ≤ 2,Θ(ρ2−αi

)α > 2.

(37)

Since the nodes are uniformly distributed in the graph, theorder of W can be computed as

W =∑

i is in O1

Wi

= Θ

(∫ √ n8m

1

(√n

8m− ρi

)Widρi

)

=

Θ(nm

)0 ≤ α ≤ 2,

Θ(n2−α2 m

α2−2

)2 < α ≤ 3,

Θ(√

nm

)α > 3.

(38)

It should be noted that the poly-logarithmic factors are ignoredin (37) and (38) for brevity. Since there are Θ(n) sources needto transmit packets into O1, the per-node throughput of thesesources can be limited by

Θ

(W

n

)=

Θ(

1m

)0 ≤ α ≤ 2,

Θ(n−

α2 m

α2−2

)2 < α ≤ 3,

Θ(√

1nm

)α > 3.

(39)

According to the Definition 1, (39) is also the per-nodethroughput bound of the network.

12

θρ =

arcsin

(√s(s−√

n8m )(s−

√n

8m+ρi)(s−ρ)

2ρ(√

n8m−ρi)

)ρi < ρ ≤

√n

8m −(√

n8m − ρi

)2,

π − arcsin

(√s(s−√

n8m )(s−

√n

8m+ρi)(s−ρ)

2ρ(√

n8m−ρi)

) √n

8m −(√

n8m − ρi

)2< ρ ≤ n

2m − ρi,

π ρ >√

n2m − ρi,

(35)

The following part proves that Scheme II can achieve theper-node throughput upper-bound (14). If 0 ≤ α < 2, noticethat one node will meet another one with probability Θ( 1

n ),the mobility model can be treated as the random i.i.d. mobilitymodel. Therefore, the per-node throughput for this case canbe derived in the same way as in [9], and the result is shownas follow.

T (n) = Θ

(1

m

). (40)

For the case α ≥ 2, the mobility model becomes differentfrom the random i.i.d. mobility model. Considering a timeinterval t which is large enough, M(t) multicast sessions arefinished in t. Since each EMST is composed of m edges, andeach edge is treated as a unicast session, there are mM(t)unicast sessions. For one unicast session of the EMST, thetransmitter meets a relay within tinput(ρe) time slots, andthe receiver meets a relay within toutput(ρe) time slots ifthe home-point distance between transmitter and receiver isρe. Owing to constant transmission range, a transmission pairwill be selected with constant probability when they are withintheir transmission range. Thus, for the case 2 ≤ α ≤ 3, thetinput(ρe) and toutput(ρe) can be calculated as follows

tinput(ρ) = Pr−1transmitter meets a relay

=

Θ(log ρe) α = 2,Θ(ρα−2

e ) , 2 < α ≤ 3,

toutput(ρe) = Pr−1receiver meets a relay

=

Θ(log ρe) α = 2,Θ(ρα−2

e ) , 2 < α ≤ 3.

(41)

Since the sum length of all the edges of one EMST is Θ(√mn)

[19], there must be∑me=1 ρe = Θ(

√mn). Recall that there are

M(t) multicast sessions, the total transmission time t mustsatisfy

t = Θ

1

n

mM(t)∑e=1

maxtinput(ρe), toutput(ρe)

(a)= O

(mM(t)

ntinput

(∑me=1 ρem

))= O

(mM(t)

ntinput

(√n

m

)),

(42)

(a) holds since tinput(ρ) is a concave function of ρ andtinput(ρ) = toutput(ρ). Consequently, based on the definitionof per-node throughput, the per-node throughput of Scheme II

for 2 ≤ α ≤ 3 can be derived as follow

T (n) = lim inft→∞

1

tnM(t)

= Ω

(lim inft→∞

1

mtinput(√

nm )

)

=

Ω(

1m log n

m

)α = 2,

Ω(n1−α2 m

α2−2

)2 < α ≤ 3.

(43)

At last, for the case α > 3, the network is similar to staticone, and the per-node throughput can be obtained in the sameway as in [8]. Thus, the result is

T (n) = Θ

(1√mn

). (44)

Based on (40), (43) and (44), it can be proved that SchemeII can achieve the per-node throughput upper-bound in (14),and therefore the per-node capacity of multicast is (14) due toits definition.

APPENDIX D: THE PROOF OF LEMMA 2

In [9], X. Wang, et al. study the delay of flooding schemefor random i.i.d. mobility model. The result shows that thedestination will receive the packet in Θ(log n) time slots fromthe beginning. In their model, all the transmissions are LDTwith probability 1. Therefore, it is necessary to know theprobability of the event of LDT in our model. Based on (12)and ρ = Θ(

√n) for LDT, the probability can be obtained as

pLDT = min1,∫ρHi,k=Θ(

√n)

p(ρHi,k)ρHi,kdρHi,k

=

Θ (1) 0 ≤ α < 2,

Θ(

1logn

)α = 2,

Θ(n

2−α2

)α > 2.

(45)

Since this lemma just focus on the packets transmitted throughLDT, the transmission distance of each packet is the same asthat in the random i.i.d. mobility model. The average delayfrom source to destination by LDT can be expressed as

DLDT = Θ

(log n · 1

pLDT

)

=

Θ (log n) 0 ≤ α < 2,

Θ(log2 n

)α = 2,

Θ(nα−22 log n

)α > 2.

(46)

13

APPENDIX E: THE PROOF OF LEMMA 3

To prove this lemma, we consider the condition that thereis an region Oρ0 of radius ρ0 centered at the home-point ofsource, and each node with home-point in Oρ0 holds a packetfrom i with probability Θ(1). After tρ0→ρ1 time slots, there isan region Oρ0+ρ1 of radius ρ0 +ρ1 centered at the home-pointof source, and each node with home-point in Oρ0+ρ1 holds apacket from i with probability Θ(1). This process is calledregion extension, which is illustrated in Fig. 9, and tρ0→ρ1 isthe region extension time from Oρ0 to Oρ0+ρ1 . It should benoted that the Oρ0+ρ1 is not the ring in Fig. 9 but the circlewhich covers Oρ0 .

0

0

0 1

Region extension

0 1

Source

Fig. 9. The region extension from Oρ0 to Oρ0+ρ1 .

The relation among tρ0→ρ1 , ρ0 and ρ1 is analyzed asfollows. Considering a node k whose home-point is in the ringOρ0+ρ1−Oρ0 and Θ(ρ1) from the edge of Oρ0 , the number ofcells within Oρ0 , whose distance from k’s home-point belongsto [ρ− 1, ρ+ 1], satisfies

Θ

(√s(s− ρ0)(s− (ρ0 + ρ1))(s− ρ)

ρ0 + ρ1

)

=

Θ(√

ρ2 − ρ21

)ρ1 = O(ρ0),

Θ(√

ρ0ρ1·√ρ2 − ρ2

1

)ρ1 = ω(ρ0),

(47)

where s = ρ0+(ρ0+ρ1)+ρ2 , and (47) holds due to Heron’s

formula. Afterwards, the probability that k meets nodes whosehome-points are within Oρ0 is calculated. If α = 2, theprobability is

pOρ0→Oρ0+ρ1

=

Θ

(min

1,

∫ ρ0+ρ1ρ1

√ρ2−ρ21ρ

−2 log ρdρ

log2 n

)ρ1 = O(ρ0),

Θ

(min

1,

∫ ρ0+ρ1ρ1

√ρ0(ρ2−ρ21)

ρ1ρ−2 log ρdρ

log2 n

)ρ1 = ω(ρ0).

(48)

The cells in Oρ0+ρ1 −Oρ0 very close to Oρ0 are ignored forthe reason that such kind of cells are much fewer than othersin order sense, and the transmission time is also smaller thanothers. Hence, the ignorance of these cells does not affect theorder of tρ0→ρ1 . Thus, the distance between cells in Oρ0 andOρ0+ρ1 − Oρ0 is considered to be from Θ(ρ1) to Θ(ρ0 +

ρ1). The region expansion time from Oρ0 to Oρ0+ρ1 can bebounded as

tOρ0→Oρ0+ρ1= O(p−1

Oρ0→Oρ0+ρ1log (ρ1 + ρ0)). (49)

There is a factor log (ρ1 + ρ0) in (49) because there areΘ(ρ1(ρ1 + ρ0)) nodes in Oρ0+ρ1 − Oρ0 . The transmissionsout of Oρ0+ρ1 and the relay to relay transmissions within thering Oρ0+ρ1 − Oρ0 during the region extension from Oρ0 toOρ0+ρ1 are ignored. For a given ρ0, if the ρ1 is too large, itmeans that too many relay to relay transmissions within thering Oρ0+ρ1 − Oρ0 are ignored, which will cause the boundof DSDT not tight enough. On the other hand, if the ρ1 is toosmall, it means that too many transmissions out of Oρ0+ρ1 areignored, which will also cause the bound of DSDT not tightenough. Hence, it is necessary to find the optimal relationbetween ρ1 and ρ0 to ensure that ignored transmissions areminimized. Thus, it is necessary to discuss tOρ0→Oρ0+ρ1

fordifferent relations between ρ0 and ρ1. Firstly, if ρ1 = ω(ρ0),

tOρ0→Oρ0+ρ1= O

(max

1,

log2 n√ρ0ρ−21 log ρ1

∫ ρ00

√ρ†dρ†

)= O

(ρ21 log2 n

ρ20

).

(50)

If ρ1 = Θ(ρ0), tOρ0→Oρ0+ρ1satisfies

tOρ0→Oρ0+ρ1= O

(log2 n

). (51)

Finally, if ρ1 = o(ρ0),

tOρ0→Oρ0+ρ1

=O

(log ρ0 max

1,

log2 n∫ ρ0+ρ12ρ1

log ρρdρ+

∫ 2ρ1ρ1

√ρ−ρ1 log ρ

ρ32

dρ

)

=O

(log ρ0 max

1,

log2 n

log ρ0 log ρ0ρ1

+ρ− 3

21 log ρ1

∫ ρ00

√ρ†dρ†

)

=O

(log2 n

log ρ0ρ1

).

(52)

Based on the above results, if the Oρ0 is expanded toOρ0+ρ1 where ρ1 = ω(ρ0), the expansion time is Θ

(ρ21 log2 n

ρ20

)in (50). However, if the expansion Oρ0 → Oρ0+ρ1 is treated asexponential growth, i.e. Oρ0 → O2ρ0 → O4ρ0 · · · → Oρ0+ρ1 ,the extension time can be calculated according to (51). Basedon (51), the expansion time is

Θ

logρ1ρ0∑

k=1

log2 n

= Θ

(log2 n log

ρ1

ρ0

). (53)

For any ρ1 = ω(ρ0), there must be log2 n log ρ1ρ0

=

o(ρ21 log2 n

ρ20

). Thus, the ignored transmissions in case ρ1 =

Θ(ρ0) is smaller than that in case ρ1 = ω(ρ0). Therefore,ρ1 = Θ(ρ0) performs better than ρ1 = ω(ρ0). Then,ρ1 = Θ(ρ0) and ρ1 = o(ρ0) are compared. If the Oρ0 isexpanded toOρ0+ρ1 where ρ1 = Θ(ρ0), it can be treated as thecombination of multiple expansions, i.e. Oρ0 → Oρ0+ρ1 →

14

Oρ0+ρ1+ρ2 · · · → Oρ0+ρ1 . Based on (52), the expansion timeis ∑

∑ρi=ρ0

log2 n

logρ(i)0ρ1

=Ω

∑∑ρi=ρ0

ρi log2 n

ρ(i)0

= Ω(log2 n), (54)

where ρ(i)0 = ρ0+

∑i−1j=1 ρj . Thus, the ignored transmissions in

case ρ1 = Θ(ρ0) is also smaller than that in case ρ1 = o(ρ0).Consequently, ρ1 = Θ(ρ0) is the optimal, and the DSDT forα = 2 is bounded by the region expansion as ρ1 = Θ(ρ0) asfollow

DSDT = O

(logn∑k=1

log2 n

)= O

(log3 n

). (55)

For the condition that α > 2, the expansion time can beobtained in the similar way, and the result is shown in (56)

tOρ0→Oρ0+ρ1=

Θ(ρα1 ρ

−20 log ρ1) ρ1 = ω(ρ0),

Θ(ρα−20 log ρ0) ρ1 = Θ(ρ0),

Θ(ρα−21 log ρ0) ρ1 = o(ρ0).

(56)

Firstly, the relations ρ1 = ω(ρ0) and ρ1 = Θ(ρ0) are com-pared. If the Oρ0 is expanded to Oρ0+ρ1 where ρ1 = ω(ρ0),it is treated as exponential growth in the same way as α = 2,and therefore the expansion time is

logρ1ρ0∑

k=1

2(α−2)kρα−20 (k + log ρ0)

= o

logρ1ρ0∑

k=1

2(α−2)kρα−20 (log

ρ1

ρ0+ log ρ0)

= o(ρα−2

1 log ρ1)

= o

(ρα1 log ρ1

ρ20

).

(57)

Thus, ρ1 = Θ(ρ0) performs better than ρ1 = ω(ρ0).Furthermore, to compare ρ1 = Θ(ρ0) and ρ1 = o(ρ0), itcan be assumed that the Oρ0 is expanded to Oρ0+ρ1 whereρ1 = Θ(ρ0). If it is treated as the combination of multipleexpansions as in the case α = 2, the expansion time satisfies

Θ

∑∑ρi=ρ0

ρα−2i log ρ0

=

ω(ρα−2

0 log ρ0) 2 < α < 3,O(ρα−2

0 log ρ0) α ≥ 3.

(58)

Hence, ρ1 = Θ(ρ0) is optimal when 2 < α < 3, and ρ1 =o(ρ0) is optimal when α ≥ 3. Moreover, for the case ρ1 =o(ρ0) and α ≥ 3, it can be easily proved that the optimal ρ1 isρ1 = Θ(1). Therefore, the bound of DSTD can be calculatedbased on the optimal method, and the results are shown in(16).

VII. ACKNOWLEDGEMENT

This work is supported by NSF China (No.61325012,61271219, 61221001, 61428205);China Ministry of Educa-tion Doctor Program(No.20130073110025); Shanghai BasicResearch Key Project (12JC1405200, 11JC1405100);ShanghaiInternational Cooperation Project: (No. 13510711300).

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Yi Qin received the B.S. and M.S. degrees inelectronic engineering from Shanghai Jiao TongUniversity, Shanghai, China, in 2009 and 2012,respectively. He is currently working toward thePh.D. degree with Shanghai Jiao Tong University,Shanghai, China.

His research interests include D2D communica-tion, D2D discovery, and Ad Hoc network scalinglaws study.

Xiaohua Tian received his B.E. and M.E. degreesin communication engineering from NorthwesternPolytechnical University, Xi’an, China, in 2003 and2006, respectively. He received the Ph.D. degree inthe Department of Electrical and Computer Engi-neering (ECE), Illinois Institute of Technology (IIT),Chicago, in Dec. 2010. He is currently an AssistantProfessor in Department of Electronic Engineeringof Shanghai Jiao Tong University, China. He wonthe Highest Standards of Academic Achievement2011 of IIT and Fieldhouse Research Fellowship

2009 of IIT, which is awarded to only one student over IIT each year. Hisresearch interests include application-oriented networking, Internet of Thingsand wireless networks. He serves as the guest editor of International Journalof Sensor Networks, publicity co-chair of WASA 2012. He also serves as theTechnical Program Committee member for Wireless Networking Symposium,Ad Hoc and Sensor Networks Symposium of IEEE GLOBECOM 2013,Ad Hoc and Sensor Networks Symposium of IEEE ICC 2013, WirelessNetworking Symposium of IEEE GLOBECOM 2012, Communications QoS,Reliability, and Modeling Symposium (CQRM) of GLOBECOM 2011, andWASA 2011.

Weijie Wu is an Assistant Professor in School ofElectronic, Information and Electrical Engineering,Shanghai Jiao Tong University. Before that, he was aresearch fellow working with Dr. Richard T.B. Ma inNational University of Singapore, and a postdoctoralfellow working with Prof John C.S. Lui at TheChinese University of Hong Kong. He obtainedhis Ph.D. degree in computer science from TheChinese University of Hong Kong in August 2012,and Bachelor’s degree in electronic & informationscience and technology from Peking University in

July 2008. When he was a Ph.D. student, he spent two months at NationalUniversity of Singapore working as a research intern. His current researchinterests are in computer networks from mathematical modelling, data an-alytics, and economic perspectives. In particular, he is recently interestedin network science (e.g., online social networks, large scale network withdata implications, etc.), network economics (e.g, game theoretic analysis oncommunication networks, pricing and incentive design in network applica-tions, etc.), and network optimization (e.g., resource allocation and pricing incloud computing, information centric network, and P2P systems). His personalinterests include table-tennis, badminton and hiking.

Xinbing Wang received the B.S. degree (with hon-s.) from the Department of Automation, ShanghaiJiaotong University, Shanghai, China, in 1998, andthe M.S. degree from the Department of Comput-er Science and Technology, Tsinghua University,Beijing, China, in 2001. He received the Ph.D.degree, major in the Department of electrical andComputer Engineering, minor in the Departmentof Mathematics, North Carolina State University,Raleigh, in 2006. Currently, he is a professor inthe Department of Electronic Engineering, Shanghai

Jiaotong University, Shanghai, China. Dr. Wang has been an associate editorfor IEEE/ACM Transactions on Networking and IEEE Transactions on MobileComputing, and the member of the Technical Program Committees of severalconferences including ACM MobiCom 2012, ACM MobiHoc 2012-2014,IEEE INFOCOM 2009-2014.