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Delay and Capacity Tradeoff Analysis forMotionCast

Xinbing Wang, Wentao Huang, Shangxing Wang, Jinbei Zhang, and Chenhui Hu

Abstract—In this paper, we define multicast for ad hoc networkthrough nodes’ mobility as MotionCast, and study the delay andcapacity tradeoffs for it. Assuming nodes move according to anindependently and identically distributed (i.i.d.) pattern and eachdesires to send packets to k distinctive destinations, we comparethe delay and capacity in two transmission protocols: one uses2-hop relay algorithm without redundancy, the other adopts thescheme of redundant packets transmissions to improve delaywhile at the expense of the capacity. In addition, we obtainthe maximum capacity and the minimum delay under certainconstraints. We find that the per-node delay and capacity for2-hop algorithm without redundancy are Θ(1/k) and Θ(n log k),respectively; and for 2-hop algorithm with redundancy they areΩ(1/k

√n log k) and Θ(

√n log k), respectively. The capacity of

the 2-hop relay algorithm without redundancy is better than themulticast capacity of static networks developed in [3] as long ask is strictly less than n in an order sense; while when k = Θ(n),mobility does not increase capacity anymore. The ratio betweendelay and capacity satisfies delay/rate ≥ O(nk log k) for thesetwo protocols, which are both smaller than that of directlyextending the fundamental tradeoff for unicast established in [1]to multicast, i.e., delay/rate ≥ O(nk2). More importantly, wehave proved that the fundamental delay-capacity tradeoff ratiofor multicast is delay/rate ≥ O(n log k), which would guide usto design better routing schemes for multicast.

Index Terms—Multicast, Capacity, Delay, Scaling Law

I. INTRODUCTION

A mobile ad hoc network (MANET) consists of a collectionof wireless mobile nodes dynamically forming a temporarynetwork without the support of any network infrastructure orcentralized control. In these networks, nodes often operate notonly as sources, but also as relays, forwarding packets forother mobile nodes. With the fast progress of computing andwireless networking technologies, there are increasing interestsand use of MANETs. Examples where they may be employedare the establishment of connections among handheld devicesor between vehicles.

Multicast is a fundamental service for supporting informa-tion communication and collaborative task completion amonga group of users and enabling cluster-based system designin a distributed environment [2]. Different from in the wirednetworks, multicast in MANETs is faced with a more chal-lenging environment. In particular, one needs to deal withnode mobility and thus frequent and possible drastic topologychanges [1]. Numerous protocols have then been proposed for

The authors are with Department of Electric Engineering, Shanghai JiaoTong University, China.E-mail: xwang8, wht, wsx, zjb, hch@sjtu.edu.cn

The early version of this paper is appeared in the Proceedings of ACMMobiHoc 2009 [21].

multicast in MANETs. They include traditional tree- or mesh-based protocols [3, 4, 5, 6], stateless protocols [7, 8], flooding-based protocols [9], location-based protocols [10] and hybridprotocols [11]. Some of them have already pointed out thatbecause links can be shared by several destinations, multicastis beneficial to improve performance comparing to multipleunicast.

However, the feasible performance gains, in terms of boththroughput capacity and delay, that can be achieved by exploit-ing multicast, as well as the resulting scaling laws in a networkwith increasing number of nodes have not been investigatedso far. In this paper, we bridge the theoretical analysis offundamental scaling laws in multicast mobile ad hoc networkswith the insights already gained through practical protocoldevelopment. By so doing, we provide a theoretical foundationto the design of intelligent communication schemes whichexploit multicast, analytically showing the potential of suchschemes in terms of capacity delay tradeoffs.

The theoretical analysis of scaling laws in wireless networksis initiated by the seminal work of Gupta and Kumar [4].Several interesting studies have later emerged aiming at estab-lishing the fundamental scaling laws for networks with multi-cast traffic. Li et al. [3][22][23] study the capacity of a staticrandom wireless ad hoc network for multicast where each nodesends packets to k − 1 destinations. They show that the per-node multicast capacity is Θ( 1√

n logn√k) when k = O( n

logn );and is Θ( 1n ) when k = Ω( n

logn ). Their results generalizeprevious capacity bounds on unicast [4] and broadcast [5].Under a more general Guassian channel model, multicastcapacity is investigated in [6] using percolation theory. Jacquetet al. [7] consider multicast capacity by accounting the ratio ofthe total number of hops for multicast and the average numberof hops for unicast. Shakkottai et al. [8] propose a comb-basedarchitecture for multicast routing which achieves the upperbound for capacity in an order sense.

In contrast to the above discussed static networks, Goss-glauser and Tse [9] for the first time have shown that a constantunicast per-node capacity can be achieved in mobile ad hocnetworks by exploiting the store-carry-forward communicationparadigm, i.e., by allowing nodes to store the packets andphysically carry them while moving around the network.Although this communication scheme incurs a tremendousaverage delay of Ω(n) [1], [10], it has laid the foundationof an entire new area of research, usually referred to as delaytolerant or disruption tolerant networks (DTNs), which hasrecently attracted a lot of attention. A typical DTN consistsof a set of fixed or mobile nodes, and is characterized by

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intermittent connectivity and frequent network partitioning,such that node mobility is essential to ensure end-to-endcommunication. Many interesting applications of DTN havebeen already envisioned and experimented upon, such asvehicular networks based on WiFi [13-16], networks based onhuman mobility [17], disaster-relief networks [18] and Internetaccess to remote villages [19].

The asymptotic capacity delay tradeoff in MANETs ex-ploiting store-carry-forward schemes has attracted significantattentions and is studied by many authors under variousmobility models. The mostly studied model is arguably thei.i.d. mobility model, where all nodes are reshuffled in anew time slot, due to its mathematical tractability. Withthis assumption, Neely and Modiano [1] present a strategyutilizing redundant packets transmissions along multiple pathsto reduce delay at the cost of capacity. They establish thenecessary tradeoff of delay/capacity ≥ O(n), and proposeschemes to achieve Θ(1),Θ(1/

√n) and Θ(1/(n log n)) per-

node capacity, when the delay constraint is Θ(n),Θ(√n)

and Θ(log n), respectively. In [16], Toumpis and Goldsimthconstruct a better scheme that can achieve a per-node capacityof Θ(n(d−1)/2/ log5/2 n) under fading channels when thedelay is bounded by O(nd). Lin and Shroff [2] later studythe fundamental capacity-delay tradeoff and identify the lim-iting factors of the existing scheduling schemes in MANETs.Recently, Ying et al. [15] propose joint coding-schedulingalgorithms to improve capacity-delay tradeoffs while Garettoand Leonardi [A20] show that under a restricted i.i.d. mobilitymodel, it is possible to exploit node heterogeneity to achieveboth constant capacity and constant delay.

To the best of our knowledge, this is the first work tostudy capacity and delay tadeoffs in MANETs with multicasttraffics. Because a key feature of multicast in MANETs isthat packets can be delivered via nodes’ mobility, we refer itas MotionCast. Intuitively, delay and capacity tradeoffs stillexist for MotionCast, but are more complicated than unicastscenarios. Since packets can be delivered through the mobilityof relay nodes, a higher per-node multicast capacity than instatic networks is expected. However, the scheduling designbecomes more difficult because of the permanent change of thenetwork topology as well as the fact that multiple destinationsfor a packet will imply a larger delay. Hence, some challengingissues raised naturally in this context are:

• What is the maximum per-node MotionCast capacity?• What is the delay for maximal capacity achieving

schemes and and what is the minimum possible delay?• What is the delay and capacity tradeoff for MotionCast?

Answering these questions would provide helpful fundamen-tal insights on the understanding and design of large scalemulticast MANETs.

In this paper, we study the scaling laws in a cell partitionedMANET with multicast traffic. To start with, we proposea 2-hop relay algorithm without redundancy. This algorithmis a generalized version of the algorithm presented in [1],and corresponds to a decoupled queuing model. Because kdestinations are associated with a source, the delay for apacket is defined as the total time needed to deliver it toall destinations. For a specific packet, we first divide nodes

other than the source into relays and destinations (referred toas non-cooperative mode). In this case, the packet may becarried to the destinations either through the relays or viathe source, but will not be passed from one destination toanother. Once a packet is sent to a relay, the relay will be incharge of delivering it to all its destinations. Otherwise, if thesource encounters a destination before a relay, it will take fullresponsibility of the rest multicast session. The MotionCastdelay and capacity are calculated under this model.

Then, we loose the constraints of our initial model bypermitting information dissemination among destinations (co-operative mode). In this scheme, we do not discriminatedestinations against the remnant nodes except the source. Wedefine the first node that a source meets as the “designatedrelay”, which in fact may possible be an intended destination.Likewise, the designated relay should carry the packet fromthe source until it delivers this packet to all the destinationsthat have not received the message. Notice that only one relayis associated to a specific packet in the 2-hop relay algorithm,and therefore after a relay is designated other destinations willmerely act as receivers for the packet and do not help transmitthe packet to other nodes. Quite counter-intuitively, we findthat there would be no gain in performance for the cooperativescheme comparing to the non-cooperative one from an ordersense.

Next, we employ redundant packets transmissions to reducethe delay. In a 2-hop relay strategy with redundancy, a sourcesends a packet to multiple relays before all the destinationsreceive the packet, which increases the chance that a desti-nation meets some of the relays at the expense of reducedcapacity. If each timeslot only one transmission from a senderto a receiver is permitted in a cell, we show that the expecteddelay in the network is no less than Ω(

√n log k). Besides,

delay of O(√n log k) is achievable with per-node capacity of

Ω(1/k√n log k).

The main results of this paper are summarized as follows.For 2-hop relay algorithm without redundancy, the capacityfor MotionCast is Θ(1/k) with average delay of Θ(n log k).Notice that the per-node capacity is better than the results ofstatic multicast scenario in [3] as long as k is strictly lessthan n in an order sense, i.e., k = o(n). For 2-hop relayalgorithm with redundancy, the capacity is Ω(1/k

√n log k)

with the delay scaling as Θ(√n log k). Thus, delay and

capacity tradeoffs emerge between these two algorithms, i.e.,we can utilize redundant packets transmissions to reduce delaybut the capacity will also decrease. The tradeoff obtained byus is better than that of directly extending the tradeoff forunicast to multicast. We have also studied the fundamentaldelay-capacity tradeoff for MotionCast and shown that thefundamental tradeoff ratio is delay/capacity ≥ Ω(n log k).

The rest of the paper is organized as follows. In Section II,we describe the network model. In Section III, we introducethe 2-hop relay algorithm without redundancy. In Section IV,the 2-hop relay algorithm with redundancy is presented. InSection V, we discuss the results and figure out the funda-mental tradeoff for multicast. Finally, we conclude in SectionVI.

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II. NETWORK MODEL

(a) Network model. (b) Traffic patternFig. 1. A cell partitioned MANET model with c cells and n mobile nodesunder multicast traffic pattern.

Cell Partitioned Network Model: The system model is basedon the cell partitioned network model exploited in [1] and[18]. Suppose the network is an unit square and there are nmobile nodes in it. Then, we divide it into c non-overlappingcells with equal size as depicted in Figure 1. We assumenodes can communicate with each other only when theyare within a same cell (to locate the nodes, please refer to[17] and the references therein), and to avoid interferencedifferent frequencies are employed among the neighboringcells.1 Additionally, to bound the interference inside each cell,we assume that the number of the cells is on the same orderas that of the nodes throughout this paper. Thus, node per celldensity d = n/c scales as Θ(1)2.

Mobility Model: Dividing time into constant duration slots,we adopt the following ideal i.i.d. mobility to model the some-times drastic topology changes in MANETs and investigatetheir impact. The initial position of each node is equallylikely to be any of the c cells independent of others. Andat the beginning of each time slot, nodes randomly chooseand move to a new cell i.i.d. over all cells in the network.Although the ideal i.i.d. mobility model may appear to be aoversimplification, it has been widely adopted in the literaturebecause of its mathematical tractability, which could providemeaningful bounds on performance. Note that the i.i.d. modelalso characterizes the maximum degree of mobility. With thehelp of mobility, packets can be carried by the nodes untilthey reach the destinations.

Traffic Pattern: We first define the source-destination re-lationships before the transmissions start. In particular, weassume the number of users n is divisible by k+1 and numberall the nodes from 1 to n. We uniformly and randomly dividethe network into different groups with each of them havingk+1 nodes. And assume packets from each node i in a specificgroup must be delivered to all the other nodes within the group.And nodes not belonging to the group can serve as relays.Hence each node i is a source node associated with k randomlyand independently chosen destination nodes D1, D2, . . . , Dk

over all the other nodes in the network. The relationships do

1It is clear that only four frequencies are enough for the whole network.2Theorem 3 and Theorem 4 will show this assumption does not vitiate

our result and can lead us to design a more simple and practical schedulingalgorithm with the purpose to achieve a good tradeoff between throughputand delay.

not change as nodes move around. Then, the sources willcommunicate data to their k destinations respectively througha common wireless channel.

Definition of Capacity: First, we define stability of the net-work. Packets are assumed to arrive at node i with probabilityλi during each slot, i.e. as a Bernoulli process of arrival rateλi packets/slot. For the fixed λi rates, the network is stable ifthere exists a scheduling algorithm so that the queue in eachnode does not increase to infinity as time goes to infinity. Thus,the per-node capacity of the network is the maximum rate λthat the network can stably support. Note that sometimes theper-node capacity is called capacity for brief.

Definition of Delay: The delay for a packet is defined as thetime it takes the packet to reach all its k destinations after itarrives at the source. The total network delay is the expectationof the average delay over all packets and all random networkconfigurations in the long term.

Definition of Redundancy: At each timeslot, if more thanone nodes are performing as relays for a packet, we saythere is redundancy in the network. Furthermore, we saythe corresponding scheduling scheme is with redundancy orredundant. Otherwise, it is without redundancy.

Definition of Cooperative: We adopt the term “cooperative”here to refer a destination can relay a packet from the source toother destinations. Otherwise, the destinations merely acceptpackets destined for them, but do not forward to others, whichis called non-cooperative mode.

Notations: In our work, we adopt the following widely usedorder notations in a sense of probability. We say that an eventoccurs with high probability (w.h.p.), if its probability tendsto 1 as n goes to infinity. Given two functions f(n) and g(n),we say that f(n) = O(g(n)) w.h.p., if there exist a constantc such that

limn→∞

P (f(n) ≤ cg(n)) = 1. (1)

If the above sign of inequality is strict, we denote f(n) =o(g(n)). Moreover, we say that f(n) = Ω(g(n)) w.h.p., ifg(n) = O(f(n)) w.h.p.. If both f(n) = Ω(g(n)) and f(n) =O(g(n)) w.h.p., then we say that f(n) = Θ(g(n)) w.h.p..

III. DELAY AND CAPACITY IN THE 2-HOP RELAYALGORITHM WITHOUT REDUNDANCY

In this section, we propose 2-hop relay algorithms withoutredundancy and compute the achievable delay and capacityboth under non-cooperative mode and cooperative mode. Then,we explore the maximum capacity and the minimum delay inthese situations.

A. Under non-cooperative mode

In this subsection, we describe a 2-hop relay algorithmwithout redundancy. Usually, a source sends a packet to oneof the relays, then the relay will distribute the packet to allits destinations. While as an initial step, we consider thenon-cooperative mode, which means a destination can not bea relay.2-hop Relay Algorithm Without Redundancy I: During atimeslot, for a cell with at least two nodes:

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Fig. 2. A decoupled queuing model of the network as seen by the packetstransmitted from a single source to multiple destinations.

1) If there exists a source-destination pair within the cell,randomly select such a pair uniformly over all possiblepairs in the cell. If the source has a new packet in thebuffer intended for the destination, transmit. If all itsdestinations have received this packet,3 then it will deletethe packet from the buffer. Otherwise, stay idle.

2) If there is no such pair, randomly assign a node assender and independently choose another node in thecell as receiver. With equal probability, choose from thefollowing two options4:

• Source-to-Relay Transmission: If the sender has anew packet one that has never been transmittedbefore, send the packet to the receiver and deleteit from the buffer. Otherwise, stay idle.

• Relay-to-Destination Transmission: If the sender hasa new packet from other node destined for thereceiver, transmit. If all the destinations who wantto get this packet have received it, it will be droppedfrom the buffer in the sender. Otherwise, stay idle.

Intuitively, since there are no redundant transmissions andthe cell partition with constant density scheme guaranteesmaximal spatial reuse, the algorithm could achieve maximalthroughput. The only reason that a constant throughput cannotbe achieved is that a single packet needs to be transmittedrepetitively for about k times to different destinations, andtherefore a Θ(1/k) throughput is feasible. Considering delay,it is intuitive for us to loosely model the network as a queueingsystem such that every source destination pair corresponds toa M/M/1 queue. The service time for a single packet, whichfollows exponential distribution, has an expectation of Θ(n),i.e., the average waiting time that two specific nodes meet.Then the total delay for a complete multicast session willroughly equals the maximum of k such i.i.d. random delays,and turns out to be Θ(n log k). In below we formally derivethe performance of the above algorithm.

The algorithm has an advanced decoupling feature betweenall n multicast sessions, as illustrated in Figure 2, where nodesare divided into destinations and relays for the packets from a

3We assume that nodes can aware this from the control information passedover a reserved bandwidth channel.

4Note that because of the traffic pattern we assume and the probabilitiesof source-destination and source-relay (or relay-destination) transmissions wecalculate, source-destination transmission does not have priority over non-source-destination transmission, i.e., they happen independently.

single source, and the packets transmissions for other sourcesare modeled just as random ON/OFF service opportunities.

Let p represent the probability of finding at least two nodesin a particular cell, and q represent the probability of findinga source-destination pair within a cell. From Appendix I, weobtain that

p = 1−(1− 1

c

)n − n

c

(1− 1

c

)n−1 (2)

q = 1−[k + 1

c

(1− 1

c

)k+

(1− 1

c

)k+1] n

k+1

(3)

When n tends to infinity, it follows p → 1− (d+ 1)e−d andq → 1−e−

kk+1d(1+ k

c )n

k+1 . Thus, if k = o(n), q → 0;5 else ifk = Θ(n), q → 1−(d+1)e−d. Intuitively, when k approachesthe same order as n, the multicast will reduces to a broadcastand the events corresponding to p and q will gradually becomeidentical. Then, we have the following theorem.

Theorem 1: Consider a cell-partitioned network (with nnodes and c cells) under the 2-hop relay algorithm withoutredundancy I, and assume that nodes change cells i.i.d. anduniformly over each cell every timeslot. If the exogenous inputstream to node i which makes the network stable is a Bernoullistream of rate λi = O(µ/k) and k = o(n), then the averagedelay Wi for the traffic of node i satisfies

EWi = O(n log k) (4)

where µ = p+q2d .

Proof: A decoupled view of the network as seen bya single source i is shown in Figure 2. Due to the i.i.d.mobility model, the source user can be represented as aBernoulli/Bernoulli queue, where in every timeslot a newpacket arrives with probability λi, and a service opportunityarises with some fixed probability µ when the packet is handedover a relay or transmitted to a destination. We first show thatthe expression µ = p+q

2d still holds.The Bernoulli nature of the server process implies that the

transmission probability µ is equal to the time average rateof transmission opportunities of source i.6 Let r1 representthe rate at which the source is scheduled to transmit directlyto one of the destinations, and r2 represent the rate at whichit is scheduled to transmit to one of its relays. The same asµ, r1 equals to the probability that the source is scheduledto transmit directly to the destination and r2 equals to theprobability that the source is scheduled to transmit to one ofits relay users. Then, we have µ = r1 + r2. Since the relayalgorithm schedules transmissions into and out of the relaynodes with equal probability, hence r2 is also equal to therate at which the relay nodes are scheduled to transmit tothe destinations. Every timeslot, the total rate of transmissionopportunities over the network is thus n(r1+2r2). Meanwhile,

5because when n → ∞ and k = o(n), q → 1− e− k

k+1d(1 + k

c)

nk+1 →

1− e−ded = 06A transmission opportunity arises when user is selected to transmit to an-

other user, and corresponds to a service opportunity in the Bernoulli/Bernoulliqueue. Such opportunities arise with probability µ every timeslot, independentof whether or not there is a packet waiting in the queue.

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a transmission opportunity occurs in any given cell withprobability p, hence,

cp = n(r1 + 2r2) (5)

Recall that q is the probability that a given cell containsa source-destination pair. Since the algorithm schedules thesingle-hop source-to-destination transmissions whenever pos-sible, the rate r1 satisfies

cq = nr1 (6)

It follows from (8) and (6) that r1 = qd , r2 = p−q

2d . The totalrate of transmissions out of the source node is thus given byµ = r1 + r2 = p+q

2d .Next, we compute the average delay for the traffic of

node i. There are two possible routings from a source toits destinations: one is the 2-hop path along “source-relay-destinations”, the other is the single-hop path from source todestinations directly. As for the first routing, packet delay iscomposed of the waiting time at source and relay. In this case,since the source can be viewed as a Bernoulli/Bernoulli queuewith input rate λi and service rate µ , it has an expectednumber of occupancy packets given by Ls =

ρ(1−λi)1−ρ , where

ρ= λi

µ . From Little’s theorem, the average waiting time inthe source is EWs = Ls

λi= 1−λi

µ−λi. Besides, this queue is

reversible, so the output process is also a Bernoulli stream ofrate λi.

Notice that our traffic pattern has defined every disjoint k+1nodes as a group and every node in this group is the source forthe other k nodes. From a more delicate point of view, a packetdelivered from a source to a relay contains not only necessarypayload, but also redundant data in its head which tells therelay which k destinations this packet should be transmittedto, shown in Figure 3-(a). Based on this information, therelay can make k similar copies, each of which contains lessredundant data in its head just indicating its own correspondingdestination. And also since a node can act as a relay to transmitpackets to other n− k − 1 destinations, we model a relay asa node that has n − k − 1 parallel subqueues (each of thembuffers the packets intended for a certain destination), shown inFigure 3-(b). Next, we will compute the input rate and outputrate of a subqueue.

A given packet from a source is transmitted to the first relaynode with probability pi = r2

µ(n−k−1) and rate λr = λipi(because with probability r2

µ the packet is delivered to a relay,and each of the n − k − 1 relay nodes are equally likely).Since there are k sources for each subqueue, every timeslot,a subqueue in this relay receives a packet with probability1 − (1 − pi)

k which can be expressed as kpi + o(kpi). Thelatter one won’t influence our results, so we omit it. Hence,the input rate of a subqueue is kλr. On the other hand,the subqueue in the relay node is scheduled for a potentialpacket transmission to a destination node with probabilityµr = r2

n−k−1 (because when it acts as a relay, it can transmitpackets to n−k−1 destinations except the source of the givenpacket and itself with equal probability). Notice that packetarrivals and transmission opportunities in a subqueue of therelay node are mutually exclusive events. It follows that the

discrete time Markov chain for queue occupancy in the relaynode can be written as a simple birth-death chain which isidentical to a continuous time M/M/1 queue with input ratekλr and service rate µr. Each destination i (1 ≤ i ≤ k)obtains the packet from the relay through such a queue, thusthe waiting time for it is an exponential distributed variablewith expectation of EW i

rd = 1/(µr − kλr).

Fig. 3. A more delicate view of a relay-destinations transmission. (a) Asubqueue in a relay receives packets from a source with rate λr . Here differentpackets are in different colors and each has a similar form as the extendedred one. (b) Each of a relay can make a packet into k similar copies, and canbe modeled as a node having n− k− 1 parallel subqueues buffering packetsintended for different destinations. In special, k subqueues associated with kdestinations of the current source are depicted in red in the figure.

The resulting waiting time Wrd for multicast is deter-mined by the maximum value among all the waiting timesW 1

rd,W2rd, ...,W

krd of these k destinations. Due to the fact that

1) all k destinations share the same arrival process and 2) theinterference constraint that a relay node can communicate withonly one destination in one time slot, W k

rd are correlatedover k. However, we can construct a set of dual random vari-ables W ′i

rd such that they are i.i.d.. They provide a slightlyalternated view of the queueing system depicted in Fig. 3, withmulti-destination reception enabled, i.e., if a source encountersmore than one destinations, the packet will be transmitted toall of them. As well, we hypothesize that the arrival processesof different destinations are independent, each with rate kλr.In the following we shall show W k

rd = W ′krd w.h.p.

Conditioning on the event that the relay encounters one ormore destinations, and denotes φ1, φ>1 as the probabilitythat exactly one or more than one destinations are reached,respectively. It is clear that φ1 = ω(φ>1) if k = o(n) andφ1 = Θ(φ>1) if k = Θ(n). Therefore µr = Θ(µ′

r), whichindicates that multi-reception does not affect service processin order sense. Similarly, also notice that the input process fora subqueue in the two queueing systems, though constructedon independent probability spaces, are the same kλr, and does

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not rely on network scale n. Due to the nature of M/M/1 queue,the waiting time W i

rd or W ′ird only depends on the input and

the service process, it is clear that W ird = Θ(W ′i

rd) w.h.p. Inother words, there exists constant ci such that W i

rd = ciW′ird

w.h.p. By Lemma 2 (see the proof in Appendix II), we obtainthat EWrd = Θ(EW ′

rd) = Θ(log k/(µr − kλr)). Thus,if the packet is delivered through the path “source-relay-destinations”, the average delay is EWs+ EWrd.

While if the packet is directly sent to the destinations by thesource, it will wait at the source for a time Ws first, then thesource distributes this packet to the remnant k−1 destinations.At this time, the source can be treated as a node having kparallel M/M/1 sub-buffers corresponding to its k destinationssimilarly. The source will copy this packet into k − 1 similarduplicates and add them into respective sub-buffers associatedwith the remnant k − 1 destinations. Also, at this time, thesource can be treated as a continuous time M/M/1 queue. Sincethe probability that the source need send packets directly todestinations is r1

µ , the incoming data rate is thus λir1µ for such

a queue. Meanwhile, the service rate at each equals to thetransmission rate r1

k between a source-destination pair. Hence,the expectation of the waiting time for each one of the k − 1destinations through such a queue is 1/( r1k − λir1

µ ). And byLemma 2 and the same method in the above calculation ofthe delay through 2-hop route, we have the expect waitingtime for the packet to reach all k − 1 remnant destinations isEWsd = Θ(log(k − 1)/( r1k − λir1

µ )).Finally, by weighting the delay occurs in both two routings,

we achieve the total network delay is

EWi =r1µ

(EWs+ EWsd

)+

r2µ

(EWs+ EWrd

)=

r2µ

(1− λi

µ− λi+Θ

(log k

r2n−k−1 − kλir2

µ(n−k−1)

))+r1µ

(1− λi

µ− λi+Θ

(log(k − 1)r1k − λir1

µ

))=

1− λi

µ− λi+Θ

((n− k − 1) log k

µ− kλi+

k log(k − 1)

µ− kλi

)(7)

To ensure the stability of the network, the incoming rateshould be less than the service rate at any stage of the network.Thus,

µ− λi > 0r1k − λir1

µ > 0r2

n−k−1 − kλir2µ(n−k−1) > 0

i.e., λi < µ/k. Besides, the total network delay is in the orderof O(n log k) for a fixed traffic loading value ρr = kλi

µ ateach relay and source.

From the above discussion, we conclude the theorem.

B. Under cooperative mode

In the above subsection, we propose a 2-hop relay algorithmwithout redundancy obtaining per-node capacity Ω(1/k) withdelay O(n log k). In this subsection we bring forward a moregeneral algorithm which does not discriminate destinationsand the nodes other than the source, i.e., under cooperative

mode. This algorithm achieves the same performance as thefirst one. Since the second algorithm is more simple thanthe first one, we adopt this algorithm and refer it as 2-hoprelay algorithm without redundancy for brief in the rest ofthe paper. It is described as follows.

2-hop Relay Algorithm Without Redundancy II: For eachcell with at least two nodes in a timeslot, a random senderand a random receiver are picked with uniform probabilityover all nodes in the cell. With equal probability, the senderis scheduled to operate in the two options below:

1) Source-to-Relay Transmission: If the sender has a newpacket one that has never been transmitted before, sendthe packet to the receiver and delete it from the buffer.Otherwise, stay idle.

2) Relay-to-Destination Transmission: If the sender haspackets received from other nodes which are destinedfor the receiver and have not been transmitted to thereceiver yet, then choose the latest one, transmit. Ifall the destinations who want to get this packet havereceived it, it will be dropped from the buffer in thesender. Otherwise, stay idle.

The algorithm simply designates the first node a sourcemeets as the relay, no matter if it is a destination. Thusaccording to the scheduling scheme, all the packets willbe delivered along a 2-hop path “source-relay-destinations”.Then, we summarize the next theorem.

Theorem 2: Consider the same assumptions for the networkas Theorem 1, under the 2-hop relay algorithm without redun-dancy II. The resulting per-node capacity and the average delayare Ω(1/k) and O(n log k), respectively, for all k ≤ n.

Proof: Since, all the packets will be delivered alonga 2-hop “source-relay-destinations” path, by using the sameanalytical method, we can know r1 = 0 and µ = r2.Meanwhile, a transmission opportunity occurs in any givencell with probability p, hence,

cp = 2nµ (8)

It follows from (8) that µ = r2 = p2d .

Thus following the same analytical steps as Theorem 1when k is strictly less than n in an order sense, we canknow that packet delay is composed of the waiting time atsource and relay. And since the source can be viewed as aBernoulli/Bernoulli queue with input rate λi and service rateµ, the average waiting time in the source is EWs = 1−λi

µ−λi.

Besides, this queue is reversible, so the output process is alsoa Bernoulli stream of rate λi.

A given packet from this output process is transmitted to thefirst relay node with probability 1

n−1 . Hence, every timeslot,this relay independently receives a packet with probabilityλr = λi

n−1 . On the other hand, the relay node is scheduledfor a potential packet transmission to a destination node withprobability µr = µ

n−2 (because when it acts as a relay, it cantransmit packets to n−2 destinations except the source of thegiven packet and itself with equal probability). Notice thatpacket arrivals and transmission opportunities are mutuallyexclusive events in the relay node.

7

When taking the 2-hop algorithm without redundancy II,the first node a source meets as the relay, no matter if it is adestination. The difference is that if the relay is a destinationnode, it need only relay the packet to the rest k−1 destinations;otherwise, it need relay the packet to all k destinations. Sincewe focus on the performance in an order sense, we omit thisdifference between these two cases and assume a relay isresponsible for delivering a new packet to its corresponding kdestinations for simplicity.

Thus at this time, when receiving a new packet from thesource, the relay node will make it into k similar duplicates.Thus, a relay can be viewed as a M/M/1 queue with input ratekλr and service rate µr. Hence the expectation of the waitingtime of each destination is 1/( µ

n−2 −kλi

n−1 ). And by lemma 2,we have the expect waiting time for the packet to reach all kdestinations is EWsd = Θ(log k/( µ

n−2 − kλi

n−1 )).Finally, we achieve the total network delay is

EWi=EWs+ EWsd

=1− λi

µ− λi+Θ

(log k

µn−2 − kλi

(n−1)

)(9)

Looking upon the asymptotic behaviors of the network delaywhen k, n → ∞, we have µ = r2 → 1−(d+1)e−d

2d , To ensurethe stability of the network, the incoming rate should be lessthan the service rate at any stage of the network. Thus,

µ− λi > 0µ

n−2 − kλi

n−1 > 0

i.e., λi <(n−1)µ(n−2)k → µ/k (n → ∞). Besides, the total network

delay is governed by (9), which is on the order of O(n log k)

for a fixed traffic loading value ρr = λi(n−2)n−1 at each relay.

From the above discussion, we conclude the theorem.

C. Maximum capacity and minimum delay

Although we have constructed the achievable delay andcapacity if no redundancy is used, open questions still leavefor the maximum capacity and the minimum delay of thisnetwork. We address these problems here by presenting thefollowing theorems.

Theorem 3: The multicast capacity of a cell partitionednetwork is O( 1

dk ) if only a pair of sender and receiver isactive in each cell per timeslot. In particular, if d = Θ(1), themulticast capacity is O(1/k).

Proof: We use hop argument to prove this result. Since forany interval [0,T], the less hops the source need send a packetto its k destinations, the more capacity it can achieve. So weassume a packet is delivered directly from a source to oneof its destinations via the 1-hop route “Source-destination”.Let Xs(T ) represent the total number of packets transferredover the network from sources to destinations via 1-hop routeduring the interval [0,T]. Fix ϵ > 0. For network stability, theremust be arbitrarily large values T such that the sum output rateis within ϵ of the total input rate

Xs(T )

T≥ nkλ− ϵ (10)

If this were not the case, the total number of packets in thenetwork would grow to infinity and hence the network wouldbe unstable. Since every transmission just needs one hop, thetotal number of packet transmissions in the network during thefirst T slots is also Xs(T ). This value must be less than orequal to the total number of transmission opportunities Y (T ),and hence,

Xs(T ) ≤ Y (T ) (11)

where Y (T ) represents the total number of cells containingat least two users in a particular timeslot, summed over alltimeslots 1, 2, . . . , T . By the law of large numbers, it is clearthat 1

T Y (T ) → cp as T → ∞, where p is the steady-stateprobability that there are two or more users within a particularcell, and is given by (2).

From (10) and (11), it follows that

λ ≤Y (T )T + ϵ

kn=

p

kd+

ϵ

kn(12)

Notice that k=O(n), thus we have λ=O( 1dk ). And if d=Θ(1),

λ=O(1/k).Theorem 4: Algorithms permitting at most one transmis-

sion in a cell at each timeslot, which do not use redundancycannot achieve an average delay of O(n log k

d ). In particular, ifd = Θ(1), EW ′

min = Θ(n log k).Proof: The minimum delay of any packet is calculated by

considering the situation where the network is empty and node1 sends a single packet to k destinations.7 Since relaying thepacket can not help reduce delay, it can be treated as havingno relay at all. Denote p′ and W ′

min as the chance that node1 meets (i.e., two nodes move into a same cell) one of thedestinations in a timeslot and the minimum amount of timeit takes the source to meet all the destinations, respectively.We have that p′ = 1/c. Since W ′

min = i means that at the(i − 1)th timeslot the source has met k − 1 destinations andat the ith timeslot it meets the last one, thus the probabilityW ′

min = i can be written as

PW ′min = i = kp′

[(1− p′)i−1 −

(k − 1

1

)(1− 2p′)i−1

+

(k − 2

2

)(1− 3p′)i−1 − · · ·

](13)

Therein the factor kp′ denotes that the last destination D′k

meets by the source can be any one of the k destinations.The first term in the latter factor infers that D′

k has not beenmet in the former i− 1 timeslots. Because the first term alsoincludes the probability that the source has not met D′

k and anyone of the other nodes from D′

1 to D′k−1, this value should be

subtracted from the first term, so the second term attached andsimilarly we have the following terms. Hence, the expectation

7By saying the network is empty we mean only node 1 has packets to sendand other nodes have no packet and stay idle.

8

of EW ′min is

EW ′min

= kp′+∞∑i=1

i

[(1− p′)i−1 −

(k − 1

1

)(1− 2p′)i−1

+

(k − 2

2

)(1− 3p′)i−1 − · · ·

]= kp′

[+∞∑i=1

i(1− p′)i−1

−(k − 1

1

)+∞∑i=1

i(1− 2p′)i−1

+

(k − 1

2

)+∞∑i=1

i(1− 3p′)i−1 − · · ·]

= kp′[1

p′2−(k − 1

1

)1

(2p′)2+

(k − 1

2

)1

(3p′)2− · · ·

]=

k

p′

[1− 1

22

(k − 1

1

)+

1

32

(k − 1

2

)− · · ·

]=log k

p′(14)

wherein Lemma 1 and the following identical relation forany | x |< 1 are exploited

+∞∑i=1

ixi−1 = (

+∞∑i=1

xi)′ =1

(1− x)2.

Finally, notice that 1/p′ = c = nd , we obtain that EW ′

min =

Θ(n log kd ). In particular, if d = Θ(1), EW ′

min =Θ(n log k). Since at any timeslot, if there are more than onedestinations in a same cell as the source, only one destinationcould be selected as the receiver, the actual delay EWminfor the packet to be delivered to all the destinations will belarger or equal than EW ′

min, which points out the theorem.

Combining these results with the delay and capacityachieved by the 2-hop relay algorithm without redundancy,we find the exact order of the delay and capacity are Θ(1/k)and Θ(n log k), respectively.

IV. DELAY AND CAPACITY IN THE 2-HOP RELAYALGORITHM WITH REDUNDANCY

In this section, we adopt redundancy to improve delay. Theidea originates from a basic notion that if we send a particularpacket to many nodes of the network, the chances that somenode holding the packet reaches a destination will increase.This approach is also implemented in [1] and [19]. We firstconsider the minimum delay of 2-hop relay algorithms withredundancy. Then, we design a protocol using redundancy toachieve the minimum delay.

A. Lower bound of delay

In this subsection, we obtain lower bound of delay if onlyone transmission from a sender to a receiver is permitted in acell in the below Theorem.

Theorem 5: There is no 2-hop algorithm with redundancycan provide an average delay lower than O(

√n log k), if only

one transmission from a sender to a receiver is permitted in acell.

Proof: To prove this result, we consider an ideal situationwhere the network is empty and only node 1 sends a singlepacket to k destinations. Clearly the optimal scheme for thesource is to send duplicate versions of the packet to new relayswhenever possible, and if there is a destination within thesame cell as the source, it will choose a destination as relay.And for a duplicate-carrying relay, it sends the packet to berelayed to the destinations as soon as it enters the same cellas a destination. Denote TN as the time required to reach thek destinations under this optimal strategy for sending a singlepacket.

In order to avoid the interdependency of the probability thatdifferent destinations obtain a packet from the source or therelay nodes, we additionally assume that all the destinationswithin a same cell as the source or a relay node can obtain thepacket during the transmission, which is referred to as a multi-destination reception style. Note that our assumption differsfrom the multi-user reception ([1]) in that usually each cell ispermitted to have a single reception except there are more thanone intended destinations within a the cell, while [1] allowsa transmitted packet to be received by all other users in thesame cell as the transmitter. Denote T ′

N as the time to reachthe k destinations when we add the multi-destination receptionassumption. It is easy to see that ETN ≥ ET ′

N.Then, let Kt represent the total number of nodes that act

as intermediate relays (including the source) at the beginningof slot t. Because the limitation of 2 hop transmission, anew relay can only be generated by the source. Hence, everytimeslot at most one node can be a new relay. Thus, we havefor all t ≥ 1:

Kt ≤ t (15)

Observe that during slots 1, 2, . . . , t there are at most Kt

nodes holding the packet and willing to help forward it to thedestinations. Hence, during this period, the probability that adestination meets at least a relay is at most 1 − (1 − 1

c )tKt .

Moreover, note that since we take the multi-destination recep-tion style, the events that every timeslot different destinationsmeet the source or a relay node are independent. Thus, theprobability that all the k destinations meet at least a relayduring this period 1, 2, . . . , t is at most

[1− (1− 1

c )tKt

]k.

We thus have

PT ′N > t ≥ 1−

[1− (1− 1

c)tKt

]k≥ 1−

[1− (1− d

n)t

2]k= 1− (1− e−

dn t2)k (n → ∞) (16)

Choosing t =√n log k/d and letting k → ∞, it yields that

PT ′N > t ≥ 1− (1− e− log k)k

= 1− (1− 1

k)k

= 1− e−1 (17)

9

Thus:

ETN ≥ ET ′N ≥ ET ′

N | T ′N > tPT ′

N > t≥ (1− e−1)

√n log k/d (18)

as k, n → ∞. From (18), we prove the theorem.

B. Scheduling scheme

In the above subsection, we consider the minimum delay ofthe network if we implement redundant packets transmissions.In this subsection, for acquiring the upper bound of thedelay, we propose a 2-hop relay algorithm with redundancyto achieve the minimum delay.

Assume each packet is labeled with a Sender Number SN,and a request number RN is delivered by the destinationto the transmitter just before transmission. In the followingalgorithm, we let each packet be retransmitted

√n log k times

to distinct relay nodes.Denoting redundancy as A, to better understand the reason

we let A = Θ(√n log k), it is intuitive to simplify a multicast

session into two phases: duplication of relays and delivery todestinations, and assume they happen in sequence. Clearlythe duration of the first phase is Θ(A). Consider the durationof the second phase, again it is convenient for us to looselymodel the network as a queueing system such that everysource destination pair corresponds to a M/M/1 queue, wherethe exponentially distributed service time has the averageΘ(n/A), i.e., the expected time that a generic relay meetsa specific destination. The overall delay for a multicastsession would then be Θ(n log k/A). To minimize delay,clearly we should let Θ(A) = Θ(n log k/A), which yieldsA = Θ(

√n log k). Interestingly, this is exactly the lower

bound of delay established in Theorem 5.

2-hop Relay Algorithm With Redundancy: In every cellwith at least two nodes, randomly select a sender and areceiver with uniform probability over all nodes in the cell.With equal probability, the sender is scheduled to operated ineither “source-to-relay” transmission, or “relay-to-destination”transmission, as described below:

1) Source-to-Relay Transmission: The sender transmitspacket SN , and does so upon every transmission op-portunity until

√n log k duplicates have been deliv-

ered to distinct relay nodes (possible be some of thedestinations), or until the k destinations have entirelyobtained SN . After such a time, the sender number isincremented to SN + 1. If the sender does not have anew packet to send, stay idle.

2) Relay-to-Destination Transmission: When a node isscheduled to transmit a relay packet to its destinations,the following handshake is taken place:

• The receiver delivers its current RN number for thepacket it desires.

• The transmitter sends packet RN to the receiver. Ifthe transmitter does not have the requested packetRN , it stays idle for that slot.

• If all k destinations have already received RN , thetransmitter will delete the packet which has SNnumber equal to RN in its buffer.

Next, we present the performance of this algorithm.Theorem 6: The 2-hop relay algorithm with redundancy

achieves the O(√n log k) delay bound, with a per-node ca-

pacity of Ω(1/(k√n log k)).

Proof: For the purpose of proving this theorem, weconsider an extreme case of the packets transmissions. Notethat when a new packet arrives at the head of its source queue,the time required for the packet to reach its k destinationsis at most TN = T1 + T2, where T1 represents the timerequired for the source to distribute

√n log k duplicates of

the packet, and T2 represents the time required to reach allthe k destinations given that

√n log k relay nodes hold the

packet. The reason behind this claim is the sub-memorylessproperty of the random variable TN ([1]), which means theresidual time of TN given that a certain number of slots havealready passed before it expires is stochastically less than theoriginal time TN .

Now we bound the expectations of T1 and T28 by taking

into account the collisions among the multiple sessions.The ET1 bound: For the duration of T1, there are at

least n −√n log k nodes who do not have the packet. Let

G represent the event that every timeslot at least one of thesenodes visits the cell of the source. Hence the probability ofevent G is at least 1 − (1 − 1

c )n−

√n log k. Given this event,

the probability that the source is chosen by the 2-hop relayalgorithm with redundancy to transmit is expressed by theproduct α1α2, representing probabilities for the followingconditionally independent events given event G: under thecondition that at last one of these n −

√n log k nodes visits

the cell of the source, α1 is the probability that the source isselected from all other nodes in the cell to be the transmitter,and α2 represents the probability that this source is chosen tooperate in “source-to-relay” transmission. From Lemma 6 in[1], we have α1 ≥ 1/(2 + d).

The probability α2 that the source operates in “source-to-relay” transmission is 1/2. Thus, every timeslot during theinterval T1, the source delivers a duplicate packet to a newnode with probability of at least ϕ, where

ϕ ≥ (1− (1− 1

c)n−

√n log k)

1

2(2 + d)→ 1− e−d

4 + 2d

The average time until a duplicate is transmitted to a newnode is thus a geometric variable with mean less than orequal to 1/ϕ. It is possible that two or more duplicatesare delivered in a single timeslot, if we enable multi-userreception. However, in the worst case,

√n log k of these times

are required, so that the average time ET1 is upper boundedby

√n log k/ϕ.

The ET2 bound: To prove the bound on ET2, let Hrepresent the event that every timeslot in which there are atleast

√n log k nodes possess the duplicates of the packet, and

note that event H is already a certainty with a probability of

8Note that the bounds on ET1 and ET2 are computed under suitablylarge n values.

10

1. The probability that one of these nodes transmits the packetto one of the destinations is given by the chain of probabilitiesθ0θ1θ2θ3. The θi values represent probabilities for the follow-ing conditionally independent events given event H: Under thecondition that there are at least

√n log k nodes possess the

duplicates of the packet in every timeslot, θ0 represents theprobability that there is at least one other node in the samecell as the destination (θ0 = 1 − (1 − 1

c )n−1 → 1 − e−d),

θ1 represents the probability that the destination is selectedas the receiver (similar to α1, we have θ1 ≥ 1/(2 + d)), θ2represents the probability that the sender is operates in “relay-to-destination” transmission (θ2 = 1/2), and θ3 represents theprobability that the sender is one of the

√n log k nodes who

possess a duplicate of the packet intended for the destination(where θ3 =

√n log k/(n − 1) ≥

√log k/n). Thus, every

timeslot, the probability that each destination receives a de-sired packet is at least 1−e−d

4+2d

√log k/n. Similar to Theorem

4, since T2 completes when all k destinations receive thepacket, the value of ET2 is thus less than or equal to thelog k times of the inverse of that quantity. Hence, we haveET2 ≤ 4+2d

1−e−d

√n log k.

Finally according to Lemma 2 in [1], we bound the total net-work delay EW = O(

√n log k), and obtain the achievable

per-node capacity under this algorithm is Ω(1/k√n log k).

V. FUNDAMENTAL DELAY AND CAPACITY TRADEOFF

In Section III and Section IV, we present algorithms bothwithout and with redundancy to fulfill the task of MotionCast.In this section, we first draw a comparison of the delayand capacity with the former results. Then we derive thefundamental delay and capacity tradeoff for multicast.

A. Results comparison

Recall that the multi-hop algorithm in [1] is based onflooding the message among the network. It could also servefor multicast. The delay and capacity tradeoffs in the 2-hoprelay algorithm without and with redundancy, together withmulti-hop relay algorithm with redundancy can be summarizedas the following table.

TABLE IDELAY AND CAPACITY TRADEOFFS IN DIFFERENT ALGORITHMS

scheme capacity delay2-hop relay w.o. redund Θ( 1

k) Θ(n log k)

2-hop relay w. redund Ω( 1k√

n log k) Θ(

√n log k)

multi-hop relay w. redund Ω( 1n logn

) Θ(logn)

Compared with the multicast capacity of static networksdeveloped in [3], we find that capacity of the 2-hop relayalgorithm without redundancy is better when k = o(n); other-wise, capacity remains the same as that of static networks, i.e.,mobility cannot increase capacity. Moreover, compared withthe results of unicast in [1], we find that capacity diminishes bya factor of 1/k and 1/k

√log k for the 2-hop relay algorithm

without and with redundancy, respectively; delay increases bya factor of log k and

√log k for the 2-hop relay algorithm

without and with redundancy, respectively. This is because weneed distribute a packet to k destinations during MotionCast.Particularly, if k = Θ(1) we find the results of unicast is aspecial case of our paper.

Furthermore, we see that delay of the 2-hop algorithm withredundancy is better than that of the 2-hop algorithm withoutredundancy, but its capacity is also smaller than that of theno redundancy algorithm when k = o(

√n). This suggests

that redundant packets transmissions can reduce delay at anexpense of the capacity. The ratio between delay and capacitysatisfies delay/rate ≥ O(nk log k) for both of these twoprotocols. However, if we fulfill the job of MotionCast bymultiple unicast from the source to each of the k destinations,we find that capacity will diminish by a factor of 1/k and delaywill increase by a factor of k for both algorithms without andwith redundancy, which infers the fundamental tradeoff forunicast established in [1] becomes delay/rate ≥ O(nk2) inMotionCast. Thus, it turns out our tradeoff is better than thatof directly extending the tradeoff for unicast to multicast.

B. Fundamental delay and capacity tradeoff for multicast

Observing Table 1 we see that the delay-capacity ratiounder these three schemes are Θ(nk log k), O(nk log k),O(n(log n)2) respectively, which lead us to suppose the gen-eral relationship between delay and capacity is that their ratiois larger than O(n log k).

Consider a network with n users, and suppose all usersreceive packets at the same rate λ. A control protocol whichmakes decisions about scheduling, routing, and packet retrans-missions is used to stabilize the network and deliver all packetsto their respective k destinations while maintaining an averagedelay less than some threshold W . We have the followingtheorem.

Theorem 7: A necessary condition for any conceivablerouting and scheduling protocol with k destinations for trans-mitting that stabilizes the network with input rates λ whilemaintaining bounded average delay W is given by:

W ≥ Θ(n log k)λ

1− kλ(19)

which equals to this following expression,λ = O(1/k), W/λ ≥ Θ(n log k)λ = ω(1/k), W ≥ Θ(n log k/k)

(20)

Proof: Suppose the input rate of each of the n sessions isλ, and there exists some stabilizing scheduling strategy whichensures a delay of W . In general, the delay of packets fromindividual sessions could be different, and we define Wi asthe resulting average delay of packets from session i. We thushave:

W =1

n

∑i

Wi (21)

Now we count the number of transmission times for session i.Every time-slot, if this packet or its copies has been transmittedto M different non-destination receivers, the count will beadded by M . We define Ri as the non-destination redundancywhich represents the final number of counting when the

11

packet finally reaches the kth destination and end its task,averaged over all packets from session i. That is, Ri is averagenumber of non-destination transmissions for a packet fromsession i. Note that all packets are eventually received bythe k destinations, so that Ri + k is the actual number oftransmissions for packets from session i, and then the averagenumber of successful packet receptions per time-slot is thusgiven by the quantity λ

∑ni=1(Ri + k). Since each of the n

users can receive at most 1 packet per time-slot, we have:

λ

n∑i=1

(Ri + k) ≤ n (22)

Now consider a single packet P which enters the networkfrom session i. This packet has an average delay of Wi andan average non-destination redundancy of Ri. Let randomvariables Wi and Ri represent the actual delay and non-destination redundancy for this packet. We have:

Wi = EWi|Ri ≤ 2RiPr[Ri ≤ 2Ri] +

EWi|Ri ≥ 2RiPr[Ri ≥ 2Ri]

≥ EWi|Ri ≤ 2RiPr[Ri ≤ 2Ri]

≥ EWi|Ri ≤ 2Ri1

2(23)

where the last inequality follows because Pr[Ri ≤ 2Ri] ≥ 12

for any nonnegative random variable Ri.Consider now a virtual system in which there are 2Ri

users initially holding packet P, and let Zm represent thetime required for one of these users to enter the same cellas the mth destination. Then let Z represent the time requiredfor these users to enter all the k destinations, so we haveZ = maxZ1, Z2, . . . , Zk. Note that the distribution of eachZm is the same as Pr[Zm > w] = (1 − ϕ)[w], in whichϕ = 1− (1− 1

c )2Ri . And thus EZm = 1

ϕ .In order to connect this variable Z to our interest Wi,

we develop another parameter W resti , which represents the

corresponding delay under the restricted scheduling policythat schedules packets as before until either the packet issuccessfully delivered to all k destinations, or the redundancyincreases to 2Ri(where no more redundant transmissions areallowed). Since this modified policy restricts redundancy to atmost 2Ri, the delay W rest

i is stochastically greater than thevariable Z, representing the delay in a virtual system with onlyone packet that is initially held by 2Ri users. In addition, asthe restricted policy is identical to the original policy wheneverRi ≤ 2Ri, hence EWi|Ri ≤ 2Ri = EW rest

i |Ri ≤ 2Ri.Finally, we introduce the last much easier calculated con-

tinuous variable Z, which is also the maximum of severalones - Z = maxZ1, Z2, . . . , Zk. For each of them has thesame distribution as Pr[Zm > w] = e−γw = (1 − ϕ)w ≤(1− ϕ)[w] = Pr[Zm > w], where γ = log 1

1−ϕ .So now we put the relationship among these three variables

clearly as follows:

W resti ≽ Z ≽ Z 9 (24)

9Because Pr[Z > w] = 1− Pr[Z ≤ w] = 1−∏k

m=1 Pr[Zm ≤ w] ≥1−

∏km=1 Pr[Zm ≤ w] = Pr[Z], and according to the definition in [20],

we have that Z is stochastically greater than Z.

Further, although Z and Z defined in our paper area little different from those defined in [1], i.e., Z =maxZ1, Z2, ..., ZK and Z = maxZ1, Z2, ..., Zk, theyalso follow claim 1 and claim 2 in [1]. Thus, we have thisuseful inequality:

EWi|Ri ≤ 2Ri ≥ infΘ

EZ|Θ ≥ infΘ

EZ|Θ (25)

where the conditional expectation is minimized over all con-ceivable events Θ(for Z, while Θ for Z) which occur withprobability greater than or equal to 1/2.

Until now we have to calculate the last value infΘ EZ|Θ.From Lemma 8 in [1], whose result been put as follows:

For any nonnegative random variable X , we have:

infΘ|Pr[Θ]≥ 1

2EX|Θ = EX|X < w2Pr[X < w] +

w(1− 2Pr[X < w]) (26)

Where w is the unique real number such that Pr[X < w] ≤ 12

and Pr[X ≤ w] ≥ 12 .

Note that in the special case when P (x) is continuous atx = w, then Pr[X < w] = Pr[X ≤ w] = 1

2 and hence weget the simpler expression:

infΘ

EZ|Θ = EZ|Z ≤ w (27)

Now recall the distribution expression of Z:

Pr[Z ≤ z] =

k∏m=1

Pr[Zm ≤ z] = (1− e−zγ)k. (28)

Then we get the value of w: w = − 1γ ln[1− ( 12 )

1k ].

Return to our question, we do the calculation as follows:

EZ|Z ≤ w =

∫ w

0x[(1− e−γx)k]

′

xdx

Pr[Z ≤ w]

= 2x(1− e−xγ)k|w0 − 2

∫ w

0

(1− e−xγ)kdx

= w − 2

∫ w

0

k∑i=0

(k

i

)(−1)ie−ixγdx

= w + 2

k∑i=0

(k

i

)(−1)i+1

∫ w

0

e−ixγdx

= −w + 2k∑

i=1

1

iγ

(k

i

)(−1)i[e−iwγ − 1]

, −w + 2X(k) (29)

Note that(ki

)=

(k−1i−1

)+

(k−1i

), we can calculate X(k) by

gradually reducing the variable k, as X(1) can be easilyobtained as X(1) = − 1

γ [e−wγ − 1].

12

X(k)−X(k − 1) =k∑

i=1

1

iγ

(k − 1

i− 1

)(−1)i[e−iwγ − 1]

=1

kγ

k∑i=0

(k

i

)(−1)i[e−iwγ − 1]

=1

kγ[(1− e−wγ)k − (1− 1)k]

=1

2kγ(30)

Wherein 1i

(k−1i−1

)= 1

k

(ki

). And continue this recursion, we

have that:

X(k) = X(1) +1

2γ(1

2+

1

3+ . . .+

1

k)

= − 1

γ[e−wγ − 1] +

1

2γ(1

2+

1

3+ . . .+

1

k)

, − 1

γ[e−wγ − 1] +

1

2γf(k) (31)

Wherein f(k) = Θ(log k). Connecting (27),(29) and (31) canwe have that:

infΘ

EZ|Θ = EZ|Z ≤ w

= −w + 2X(k)

= − 1

γln[1− (

1

2)

1k ]− 2

γ(1

2)

1k +

1

γf(k)

, 1

γg(k) (32)

Wherein g(k) = Θ(log k). From the definitions of γ and ϕ,we have γ = log(1/(1 − 1

c )2Ri) = 2Ri log(1 + 1

c−1 ). Sincelog(1 + x) ≤ x for any x, we have γ ≤ 2Ri/(c − 1). Thenusing (23), (25) and (32) in (21) yields:

W =1

n

n∑i=1

W i ≥ 1

2n

n∑i=1

1

γg(k)

≥ g(k)

2n

n∑i=1

c− 1

2Ri

≥ g(k)(c− 1)1

4n

n∑i=1

1

Ri

≥ g(k)c− 1

4

11n

∑ni=1 Ri

(33)

Where (33) follows from Jensen’s inequality, noting that thefunction f(R) = 1

R is convex, and hence 1n

∑ni=1 f(Ri) ≥

f( 1n∑n

i=1 Ri). Combining (33) and (22), we have:

W ≥ g(k)c− 1

4

λ

1− kλ= Θ(n log k)

λ

1− kλ(34)

wherein c has the same order as n. Proving the theorem.We notice that Ri >= 0 in inequality (22), thus λ <

1k . Divide λ on both sides of formula (1), we get W

λ ≥Θ(n log k) 1

1−kλ . Since λ = o(1/k), i.e. 1 − kλ remain aconstant as n and k growing into infinity, we get

W

λ≥ Θ(n log k) (35)

Finally, we have the following corollary.Corollary 1: For any scheduling algorithm in the network

with n nodes moving according to i.i.d. pattern, and eachdesire to send its data to k distinct destination nodes, theachievable capacity λ and delay W satisfy the fundamentalrelationship: W

λ = Ω(n log k).

VI. CONCLUSION AND FUTURE WORK

In this paper, we study delay and capacity tradeoffs forMotionCast. We utilize redundant packets transmissions torealize the tradeoff, and present the performance of the 2-hop relay algorithm without and with redundancy respectively.We find that the capacity of the 2-hop relay algorithm with-out redundancy is better than that of static networks whenk = o(n). And our tradeoff is better than that of directlyextending the tradeoff for unicast to multicast. Moreover, weprove that the fundamental delay-capacity tradeoff ratio formulticast is Ω(n log k). We have not taken into account themulti-hop transmission schemes and the effect of differentmobility patterns yet, which could be a future work.

Moreover, the results in this paper are derived theoreticallyand it would be an interesting work to validate the results inreal experimental or simulation examinations. Also, this is achallenging work since the network is a large scale one andall the nodes in the network keep on moving. Since therehas been some work in the area of DTN trying to implementexperiments or simulate situations and some excited results areobtained, it would be very interesting and promising to inves-tigate the capacity and delay under a realistic circumstance.This could be our future work as well.

ACKNOWLEDGMENT

We deeply appreciate the valuable and constructive com-ments from the anonymous reviewers, and editor. This workis supported by the National Basic Research Program ofChina (973 Program) (No. 2011CB302701), NSF China (No.60832005); China Ministry of Education Fok Ying Tung Fund(No. 122002); Qualcomm Research Grant.

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13

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APPENDIX I – THE DERIVATION OF p AND q

Since p represents the probability of finding at least twonodes in a particular cell, the opposite event of it is there isno node (and this happens with a probability of (1− 1

c )n) or

only one node in the cell (this occurs with a probability ofnc (1−

1c )

n−1, where n infers that the node in the cell can beany one among all n nodes of the network). Thus, we havethe expression of (2).

As for q, it represents the probability of finding a source-destination pair within a cell. Note that in our traffic pattern,we suppose the number of nodes n is divisible by k+1 and uni-formly and randomly divide the network into different groupswith each of them having k + 1 nodes. Also assume packetsfrom each node i in a specific group must be delivered to allthe other nodes within the group. Thus, any two nodes withina same group is a pair of source-destination. The probabilitythat there is not any source-destination pair belonging to anygroup within a particular cell is k+1

c (1 − 1c )

k + (1 − 1c )

k+1.Since each group is independent with others, the probabilitythat there is not any source-destination pair in the cell is thus

nk+1 th power of the above quantity. Hence, the probability ofthe inverse event q is given by (3).

APPENDIX II – USEFUL LEMMAS

Here we present useful lemmas in this paper.

Lemma 1:k∑

i=1

(−1)i−1

i

(ki

)= ln(k + 1) + r, where k ≥ 1

and r is Euler constant.

Proof: Denote the left-hand-side of the equation by A(k),

then we have A(k−1) =k∑

i=1

(−1)i−1

i

(k−1i

). Notice that

(ki

)=(

k−1i

)+(k−1i−1

), it follows

A(k)−A(k − 1) =

k∑i=1

(−1)i−1

i

(k − 1

i− 1

)

=1

k

k∑i=1

(−1)i−1

(k

i

)(36)

Recall that (1 − 1)k =k∑

i=0

(−1)i(ki

)= 0, hence we obtain

k∑i=1

(−1)i−1(ki

)= −

k∑i=1

(−1)i(ki

)= −

[ k∑i=0

(−1)i(ki

)−1

]= 1.

Combining with (36), we get A(k)−A(k − 1) = 1k , then

A(k) = A(1) +

k∑i=2

[A(k)−A(k − 1)

]= 1 +

k∑i=2

1

k=

k∑i=1

1

k(37)

Since the right-hand-side of (37) is the harmonic series, thislemma holds.

Lemma 2: Suppose X1, X2, ..., Xk are continuous i.i.d.exponential variables with expectation of 1/a, and de-note Xmax = maxX1, X2, ..., Xk, then EXmax =Θ(log k/a) (for simplicity, we can treat EXmax just aslog k/a), where k ≥ 1.

Proof: Consider the cdf of Xmax,

FXmax(t) = PXmax ≤ t = (1− e−at)k (38)

Thus, the pdf of Xmax can be expressed as

fXmax(t) =dFXmax(t)

dt= k(1− e−at)k−1 · ae−at (39)

14

Then, we obtain

EXmax =

∫ ∞

0

k(1− e−at)k−1ae−at · tdt

= ka

∫ ∞

0

k−1∑i=0

(k − 1

i

)(−1)ie−a(i+1)t · tdt

=k−1∑i=0

ka

(k − 1

i

)(−1)i

1[a(i+ 1)

]2=

k∑i=1

ka

(k − 1

i− 1

)(−1)i−1 1

a2i2

=k

a

k∑i=1

(−1)i−1

i2

(k − 1

i− 1

)

=1

a

k∑i=1

(−1)i−1

i

(k

i

)(40)

According to Lemma 1, we conclude this lemma.

Xinbing Wang received the B.S. degree (with hons.)from the Department of Automation, Shanghai Jiao-tong University, Shanghai, China, in 1998, and theM.S. degree from the Department of Computer Sci-ence and Technology, Tsinghua University, Beijing,China, in 2001. He received the Ph.D. degree, majorin the Department of electrical and Computer Engi-neering, minor in the Department of Mathematics,North Carolina State University, Raleigh, in 2006.Currently, he is a faculty member in the Departmentof Electronic Engineering, Shanghai Jiaotong Uni-

versity, Shanghai, China. His research interests include resource allocationand management in mobile and wireless networks, TCP asymptotics analysis,wireless capacity, cross layer call admission control, asymptotics analysisof hybrid systems, and congestion control over wireless ad hoc and sensornetworks. Dr. Wang has been a member of the Technical Program Committeesof several conferences including IEEE INFOCOM 2009-2011, IEEE ICC2007-2011, IEEE Globecom 2007-2011.

Wentao Huang received B.S. degree from NanjingUniversity of Posts and Telecommunications, China,in 2008. He is currently a M.S. student in theDepartment of Electronic Engineering, at ShanghaiJiao Tong University. His current research interestsinclude distributed system, mobile computing, net-work security.

Shangxing Wang received B.S. degree in Telecom-munication Engineering from Xidian University,Xi’an, China, in 2010. She is currently a Ph.D. stu-dent in the Department of Electronic Engineering, atShanghai Jiao Tong University. Her current researchinterests include networking, Ad hoc network andwireless communication.

Jinbei Zhang received B.S. degree in EletronicEngineering from Xidian University, Xi’an, China,in 2010. He is currently a Ph.D. student in theDepartment of Electronic Engineering, at ShanghaiJiao Tong University. His current research interestsinclude network capacity, scheduling and connectiv-ity.

Chenhui Hu received B.S. and M.S. degree fromthe Department of Electrionic Engineering, ShanghaiJiaotong University, Shanghai, China, in 2007 and2010 respectively. From 2007 to 2010, he was doingresearch in the Institute of Wireless CommunicationTechnology (IWCT) in Shanghai Jiao Tong Univer-sity, supervised by Prof. Xinbing Wang and YouyunXu. His research interests include wireless capacityand connectivity, asymptotic analysis of mobile adhoc networks, multicast, distributed MIMO, perco-lation theory. He is pursuing his Ph.D degree in the

School of Engineering and Applied Sciences in Harvard, 2010.