optimal delay–throughput tradeoffs in mobile ad hoc networks

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 9, SEPTEMBER 2008 4119

Optimal Delay–Throughput Tradeoffs in MobileAd Hoc Networks

Lei Ying, Member, IEEE, Sichao Yang, and R. Srikant, Fellow, IEEE

Abstract—In this paper, we investigate the delay–throughputtradeoffs in mobile ad-hoc networks. We consider four nodemobility models: 1) two-dimensional independent and identicallydistributed (i.i.d.) mobility, 2) two-dimensional hybrid randomwalk, 3) one-dimensional i.i.d. mobility, and 4) one-dimensionalhybrid random walk. Two mobility time scales are included inthis paper. i) Fast mobility, where node mobility is at the sametime scale as data transmissions. ii) Slow mobility, where nodemobility is assumed to occur at a much slower time scale than datatransmissions. Given a delay constraint D, we first characterizethe maximum throughput per source–destination (S-D) pair foreach of the four mobility models with fast or slow mobiles. Wethen develop joint coding–scheduling algorithms to achieve theoptimal delay–throughput tradeoffs.

Index Terms—Delay–throughput tradeoffs, hybrid randomwalk models, independent and identically distributed (i.i.d.) mo-bility models, mobile ad hoc networks, rate-less codes, scalinglaws.

I. NOTATIONS

T HE following notations are used throughout this paper.Given nonnegative functions and :

(1) means there exist positive constantsand such that for all ;

(2) means there exist positive constantsand such that for all ; namely,

;(3) means that both and

hold;(4) means that ;(5) means that ;

namely, .

II. INTRODUCTION

In this paper, we investigate the delay–throughput tradeoffsin a mobile ad hoc network with mobiles. The throughput of

Manuscript received May 2, 2007; revised December 26, 2007. PublishedAugust 27, 2008 (projected). This work was supported in part by a VodafoneFellowship and a National Science Foundation under Grants CNS 05-19691,CCF 06-34891, CNS 07-21286, CCF 03-25673. The material in this paper waspresented at WiOpt 2007, Limassol, Cyprus, April 2007 and the ITA Workshop,San Diego, CA, January2007.

L. Ying is with the Department of Electrical and Computer Engineering, IowaState University, Ames, IA 50011 USA (e-mail: [email protected]).

S. Yang is with Qualcomm Flarion Technologies, Bedminster, NJ 07921 USA(e-mail: [email protected]).

R. Srikant is with the Department of Electrical and Computer Engineering andCoordinated Science Laboratory, University of Illinois at Urbana-Champaign,Urbana, IL 61801 USA (e-mail: [email protected]).

Communicated by E. Modiano, Associate Editor for Communication Net-works.

Color versions of Figures 1, 2, and 9 in this paper are available online athttp://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIT.2008.928247

a random wireless network with static nodes and randomsource–destination (S-D) pairs was studied by Gupta and Kumar[1]. They showed that the maximum throughput per S-D pairis , and proposed a scheduling scheme achieving athroughput of per S-D pair. One of the rea-sons that the throughput decreases with is that each successfultransmission from source to destination needs to takehops. Later, Grossglauser and Tse [2] considered mobile adhoc networks, and showed that throughput per S-D pairis achievable. The idea is to deliver a packet to its destinationonly when it is within distance from the destination.However, packets have to tolerate large delays to achieve thisthroughput. Since then, a number of papers have studied thetradeoffs between throughput and delay.

We first review the delay–throughput tradeoff results for in-dependent and identically distributed (i.i.d.) mobility models.Neely and Modiano [3] studied the i.i.d. mobility model wherethe positions of nodes are totally reshuffled from one time slotto another, and showed that the mean delay of Grossglauserand Tse’s algorithm is In the same paper, they also pro-posed an algorithm which generates multiple copies of each datapacket to reduce the mean delay. Since more transmissions arerequired when we generate multiple copies, the throughput perS-D decreases with the number of copies per data packet. Thedelay–throughput tradeoff is shown to be in [3],where is the throughput per S-D pair, and is the number oftime slots taken to deliver packets from source to destination.

In [3], fast mobility is assumed. A different time scaleof mobility, slow mobility, was considered by Toumpis andGoldsmith in [4], and Lin and Shroff in [5]. For slow mo-biles, node mobility is assumed to be much slower than datatransmissions. So the packet size can be scaled down asincreases, and multihop transmissions are feasible in singletime slot. The delay–throughput tradeoff was shown to be

in [4]. A better tradeoff was obtainedin [5], where the maximum throughput per S-D pair for meandelay was shown to be and ascheme was proposed to achieve a tradeoff of

Besides the i.i.d. mobility model, other mobility models havealso been studied in the literature. The random walk modelwas introduced by El Gamal et al. in [6], and later studied in[7]–[9]. In [7], [8], the throughput per S-D pair is shown tobe for , and for

, where [7] focused on the slow mobilityand [8] focused on the fast mobility. Other mobility models,

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4120 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 9, SEPTEMBER 2008

like Brownian motion, one-dimensional mobility, and hybridrandom walk models have been studied in [9]–[12].

Although the delay–throughput tradeoff has beenwidely studied for various mobility models, the optimaldelay–throughput tradeoff has not yet been established exceptfor two cases of mobility models [7], [8], [10]. In this paper,we study four different node mobility models. First, we studythe two-dimensional i.i.d. mobility model, where the nodes aretotally reshuffled at each time slot. Then we extend the resultsto the two-dimensional hybrid random walk model, which wasintroduced by Sharma et al. in [9]. Under the two-dimensionalhybrid random walk model, the unit torus is divided into

small-squares, and mobiles can move from the currentsmall-square to one of its eight adjacent small-squares at thebeginning of each time slot. Since the distance each mobilecan move is at most at each time slot, we can usedifferent values of to model mobiles with different speeds.Note that the two-dimensional hybrid random walk model isthe same as the two-dimensional i.i.d. mobility model when

. However, the Markovian mobility dynamics of therandom walk requires a different set of tools and as a result, thetradeoffs of the two-dimensional hybrid random walk modelare applicable only when .

We also study one-dimensional mobility models. Thesemodels are motivated by certain types of delay-tolerant net-works [13], in which a satellite subnetwork is used to connectlocal wireless networks outside of the Internet. Since thesatellites move in fixed orbits, they can be modeled as one-di-mensional mobilities on a two-dimensional plane. Motivatedby such a delay-tolerant network, we consider one-dimensionalmobility model where nodes move horizontally and the other

nodes move vertically. Since the node mobility is restrictedto one dimension, sources have more information about thepositions of destinations compared with the two-dimensionalmobility models. We will see that the throughput is improvedin this case; for example, under the one-dimensional i.i.d.mobility model with fast mobiles, the tradeoff will be shown tobe , which is better than , the tradeoffunder the two-dimensional i.i.d. mobility model with fastmobiles.

We summarize our main results as follows.(1) Two-dimensional i.i.d. mobility models:

(i) Under the fast mobility assumption, it is shownthat the maximum throughput per S-D pair is

under a delay constraint . Ajoint coding–scheduling algorithm is presentedto achieve the maximum throughput when isboth and

(ii) Under the slow mobility assumption, it is shownthat the maximum throughput per S-D pair is

given a delay constraint . Wepropose a joint coding–scheduling algorithm toachieve the maximum throughput when isboth and .

(2) Two-dimensional hybrid random walk models:(i) Under the fast mobility assumption, it is

shown that the maximum throughput per

S-D pair is when and, where is the step size

of the random walk. A joint coding–schedulingalgorithm is proposed to achieve the maximumthroughput when and is both

and.

(ii) Under the slow mobility assumption, it isshown that the maximum throughput perS-D pair is when and

, and a joint coding–sched-uling algorithm is proposed to achieve themaximum throughput when and isboth and .

(3) One-dimensional i.i.d. mobility models:(i) Under the fast mobility assumption, it is shown

that the maximum throughput per S-D isgiven a delay constraint . A

joint coding–scheduling algorithm is proposedto achieve the maximum throughput when isboth and .

(ii) Under the slow mobility assumption, it is shownthat the maximum throughput per S-D pair is

. A joint coding–scheduling al-gorithm is proposed to achieve the maximumthroughput when is .

(4) One-dimensional hybrid random walk models:(i) Under the fast mobility assumption, it is

shown that the maximum throughput per S-Dpair is when and

, and a joint coding–schedulingalgorithm is proposed to achieve the maximumthroughput when and is both

and.

(ii) Under the slow mobility assumption, it isshown that the maximum throughput per S-Dpair is when and

, and a joint coding–schedulingalgorithm is proposed to achieve the maximumthroughput when and is both

and .Note that the optimal delay–throughput tradeoff are establishedunder some conditions on . When these conditions are notmet, the tradeoff is still unknown in general, though a tradeoff ofthe two-dimensional hybrid random walk model with slow mo-biles has been established under an assumption regarding packetreplication in [9]. We also would like to mention that when thestep size of the two-dimensional hybrid random walk is ,our two-dimensional hybrid random walk model is identical tothe random walk model studied in [7], [8], where the optimaldelay–throughput tradeoff has been obtained. Our results do notapply to this case since the set of allowed values for becomesempty in that case (see (2) (i) above).

We also would like to mention that there is a very recent resultby Ozgur et al. [14] where they showed a throughput ofper S-D pair is achievable using node cooperation and mul-

YING et al.: OPTIMAL DELAY–THROUGHPUT TRADEOFFS IN MOBILE Ad Hoc NETWORKS 4121

Fig. 1. Two-dimensional random walk model.

tiple-input multiple-output (MIMO) communication; see alsothe earlier paper by Aeron and Saligrama in [15]. These schemesrequire sophisticated signal processing techniques, not consid-ered in this paper.

III. MODEL

In this section, we first present the mobility and wireless inter-ference models used in this paper. Then the definitions of delayand throughput are provided.

Mobile ad hoc network model: Consider an ad hoc networkwhere wireless mobile nodes are positioned in a unit square. Theunit square is assumed to be a torus, i.e., the top and bottomedges are assumed to touch each other and similarly the left andright edges also are assumed to touch each other. Further assumethat the time is slotted, we study following mobility models inthis paper.

(1) Two-dimensional i.i.d. mobility model: Our two-di-mensional i.i.d. mobility model is defined as follows.

(i) There are wireless mobile nodes positionedon a unit square. At each time slot, the nodesare uniformly, randomly positioned in the unitsquare.

(ii) The node positions are independent of eachother, and independent from time slot to timeslot. So the nodes are totally reshuffled at eachtime slot.

(iii) There are S-D pairs in the network. Each nodeis both a source and a destination. Without lossof generality, we assume that the destination ofnode is node , and the destination of node

is node 1.(2) Two-dimensional random walk model: Consider a

unit square which is further divided into squares ofequal size. Each of the smaller square will be called anRW-cell (random walk cell), and indexed bywhere . A node which is in oneRW-cell at a time slot moves to one of its eight adjacentRW-cells or stays in the same RW-cell in the next timeslot with each move being equally likely as in Fig. 1.Two RW-cells are said to be adjacent if they share a

Fig. 2. One-dimensional random walk model.

common point. The node position within the RW-cell israndomly uniformly selected.Remark: In our joint coding-scheduling algorithms, theunit square will be divided into cells, and only thosemobiles in the same cell are allowed to communicatewith each other. These cells, used for scheduling andcommunication, are different from the RW-cells definedabove.

(3) One-dimensional i.i.d. mobility model: Our one-di-mensional i.i.d. mobility model is defined as follows.

(i) There are nodes in the network. Amongthem, nodes, named H-nodes, move horizon-tally; and the other nodes, named V-nodes,move vertically.

(ii) Letting denote the position of node . Ifnode is an H-node, is fixed and is a valuerandomly uniformly chosen from . Wealso assume that H-nodes are evenly distributedvertically, so takes values .V-nodes have similar properties.

(iii) Assume that source and destination are thesame type of nodes. Also assume that node isan H-node if is odd, and a V-node if is even.Further, assume that the destination of nodeis node , the destination of nodeis node , and the destination of node isnode .

(iv) The orbit distance of two H(V)-nodes is definedto be the vertical (horizontal) distance of thetwo nodes.

(4) One-dimensional random walk model: Each orbit is di-vided into RW-intervals (random walk interval). Ateach time slot, a node moves into one of two adjacentRW-intervals or stays at the current RW-interval (as inFig. 2). The node position in the RW-interval is ran-domly, uniformly selected.


Communication model: We assume the protocol model in-troduced in [1] in this paper. Let denote the Euclideandistance between node and node , and denote the trans-mission radius of node . A transmission from node can besuccessfully received at node if and only if following two con-ditions hold:

(i) ;(ii) for each node which

transmits at the same time, where is a protocol-speci-fied guard-zone to prevent interference.

We further assume that at each time slot, at most bits can betransmitted in a successful transmission.

Time scale of mobility: Two time scales of mobility are con-sidered in this paper.

(1) Fast mobility: The mobility of nodes is at the same timescale as the data transmission, so is a constant inde-pendent of and only one-hop transmissions are fea-sible in single time slot.

(2) Slow mobility: The mobility of nodes is much slowerthan the wireless transmission, so . Underthis assumption, the packet size can be scaled as

for to guarantee -hoptransmissions are feasible in single time slot.

Delay and throughput: We consider hard delay constraintsin this paper. Given a delay constraint , a packet is said tobe successfully delivered if the destination obtains the packetwithin time slots after it is sent out from the source.

Let denote the number of bits successfully delivered tothe destination of node in time interval . A throughput of

per S-D pair is said to be feasible under the delay constraintand loss probability constraint if there exists such thatfor any , there exists a coding, routing and schedulingalgorithm with the property that each bit transmitted by a sourceis received at its destination with probability at least , and

(1)

IV. MAIN RESULTS AND SOME INTUITION FOR THE I.I.D.MOBILITY MODELS

Recall that our objective is to maximize throughput in a wire-less network subject to a delay constraint and a wireless interfer-ence constraint. More precisely, the constraints can be viewedas follows.

(1) Wireless interference: Throughput is limited due to thefact that transmissions interfere with each other.

(2) Mobility: A packet may not be delivered to its destina-tion before the delay deadline since neither the packet’ssource nor any relay node may get close enough to thedestination.

In this section, we present some heuristic arguments to obtain anupper bound on the maximum throughput subject to these twoconstraints and derive some key results. While the heuristics arefar from precise derivations of the optimal delay–throughputtradeoffs, they may be useful in understanding the main results.In addition, the heuristic arguments provide the right orderfor the “hitting distance” (to be defined later) which plays

Fig. 3. Virtual-channel representation for the two-dimensional i.i.d. mobilitymodel with fast mobiles.

a critical role in the optimal algorithms used to achieve thedelay–throughput tradeoffs.

First, consider the two-dimensional i.i.d. mobility model withfast mobiles. We say that a packet hits its destination at time slot

if the distance between the packet and its destination is lessthan or equal to . Under the two-dimensional i.i.d. mobilitymodel, a packet hits its destination with probability at eachtime slot. So, given a delay constraint , the probability that apacket hits its destination in one of time slots is

Furthermore, under the fast mobility, only one-hop transmis-sions are feasible at each time slot. So the transmission radiusneeds to be at least to deliver packets to their destinationswhen their distance is . Assume that all nodes use a commontransmission radius and that all nodes wish to transmit ateach time slot, then each node has fraction of timeto transmit, and the throughput per S-D pair is no more than

where is a positive constant independent of .Thus, the network can be regarded as a system where there aretwo virtual channels between each S-D pair as in Fig. 3. Thepackets are first sent over the erasure channel with erasure prob-ability

and then over the reliable channel with rate

bits per time slot. Each source can transmit at most bits pertime slot on average. So in this virtual system, the maximumthroughput of a S-D pair is

and the corresponding optimal hitting distancewhere .

To achieve this throughput, we first need to use the optimal. Furthermore, a coding scheme achieving the capacity of

the erasure channel is needed. Since the erasure probability isdetermined by and , which are different under differentdelay constraints, rate-less codes become a reasonable choice.If the encoding/decoding time is negligible compared to thedelay constraint , then any rate-less codes could be usedin the algorithms presented in this paper. We will use Raptorcodes because Raptor codes have linear encoding/decodingcomplexity, which guarantees that the encoding/decoding timewill not affect our order results even if it is not negligible. We


also note that the idea of using coding is one of the keys toachieve the optimal delay–throughput tradeoffs. In some of theprevious works, small delays are achieved by broadcasting eachdata packet to multiple relay nodes. From our virtual channelrepresentation, such an approach is equivalent to using the rep-etition coding over the erasure channel, which is not optimal.We would like to mention that the idea of using coding toimprove reliability of packet delivery has also been consideredindependently by Shah and Shakkottai in [16] for ad hoc sensornetworks in a different context.

Our first result is as follows: Under the two-dimensional i.i.d.mobility model with fast mobiles, the throughput per S-D pairis given delay constraint . When is both

and , this throughput can be achieved using a jointcoding-scheduling algorithm.

Note that the heuristic arguments leading up to the above re-sult have many limitations. For example, it suggests that one canwait for the source to hit the destination to deliver the packet. Inreality, such a scheme will not work since we deliver only onepacket to the destination during each encounter between the S-Dpair. Thus, other packets at the source which are not deliveredmay violate their delay constraints. This problem in the heuristicargument is because it assumes that we have an independent era-sure channel for each packet despite the fact that the transmittingnode is the same source. Despite the limitations, the heuristicargument surprisingly captures the delay–throughput tradeoffand the optimal hitting distance correctly up to the right order.In practice, the bound is achievable by exploiting the broadcastnature of the wireless channel to transmit each packet to severalrelay nodes and allowing relay nodes to independently attemptto deliver the packet the destination.

Next consider the two-dimensional i.i.d. mobility model withslow mobiles. Since multihop transmissions are feasible at eachtime slot, using a precise version of the result [1] which wasobtained in [17], the maximum throughput per S-D pair underthe slow mobility assumption is

where is a positive constant independent of . We provide acrude version of the argument from [1] here for ease of read-ability. Suppose each node uses a transmission radius and thedistance between a S-D pair is , then each bit has to travelhops. The number of bit hops needed to satisfy a throughput re-quirement of bits/slot/node in slots is . Due to theinterference model, the number of simultaneous transmissionspossible in one time slot is for some constant .Thus, we need

or

Intuitively, since the total area is and the number of nodes is, the smallest radius of transmission that can be used while

ensuring connectivity is given by , so

Fig. 4. Virtual-channel representation for the two-dimensional i.i.d. mobilitymodel with slow mobiles.

Fig. 5. Virtual-channel representation for the one-dimensional i.i.d. mobilitymodel with fast mobiles.

That this is indeed achievable in an order sense is proved in [17],and therefore, we take to be where .Then the virtual channels between an S-D pair are as depictedin Fig. 4. In this virtual system, the maximum throughput of aS-D pair is

and the optimal hitting distance where. This throughput can also be achieved using a joint

coding–scheduling scheme.The main result is as follows: Under the two-dimensional

i.i.d. mobility model with slow mobiles, the throughput per S-Dpair is given a delay constraint . Thisthroughput can be achieved using a joint coding–schedulingscheme when is both (1) and .

We next consider the one-dimensional i.i.d. mobility model.Note that we have assumed that the orbits of the source and thedestination are parallel to each other. Thus, the probability thata packet hits its destination is at most at each time slot, andthe probability of no hitting during consecutive time slots is

For fast mobiles, the throughput per S-D pair iswith a common transmission radius . Thus, the two virtualchannels are as depicted in Fig. 5. In this virtual system, themaximum throughput is

The main result is as follows: Under the one-dimensionali.i.d. mobility with fast mobiles, the throughput per S-D pair is

given a delay constraint . This throughputcan be achieved using a joint coding–scheduling algorithmwhen is both and .


Fig. 6. Virtual-channel representation for the one-dimensional i.i.d. mobilitymodel with slow mobiles.

For the slow mobility case, the virtual channels between anS-D pair are as shown in Fig. 6. In this virtual system, the max-imum throughput is

The main result is as follows: Under the one-dimensionali.i.d. mobility model with slow mobiles, the throughput per S-Dpair is given a delay constraint . Thisthroughput can be achieved using a joint coding–scheduling al-gorithm when .

For the hybrid random walk models, similar results can beobtained by using the same intuition. However, the delays haveto be larger to allow a packet to reach its destination since amobile can only move a certain distance from its current positionat one time slot.

As stated before, the crude virtual channel representationused in this section surprisingly yields the correct results.However, they do not form the basis of the proofs in the rest ofthe paper. Several assumptions have been made in deriving thevirtual channel representation.

(1) The hitting events for various packets are assumed to beindependent, which is difficult to ensure since the samenode may act as a relay for multiple packets.

(2) It assumes a fixed hitting distance which is not reason-able to obtain an upper bound on the throughput. Anupper bound must be scheme-independent.

In view of these limitations, we use the virtual channel modelto only provide some insight into the results and the hitting dis-tance we should use in the achievable algorithms, but rigorousproofs of the main results are provided in subsequent sections.

V. TWO-DIMENSIONAL I.I.D. MOBILITY MODEL,FAST MOBILES

In this section, we investigate the two-dimensional i.i.d.mobility model with fast mobiles. Assuming that all mobileshave wireless communication and coding capability, we inves-tigate the maximum throughput the network can achieve byusing relaying and coding to recover packet loss as discussedin the heuristic arguments. Given a delay constraint , we willfirst prove that the maximum throughput per S-D pair whichcan be supported by the network is . Then a jointcoding–scheduling scheme will be proposed to achieve themaximum throughput when is both and .

A. Upper Bound

In this subsection, we show the maximum throughput the net-work can support without network coding, i.e., under the fol-lowing assumption.

Assumption 1: Packets destined for different nodes cannot beencoded together. Further, we assume that coding is only usedto recover from erasures and not for data compression. Specifi-cally, we assume that at least coded packets are necessary torecover data packets, where all packets (coded or uncoded)are assumed to be of the same size.

Assumption 1 is the only significant restriction imposed oncoding, routing and scheduling schemes. We also make the fol-lowing assumption.

Assumption 2: A new coded packet is generated right beforethe packet is sent out. The node generating the coded packetdoes not store the packet in its buffer.

Assumption 2 is not restrictive since the information con-tained in the new packet is already available at the node.

Assumption 3: Once a node receives a packet (coded or un-coded), the packet is not discarded by the node till its deadlineexpires.

Assumption 3 is not restrictive since we are studying an upperbound on the throughput in this section.

Next we introduce the following notations, which will be usedin our proof:

• : Index of a bit stored in the network. Bit could be eithera bit of a data packet or a bit of a coded packet. If a nodegenerates a copy of a packet to be stored in another node,then the bits in the copy are given different indices than thebits in the original packet.

• : The destination of bit .• : The node storing bit .• : The time slot at which bit is generated.• : If bit is delivered to its destination, then is the

transmission radius used to deliver .• : The set of all bits stored at relay nodes at time slot

. We do not include bits that are still in their source nodein defining .

• .

Assume that the delay constraint is , and a data packet isprocessed by the source node at time slot . Then the datapacket is said to be active from time slot to . Abit is said to be active if at least one data packet encoded intothe packet containing bit has not expired. It is easy to see thatany bit expires at most time slots after the bit is generated.Also a bit is said to be good if it is active when delivered to itsdestination. Now let denote the number of good bits de-livered to destinations in . Without loss of generality, weassume good bits are indexed from to . Note that expiredbits might help decode good source bits but would not contributeto the total throughput, so we have

(2)


where is the number of good source bits successfully re-covered at destinations.

Next we present three fundamental constraints. In the fol-lowing lemma, inequalities (3) and (4) hold since the totalnumber of bits transmitted or received in time slots cannotexceed . Inequality (5) holds since under the protocolmodel, discs of radius around the receivers should bemutually disjoint from each other.

Lemma 1: For any mobility model, the following inequalitieshold:

(3)

(4)

(5)

where is the cardinality of the set .Proof: Since each node can transmit at most bits per

time slot, the total number of bits transmitted in time slotsis less than which implies inequalities (3) and (4). In-equality (5) was proved in [18].

We first consider the scenario where packet relaying is notallowed, i.e., packets need to be directly transmitted fromsources to destinations. In the following lemma, we show thatthe throughput in this case is at most even withoutthe delay constraint.

Lemma 2: Consider the two-dimensional i.i.d. mobilitymodel with fast mobiles. Suppose that packets have to bedirectly transmitted from sources to destinations, then

(6)

Proof: First from the Cauchy–Schwarz inequality and in-equality (5), we have

which implies

(7)

This gives an upper bound on the expected distance traveled.Next, we bound the total number of times that each mobile getswithin a distance of its destination for . From thei.i.d. mobility assumption, we have that for any , , and

which implies

Since at most bits can be transmitted at each time slot, wefurther have

Taking expectation on both sides of above inequality, we obtain

(8)

Now using Jensen’s inequality and inequalities (7) and (8),we can conclude that

(9)

Note that inequality (9) holds for any . We choose

which is less than since . Substituting intoinequality (9), we have

which implies that

The lemma then follows from inequality (2).

Next we investigate the maximum throughput the networkcan support using coding, routing, and scheduling schemes. Wehave obtained an upper bound on the number of bits directlytransmitted from sources to destinations in Lemma 2. To boundthe maximum throughput with relaying, we will calculate thenumber of bits transmitted from relays to destinations in thefollowing analysis.

Theorem 3: Consider the two-dimensional i.i.d. mobilitymodel with fast mobiles, and assume that Assumptions 1–3hold. Then given a delay constraint , we have that

(10)

Proof: In the proof of the theorem, we treat active bits atrelays and active bits at sources differently since we can boundthe number of active bits at relays using inequality (4), whilethe number of active bits at sources could be larger. Let


denote the number of good bits delivered directly from relaysto destinations in . Without loss of generality, we assumethese good bits are indexed from to . Similar to in-equality (7), we first have

(11)

Let denote the minimum distance between node and nodefrom time slot to time slot , i.e.,

Then for any and any bit , we have

which implies

Furthermore, we have

which implies that

(12)

Thus, we can conclude that

(13)

where the last inequality follows from inequality (12).Now using Jensen’s inequality and inequalities (11) and (13),

we have that for any

(14)

Substituting

into inequality (14), we can conclude that

(15)

The theorem follows from inequalities (6), (15), and (2).

From Theorem 3, we can conclude that the throughput perS-D is given a delay constraint .

B. Joint Coding–Scheduling Algorithm

In Section IV, we motivated the need to first encode datapackets. In this subsection, we use Raptor codes and proposea joint coding–scheduling scheme to achieve the maximumthroughput obtained in Theorem 3.

Motivated by the heuristic argument in Section IV, we di-vide the unit square into square cells with each side of lengthequal to , which is of the same order as the optimal hit-ting distance. In our scheme, we will allow final delivery of apacket to its destination only when a relay carrying the packetis in the same cell as the destination. Thus, a packet is deliv-ered only when the relay and destination are within a distanceof , which is also the same as the hitting distance cal-culated in Section IV except for a constant factor which doesnot play a role in the order calculations. The mean number ofnodes in each cell will be denoted by and is equal to .The transmission radius of each node is chosen to beso that any two nodes within a cell can communicate with eachother. This means that, given the interference constraint, twonodes in a cell can communicate if all nodes in cells within afixed distance from the given cell stay silent. Each time slot isfurther divided into minislots and each cell is guaranteed tobe active in at least one minislot within each time slot. Assume

. The reason we use nine minislots is that if a node in acell is active, then no other nodes in any of its neighboring eightcells can be active, but nodes outside this neighborhood can beactive. Further, we denote the packet size to be so thattwo packets can be transmitted in each minislot.

A cell is said to be a good cell at time if the number of nodesin the cell is between and . We also defineand categorize packets into four different types.

• Data packets: Uncoded data packets that have to be trans-mitted by the sources and received by the destinations.

• Coded packets: Packets generated by Raptor codes. We letdenote the th coded packet of node .

• Duplicate packets: Each coded packet could be broadcastto other nodes to generate multiple copies, called duplicatepackets. We let denote a copy of carried bynode , and to denote the set of all copies of codedpacket .

• Deliverable packets: Duplicate packets that happen to bein the same cell as from their destinations.

We now describe our coding/scheduling algorithm, which

guarantees a throughput of per S-D pair given a delayconstraint . Here, we consider a delay constraint insteadof to simplify notations, which obviously does not affect our


Fig. 7. Time structure of Algorithm I.

Fig. 8. Outline of Algorithm I.

order results. The time structure and procedure of Algorithm Iare illustrated in Figs. 7 and 8.

Joint Coding-Scheduling Algorithm I: We group everytime slots into a super-time slot. At each super-time slot, thenodes transmit packets as follows.

(1) Raptor encoding: Each source takes datapackets, and uses Raptor codes to generate codedpackets.

(2) Broadcasting: This step consists of time slots. Ateach time slot, the nodes executes the following tasks:

(i) In each good cell, one node is randomly se-lected. If the selected node has not alreadytransmitted all of its coded packets, thenit broadcasts a coded packet that was not previ-ously transmitted to other nodes in thecell during the minislot allocated to that cell.Recall that our choice of packet size allowsone node in every good cell to transmit duringevery time slot.

(ii) All nodes check the duplicate packets they have.If more than one duplicate packets have thesame destination, select one at random to keepand drop the others.

(3) Receiving: This step consists of time slots. At eachtime slot, if a cell contains no more than two deliver-

able packets, the deliverable packets are delivered totheir destinations using one-hop transmissions duringthe minislot allocated to that cell; otherwise, no node inthe cell attempts to transmit. At the end of this step, allundelivered packets are dropped. The destinations de-code the received coded packets using Raptor decoding.

Note that in describing the algorithm, we did not account forthe delays in Raptor encoding and decoding. However, Raptorcodes have linear encoding and decoding complexity. Hence,even if these delays are taken into account, our order results willnot change.

Theorem 4: Consider Joint Coding-Scheduling Algorithm I.Suppose is both and , and the delay constraint is

. Then given any , there exists such that for any, every data packet sent out can be recovered at the

destination with probability at least , and furthermore

(16)

Proof: Let denote the th super-time slot. In Ap-pendix D, we show that the following events ((17)–(19))happen with high probability.

Broadcasting: At least coded packets from asource are successfully duplicated after the broadcasting stepwith high probability, where a coded packet is said to be suc-cessfully duplicated if the packet is in at least distinctrelay nodes. Letting denote the number of coded packetsof source which are successfully duplicated in super-time slot

, we first have that there exists such that for any

(17)

Receiving: At least distinct coded packets froma source are delivered to its destination after the receiving stepwith high probability. Letting denote the number of dis-tinct coded packets delivered to destination in super-timeslot , we have that there exists such that for all

(18)

Decoding: The data packets from a source are re-covered with high probability. Letting denote the eventsuch that all data packets are fully recovered by des-tination , we have that

(19)


for some .Recall that and , so

Combining inequalities (17)–(19), we can conclude that for any, there exists such that for

(20)

which implies that every data packet sent out can be recoveredwith probability at least . Since , from theChernoff bound (see Lemma 20 provided in Appendix C forconvenience), we can conclude that for

where we choose in Lemma 21. Note that

implies at least

bits are successfully transmitted from node to node insuper-time slots. Since each super-time slot consists of timeslots, we can conclude that for

which implies that, for a fixed

VI. TWO-DIMENSIONAL I.I.D. MOBILITY MODEL,SLOW MOBILES

In this section, we investigate the two-dimensional i.i.d. mo-bility model with slow mobiles. Given a delay constraint ,we first prove the maximum throughput per S-D pair whichcan be supported by the network is . Then a jointcoding–scheduling scheme is proposed to achieve the maximumthroughput.

A. Upper Bound

Let denote the time slot in which bit is delivered to itsdestination. Under the slow mobility, the delivery in coulduse multihop transmissions, so we further define following no-tations.

• : The number hops bit travels in time slot .• : The Euclidean distance bit travels in time slot .

• : The transmission radius used in hop for .Similar to Lemma 1, we have following results.

Lemma 5: For any mobility model, the following inequalitieshold

(21)

(22)

Similar to the fast mobility cases, we first consider thethroughput under the assumption that the packets can only bedelivered to destinations from sources.

Lemma 6: Consider the two-dimensional i.i.d. mobilitymodel with slow mobiles. Suppose that packets have to bedirectly transmitted to destinations from sources, then

(23)

Proof: First from the Cauchy–Schwartz inequality andLemma 5, we have that

which implies

(24)

since . The rest of the proof is same as theproof of Lemma 2.

Theorem 7: Consider the two-dimensional i.i.d. mobilitymodel with slow mobiles, and assume that Assumptions 1–3hold. Then given a delay constraint , we have

(25)

Proof: Based on inequality (24) and the argument of The-orem 3.

From Theorem 7, we can conclude that the throughput perS-D is given a delay constraint .


In this subsection, we propose a joint coding–schedulingscheme to achieve the throughput suggested in Theorem 7. In


Fig. 9. Multihop delivery in the receiving step of Algorithm II.

the receiving step, we divide the unit square into square cellswith each side of length equal to , which is of the sameorder as the optimal hitting distance obtained in Section IV.The mean number of nodes in each cell will be denoted byand is equal to . The packet size is chosen to be

so that at each time slot, all nodes in a good cell can transmitone packet to some other node in the same cell by using thehighway algorithm proposed in [17] (see Appendix B), where

is a constant independent of , and multiple packets can bedelivered as illustrated in Fig. 9. In the broadcasting step, theunit square is divided into square cells with each side of lengthequal to . The mean number of nodes in each will bedenoted by and is equal to . In the broadcasting step,the transmission radius of each nodes is chosen to be .Note the packet size is

So in the broadcasting step, all nodes in a good cell could bescheduled to broadcast one coded packet at one minislot. Alsonote that .

Joint Coding–Scheduling Algorithm II: We group everytime slots into a super-time slot. At each super-time slot,

the nodes transmit packets as follows.(1) Raptor encoding: Each source takes data

packets, and uses Raptor codes to generate codedpackets.

(2) Broadcasting: The unit square is divided into a regularlattice with cells. This step consists of timeslots. At each time slot, the nodes execute the followingtasks.

(i) In each good cell, the nodes take their turnsto broadcast a coded packet to othernodes in the cell. We use the same definition ofa good cell as in Algorithm I, i.e., the number

of nodes in a good cell should be with a factorof the mean, where the factor is required to liein the interval .

(ii) All nodes check the duplicate packets they have.If more than one duplicate packet is destinedto the same destination, randomly keep one anddrop the others.

(3) Receiving: The unit square is divided into a regular lat-tice with cells. This step consists of timeslots. At each time slot, the nodes in a good cell exe-cute the following tasks in the minislot allocated to thatcell.

(i) Each node containing deliverable packets ran-domly selects a deliverable packet, and sends arequest to the corresponding destination.

(ii) Each destination only accepts one request andrefuses the others.

(iii) The nodes whose requests are acceptedtransmit the deliverable packets to their desti-nations using the highway algorithm proposedin [17].

At the end of this step, all undelivered packets aredropped. Destinations use Raptor decoding to obtainthe source packets. Note that one requires some over-head in obtaining route to the destination to performStep (3) (i) above. As in previous works, we assumethat this overhead is small since one can transmit manypackets in each time slot, under the slow mobility as-sumption.

Theorem 8: Consider Joint Coding–Scheduling Algorithm II.Suppose is both and , and the delay constraint is

. Then given any , there exists such that for any ,every data packet sent out can be recovered at the destinationwith probability at least , and furthermore

(26)

Proof: Similar to (17), we can first obtain the followingresult.

Broadcasting: At least coded packets from a sourceare successfully duplicated after the broadcasting step with highprobability, i.e.,

(27)

where a coded packet is said to be successfully duplicated if thepacket is in distinct relay nodes.

In Appendix E, we also show that the following event happenswith high probability.

Receiving: At least distinct coded packets from a sourceare delivered to its destination after the receiving step with highprobability, i.e.,

(28)

Based on inequalities (27) and (28), the theorem can beproved by following the argument in Theorem 4.


VII. TWO-DIMENSIONAL HYBRID RANDOM WALK MODEL,FAST MOBILES

In this section, we study the two-dimensional hybrid randomwalk model with fast mobiles. We will obtain the max-imum throughput for , and then showthat the maximum throughput can be achieved using JointCoding–Scheduling Algorithm I under some additional con-straints on .

A. Upper Bound

Theorem 9: Consider the two-dimensional hybrid randomwalk model with fast mobiles. Assume that step-sizeand Assumptions 1–3 hold. Then given delay constraint

, we have

(29)

Proof: For any , it is shown in Appendix Gthat

(30)

Inequality (29) then follows from the proof of Theorem 3.


From Theorem 9, we can see that the optimal delay–throughput tradeoffs of the two-dimensional hybrid randomwalk model with fast mobiles is similar to the one of the two-di-mensional i.i.d. mobility model. It motivates us to considerJoint Coding–Scheduling Algorithm I. We will show that theoptimal tradeoff can be achieved using Joint Coding–Sched-uling Algorithm I with the following modifications:data packets are coded into coded packets.

Theorem 10: Consider the two-dimensional hybrid randomwalk model with fast mobiles. Suppose that is , is both

and ,and the delay constraint is . Then given any there exists

such that for any , every data packet sent out can berecovered at the destination with probability at least , and

by using the modified Joint Coding–Scheduling Algorithm I.Proof: We consider one super-time slot which consists of

time slots, and calculate the probability that thedata packets from node are fully recovered at the destination,where is the mean number of nodes in each cell.In Appendix H, we show that the following events happen withhigh probability.

Node distribution: All cells are good during the entire super-time slot with high probability. Letting denote this event, wehave

(31)

Broadcasting: At least coded packets from asource are successfully duplicated after the broadcasting stepwith high probability, i.e.,

(32)

where a coded packet is said to be successfully duplicated if thepacket is in at least distinct relay nodes.

Receiving: At least distinct coded packets froma source are delivered to its destination after the receiving stepwith high probability, i.e.,

(33)

From inequalities (31), (32), and (33), we can conclude thatunder the modified Joint Coding–Scheduling Algorithm I, ateach super-time slot, the data packets can be suc-cessfully recovered with probability at least

The rest of the proof follows from the proof of Theorem 4.

VIII. TWO-DIMENSIONAL HYBRID RANDOM WALK MODEL,SLOW MOBILES

In this section, we study the two-dimensional hybrid randomwalk model with fast mobiles. We will obtain the max-imum throughput for , and then showthat the maximum throughput can be achieved using JointCoding–Scheduling Algorithm II under some additional con-straints on .

A. Upper Bound

Theorem 11: Consider the two-dimensional hybrid randomwalk model with slow mobiles. Assume that step sizeand Assumptions 1–3 hold. Then given delay constraint

, we have

(34)

Proof: Follow inequality (30) and the proof of Theorem 7.


We will show that the optimal tradeoff can be achieved usingJoint Coding–Scheduling Algorithm II with the following mod-ification: data packets are coded to coded packets.

Theorem 12: Consider the two-dimensional hybrid randomwalk models with slow mobiles. Suppose that is andis both and , and the delayconstraints is . Then under the slow mobility model, givenany there exists such that for any , every data packet


sent out can be recovered at the destination with probability atleast , and

(35)by using the modified Joint Coding–Scheduling Algorithm II.

Proof: We consider one super-time slot which consists oftime slots, and calculate the probability that the data

packets from node are fully recovered at the destination. Sim-ilar to Theorem 10, it is easy to verify that the following eventshappen with high probability.

Node distribution: All cells are good during the entire super-time slot with high probability, i.e.,

Broadcasting: At least coded packets from a sourceare successfully duplicated after the broadcasting step with highprobability. Specifically, we have

(36)

where a coded packet is said to be successfully duplicated if itis in distinct relay nodes.

Receiving: At least distinct coded packets from a sourceare delivered to its destination after the receiving step with highprobability. Specifically, we have

(37)

From inequalities (31), (36), and (37) we can conclude thatunder the modified Joint Coding–Scheduling Algorithm II, ateach super-time slot, the data packets can be successfullyrecovered with probability at least

and the theorem holds.

IX. ONE-DIMENSIONAL I.I.D. MOBILITY MODEL,FAST MOBILES

In this section, we study one-dimensional i.i.d. mobilitymodel with fast mobiles.

A. Upper Bound

Theorem 13: Consider the one-dimensional i.i.d. mobilitymodel with fast mobiles. Assume that Assumptions 1–3 hold,then

Proof: Recall that is the minimum distance betweennode and node from time slot to . If the orbitsof node and are vertical to each other, then holdsonly if at some time slot , node and are in the square withside length as in Fig. 10. In this case, we have

Fig. 10. Two vertical orbits.

If the orbits of node and are parallel to each other, then itis easy to verify that

Thus, for , we can conclude that

(38)

The rest of the proof follows from the proof of Theorem 3.


Choose

We divide the unit square into horizontal rectangles,named as H-rectangles; and vertical rectangles, namedas V-rectangles as in Fig. 11. A packet is said to be destinedto a rectangle if the orbit of its destination is contained in therectangle.

The algorithms for the one-dimensional i.i.d. mobility modelhave four steps. The first step is the Raptor encoding. The secondstep is the broadcasting. In this step, the H(V)-nodes broad-cast coded packets to V(H)-nodes. The third step is the trans-porting, where the V(H)-nodes transport the H(V)-packets tothe H(V)-rectangles containing the orbits of corresponding des-tinations, and then broadcast packets to the H(V)-nodes whoseorbits are contained in the rectangles. After the third step, all du-plicate packets are carried by the nodes that move parallel withthe destinations and their orbit distance is less than . Thefourth step is the receiving, where the packets are delivered tothe destinations. The broadcasting and transporting steps are asshown in Fig. 11.

Since duplicate copies are generated in both the broadcastingand the transporting, to distinguish them, we name the duplicatepackets generated at the broadcasting as B-duplicate packets,and the duplicate packets generated at the transporting as T-du-plicate packets. Also, we say a B-duplicate packet is transport-


Fig. 11. Broadcasting and transporting in Algorithm III.

able if it is in the rectangle containing the orbit of the destinationof the packet.

Consider a cell with area and let denote thenumber of H(V)-nodes in the cell. For the one-dimensional mo-bility model, a cell is said to be a good cell at time slot if

Next we present the Joint Coding–Scheduling Algorithm III,which achieves the maximum throughput obtain in Theorem 13.

Joint Coding–Scheduling Algorithm III: The unit squareis divided into a regular lattice with cells, and the packetsize is chosen to be . We group every time slots intoa super-time slot. At each super-time slot, the nodes transmitpackets as follows.

(1) Raptor encoding: Each source takes datapackets, and uses the Raptor codes to generatecoded packets.

(2) Broadcasting: This step consists of time slots. Ateach time slot, the nodes execute the following tasks.

(i) In each good cell, one H-node and oneV-node are randomly selected. If the selectedH(V)-node has never been in the current cellbefore and not already transmitted all of its

coded packets, then it broadcasts acoded packet that was not previous transmittedto V(H)-nodes in the cell during theminislot allocated to that cell.

(ii) All nodes check the duplicate packets they have.If more than one B-duplicate packets are des-tined to the same rectangle, randomly keep oneand drop the others.

(3) Transporting: This step consists of time slots. Ateach time slot, the nodes do the following.

(i) Suppose that node is a V-node,and carriesB-duplicate packet . Node broadcasts

to H-nodes in the same cell iffollowing conditions hold: a) Node is in agood cell; b) B-duplicate packet is theonly transportable H-packet in the cell.

(ii) Each node checks the T-duplicate packets itcarries. If more than one T-duplicate packet hasthe same destination, randomly keep one anddrop the others.

(4) Receiving: This step consists of time slots. At eachtime slot, if a cell contains no more than two deliver-able packets, the deliverable packets are delivered totheir destinations using one-hop transmissions duringthe minislot allocated to that cell; otherwise, no node inthe cell attempts to transmit. At the end of this step, allundelivered packets are dropped. The destinations de-code the received coded packets using Raptor decoding.

Theorem 14: Consider Joint Coding–Scheduling Algo-rithm III. Suppose is and , and thedelay constraint is . Then given any , there existssuch that for any , every data packet sent out can be


recovered at the destination with probability at least , andfurthermore

Proof: Consider one super-time slot and let denote theevent that all cells are good in the super-time slot. The proof isbased the fact that the following events happen with high prob-ability.

Node distribution: All cells are good during the entire super-time slot with high probability. Specifically, it is easy to verifythat

(39)

Broadcasting: At least coded packets from asource are successfully duplicated after the broadcasting withhigh probability, where a coded packet is said to be success-fully duplicated if it has at least B-duplicate packets.Specifically, we have

(40)

Transporting: At least coded packets from asource are successfully transported after the transporting withhigh probability, where a coded packet is said to be successfullytransported if it has at least T-duplicate copies. Letting

denote the number of successfully transported packets fromnode , we have

(41)

Receiving: At least distinct coded packetsfrom a source are delivered to its destination after the receiving.Specifically, we have

(42)

Inequality (39) is easy to show, the proof of inequality (42) issimilar to the proof of inequality (18), and inequalities (40) and(41) are proved in Appendix A.

From the facts above, we can conclude that the probabilitythat the data packets are fully recovered in onesuper-time slot is at least


X. ONE-DIMENSIONAL I.I.D. MOBILITY MODEL,SLOW MOBILES

A. Upper Bound

Theorem 15: Consider the one-dimensional i.i.d. mobilitymodel with slow mobiles. Assume that Assumptions 1–3 hold,then

Proof: Follow from inequality (38) and the proof of The-orem 7.


In this subsection, we propose an algorithm which achievesthe delay–throughput tradeoff obtained in Theorem 15. Firstchoose

and scale the packet size to be

Further, we divide the unit square into horizontal rect-angles, named as H-rectangles; and vertical rectangles,named V-rectangles.

Joint Coding–Scheduling Algorithm IV: We group everytime slots into a super-time slot. At each super-time slot,

the packets are coded and transmitted as follows.(1) Raptor Encoding: Each source takes data

packets, and uses the Raptor codes to generate codedpackets.

(2) Broadcasting: The unit square is divided into a regularlattice with cells. This step consists of timeslots. At each time slot, the nodes execute the followingtasks.

(i) The nodes in good cells take their turns tobroadcast. If node is an H(V)-node andhas never been in the current cell before, itrandomly selects V(H)-nodes andbroadcasts a coded packet to them.

(ii) Each node checks B-duplicate packets it carries.If there are multiple B-duplicate packets des-tined to a same rectangle, randomly pick oneand drop the others.

(3) Transporting: The unit square is divided into a regularlattice with cells. This step consists of timeslots. At each time slot, the nodes do the following.

(i) Suppose node carries duplicate packet, which is an H-packet. If node is in

a good cell and is transportable, nodebroadcasts the packet to H-nodes in

the cell.(ii) Each node checks the T-duplicate packets it car-

ries. If there is more than one T-duplicate packetdestined to the same destination, randomly pickone and drop the others.

(4) Receiving: The unit square is divided into a regular lat-tice with cells. This step consists of timeslots. At each time slot, the nodes in good cells do thefollowing at the minislot allocated to their cells.

(i) The nodes which contain deliverable packetsrandomly pick one deliverable packet and senda request to the corresponding destination.


(ii) For each destination, it accepts only one re-quest.

(iii) The nodes whose requests are accepted transmitthe deliverable packets to their destinationsusing the highway algorithm proposed in [17].

At the end of this step, all undelivered duplicate packetsare dropped. Destinations use Raptor decoding to de-code the received coded packets

Theorem 16: Consider Joint Coding–Scheduling Algo-rithm IV. Suppose is both and , and thedelay constraint is . Then given any , there exists

such that for any , every data packet sent out canbe recovered at the destination with probability , andfurthermore

Proof: Following the analysis of Theorem 14, it can beshown that the following events happen with high probability.

Node distribution: All cells are good during the entire super-time slot with high probability, i.e.,

(43)

Broadcasting: At least coded packets from a sourceare successfully duplicated after the broadcasting with highprobability, where a coded packet is said to be successfullyduplicated if it has at least B-duplicate packets. Specifi-cally, we have

(44)

Transporting: At least coded packets from a sourceare successfully transported after the transporting with highprobability, where a coded packet is said to be successfullytransported if it has at least T-duplicate copies. Specifi-cally, we have

(45)

Receiving: At least distinct coded packets from asource are delivered to its destination after the receiving.Specifically, we have

(46)

Thus, the probability that the data packets are fullyrecovered in one super-time slot is at least


XI. ONE-DIMENSIONAL HYBRID RANDOM WALK MODELS,FAST AND SLOW MOBILES

In this section, we present the optimal delay–throughputtradeoffs of the one-dimensional hybrid random walk models.

The results can be established by following the analysis of theone-dimensional i.i.d. mobility models, where the techniquedifficulties caused by the random walk can be tackled bythe Azuma–Hoeffding inequality as in the two-dimensionalrandom walk models. The details are omitted here for brevity.

Theorem 17: Consider the one-dimensional hybrid randomwalk model and assume that Assumptions 1–3 hold. Then for

and , we have following results.(1) For fast mobiles

(47)

When and is bothand

, Joint Coding–Scheduling AlgorithmIII can be used to achieve a throughput the same as (47)except for a constant factor.

(2) For slow mobiles

(48)

When and is bothand , Joint Coding–Scheduling Algo-rithm IV can be used to achieve a throughput the sameas (48) except for a constant factor.

XII. CONCLUSION

In this paper, we investigated the optimal delay–throughputtradeoff in mobile ad hoc networks. The optimal tradeoffshave been established under some conditions on delay .When these conditions are not met, the optimal tradeoffs arestill unknown in general. We would like to comment that thekey to establishing the optimal delay–throughput tradeoff isto obtain , the probability that node hits nodein one of consecutive time slots given a hitting distance

. For example, under the two-dimensional hybrid randomwalk model, the upper bound was obtained under the condition

since it was the condition under whichwe established an upper bound on (inequality (30)).Further, the maximum throughput was shown to be achievableunder a more restrict conditionsince it was the condition under which we established a lowerbound on (inequality (65)). Thus, if we can findtechniques to compute without using the restrictson , then the delay–throughput tradeoffs can be characterizedmore generally. This is a topic for future research.

APPENDIX ARAPTOR CODES

Raptor codes were proposed by Shokrollahi in [19] andthey have low coding complexity. Suppose we have an erasurechannel with erasure probability . To transmit sourcepackets over this erasure channel, Raptor codesfirst generate coded packets as follows.

Raptor Encoding:(1) Use a right-regular low-definition parity-check

(LDPC) code to encode the source packets and


generate precoded packets,where for some .

(2) Use Luby transform (LT) encoding [20] with mod-ified robust solition distribution [19] to generatecoded packets.

After receiving coded packets, the source packetsare decoded as follows.

Raptor Decoding:(1) Use LT decoding [20] to decode at least

precoded packets, where .(2) Use brief propagation for the right-regular LDPC

code to decode the original packets.

We say packets are successfully delivered if for. The following result is presented in [19].

Lemma 18: The receiver can correctly decode the datapackets with probability at least for some

after it obtains coded packets generated by Raptorcodes. The number of operations used for encoding and de-coding is .

The algorithms proposed in this paper transmit datapackets in time slots. Thus, the time taken by Raptor en-coding/decoding is , which is at most at the same order asthe delay . So we omit the delay due to coding in our analysis.

APPENDIX BTHROUGHPUT OF STATIC WIRELESS NETWORKS

The throughput of a random wireless network with staticnodes and random S-D pairs is introduced by Gupta andKumar [1]. They showed that the maximum throughput per S-Dpair is , and proposed a scheduling scheme achievinga throughput of per S-D pair. This gapwas later closed by Franceschetti et al. in [17] where the authorsshowed a throughput of per S-D pair is achievable.The result is obtained under the physical interference models.However, it can be easily extended to the protocol model byusing the same algorithm.

Lemma 19: In a random wireless network with staticnodes and S-D pairs, a throughput of

bits/time slot per S-D pair is achievable, where is a positiveconstant independent of .

Suppose the nodes use a common transmission radius. The key idea of [17] is to construct disjoint paths

traversing the network vertically and horizontally. These pathsare called highways in [17], and a throughput of perS-D pair is achievable by transmitting data throughput thesehighways. We call this algorithm a highway algorithm.

APPENDIX CPROBABILITY RESULTS

In this appendix, we present some standard results in proba-bility for the reader’s convenience. In addition, we also presentsome variations of standard results which do not seem to beavailable in any book to best of our knowledge.

The following lemma is a standard result in probability, whichwe provide here for convenience.

Lemma 20: Let be independent – randomvariables such that . Then, the following Cher-noff bounds hold:

(49)

(50)

Proof: A detailed proof can be found in [21].

The next lemmas are variations of standard balls-and-binsproblems. However, we have not seen the results for the par-ticular variation that we need in this paper. So we present thelemmas along with brief proofs below.

Lemma 21: Assume we have bins. At each time, choosebins and drop one ball in each of them. Repeat this times.

Using to denote the number of bins containing at least oneball, the following inequality holds for sufficiently large :

(51)

where .Proof: At each time, bin receives a ball with probability. We let denote the number of balls in bin . Now con-

sider a related balls-and-bins problem where the ball droppingprocedure is replaced by a certain number of trials as dictated bya Poisson random variable. Specifically, define to be a Poissonrandom variable with mean , and repeat the ball dropping pro-cedure times. Let denote the number of balls in bin in thiscase. It is easy to see that are i.i.d. Poisson random vari-ables with mean . So we can conclude

Since

from Lemma 20, we have

The above idea of using a Poisson number of trials to boundthe probability of the occurrence of an event in a fixed numberof trials is called the Poisson heuristic in [21].


Lemma 22: Suppose balls are independently droppedinto bins and one trash can. After a ball is dropped, the prob-ability in the trash can is , and the probability in a specificbin is . Using to denote the number of bins containingat least one ball, the following inequality holds for sufficientlylarge :

(52)

where .Proof: Let denote the number of balls in bin . Next de-

fine to be a Poisson random variable with mean . We considerthe case such that balls are independently dropped in bins.Using to be number of balls in bin in this case, it is easy tosee that are i.i.d. Poisson random variables with mean .So we have

Since

from Lemma 20, we have

which implies for sufficiently large

Next we introduce the Azuma–Hoeffding inequality.

Lemma 23: Suppose that are independentrandom variables, and there exists a constant such that

satisfies the following condition forany and any set of values and :

Then we have

Proof: A detailed proof can be found in [21].

APPENDIX DPROOF OF INEQUALITIES (17), (18), AND (19)

Proof of Inequality (17): Let denote the event thatnode broadcasts a coded packet at time slot . So occurswhen following two conditions hold:

(1) the cell node in is a good cell;(2) node is selected to broadcast.

Since the nodes are uniformly randomly positioned, from theChernoff bound we have

which implies that there exists such that for any

Then from the Chernoff bound again, we have

(53)

for . Thus, with high probability, more thancoded packets are broadcast, and each broadcast generates

copies.Duplicate packets might be dropped at step (ii) of the broad-

casting. We next calculate the number of duplicate packets ofnode left after the broadcasting. Assume node broadcastscoded packets, so . Then the number of duplicatepackets left after the broadcasting is the same as the number ofnonempty bins of the following balls-and-bins problem, wherethe bins represent the mobile nodes other than node , and theballs represent the duplicate packets broadcast from node .

Balls-and-Bins Problem: Assume we have bins. Ateach time slot, we select bins and drop one ball in eachof them. Repeat this times.

Using to denote this number, from Lemma 21, we have

where

Using the fact for any , we get

where the last inequality holds for for some since. Thus, choose and we can con-

clude for

(54)


Recall a coded packet is said to be successfully duplicated ifit has at least copies at the end of the broadcasting. In-equality (54) implies for

since, otherwise, less than duplicate packets are leftin the network. Thus, we can conclude that for

(55)

Letting , inequality (17) follows from in-equalities (53) and (55) for .

Proof of Inequality (18): Assume coded packetsare successfully duplicated.

We let denote the event that coded packet isdelivered at time slot . Then will definitely occur ifboth the following conditions hold.

(1) One and only one duplicate packet of becomes adeliverable packet. Let denote this event. As-sume the duplicate packet is , i.e., node con-tains packet .

(2) There are no other deliverable packets in the cell con-taining node except packet and one possibleduplicate packet to node carried by node . Let

denote this event.Note that duplicate packets of node are carried by different

nodes, and their mobilities are independent. Now assume thereare copies of in the entire network, then

Note that andif is successfully duplicated. So for a suc-

cessfully duplicated packet, there exists such that for any

Suppose we have nodes in the cell containing node ; fromthe Chernoff bound, we have

Note that condition (2) is equivalent to the following event:Given node and node in the cell, no more deliverablepackets appear when we put another nodes into the cell.Now given nodes in the cell, the probability that no more de-liverable packet appears when we put another node is at least

This holds due to the following two facts.(i) The new node should not be the destination of any du-

plicate packets already in the cell (there are at mostduplicate packets already in the cell).

(ii) The duplicate packets carried by the new node are notdestined for any of the existing nodes. Note that eachsource has no more than duplicate packets, so thereare at most nodes which carry the duplicate packettoward the existing nodes.

Note that , so there exists such that for any

So we can conclude that for any

(56)

which implies that at each time slot, a successfully dupli-cated packet will be delivered with probability at least

. Note at each time slot, only one coded packetcan be delivered to the destination of node . So the numberof distinct coded packets delivered to the destination of node

is the same as the number of nonempty bins of followingballs-and-bins problem, where the bins represent the distinctcoded packets, the balls represent successful deliveries, and aball is dropped in a specific bin means the corresponding codedpacket is delivered to the destination.

Balls-and-Bins Problem: Suppose we havebins and one trash can. At each time slot, we drop a ball. Eachbin receives the ball with probability , and the trashcan receives the ball with probability , where

Repeat this times, i.e., balls are dropped.Let denote nonempty bins of the above balls-and-bins

problem and choose . From Lemma 21, we have

and inequality (18) holds for , where.

Proof of Inequality (19): Inequality (19) follows fromLemma 18 on the error probability of Raptor codes.

APPENDIX EPROOF OF INEQUALITY (28)

Proof: Assume that coded packetsare successfully duplicated. Note that duplicate packets from acommon source are carried by different nodes; and the mobili-ties of those nodes are independent. So will be definitelydelivered to its destination if both the following conditionshold.


(1) A copy of is the only deliverable packet for desti-nation in time slot . Let denote this event;and assume that the duplicate packet is in node .

(2) Node has no other deliverable packet, and the cell con-taining node is good. Let denote this event.

Let denote the number of copies of . Since eachsource has at most duplicate packets in the network,we have

It is easy to verify that

Thus, for successfully duplicated packet , i.e.,, we can conclude that

(57)

holds for sufficiently large . Note that each node carries at mostduplicate packets, so we further have

(58)

where is the lower bound on the probabilitythat all packets in node except are undeliverable, and

is the probability that the cell is good.From inequalities (57) and (58), we can conclude that for suf-

ficiently large

Inequality (28) can be proved by the balls-and-bins argumentused to show inequality (18).

APPENDIX FPROPERTIES OF RANDOM WALK

Consider following two random walks.(1) One-dimensional random walk: A random walk on a

circle with unit length and points. At each time slot,a node moves to one point left, one point right or doesnot move with equal probability as in Fig. 12.

(2) Two-dimensional random walk: A random walk on aunit torus with points. At each time slot, a nodemoves to one of eight neighbors or does notmove withequal probability as in Fig. 13.

We introduce following definitions.• Transition matrix where is the proba-

bility of moving from point to point .• Stationary distribution : A vector which satisfies the

equation .

Fig. 12. One-dimensional random walk.

Fig. 13. Two-dimensional random walk.

• Hitting time : Time taken for a node to move frompoint to point .

• Mixing time

where is the th entry of .

Lemma 24: For the one-dimensional random walk, wehave

• ;• .For the two-dimensional random walk, we have• ;• .

Proof: Please refer to [22] for the one-dimensionalrandom walk. The proof of the hitting time of the two-dimen-sional random walk is presented in Lemma 13 in [7], and themixing time result holds since the two-dimensional randomwalk can be regarded as two independent one-dimensionalrandom walks.


Lemma 25: Let denote the number of visits topoint in time slots starting from point and ending at point

. If , we have

(59)

where for the one-dimensional random walk andfor the two-dimensional random walk. Furthermore, if

where (1), then we have

(60)

Proof: First we have

where is the time duration between th visits to pointand th visits to point . Taking the expectation on bothsides, we have

which implies

Inequality (59) follows from the facts that

and

Next, let denote the position of the node at timeslot denote for

, and denote for

where .

Further, let denote the number of visits to point

given . It is easy to see that for any there exists a functionsuch that

where are mutually independent given . Note thatcontains the position information from time slotto time slot , so

Next note that , so from inequality(59), we can conclude that for any , , and

which implies that

holds for any . Then from the Azuma-Hoeffding inequality(Lemma 23), we have that

for any , and inequality (60) holds.

APPENDIX GPROOF OF INEQUALITY (30)

Let denote the number of time slots, from to ,at which node and are in the same RW-cell or neighboringRW-cells. Then for any , we have

where the first inequality follows from the fact that the nodeposition within an RW-cell is randomly uniformly selected, andthe last inequality follows from the Jensen’s inequality.

Next we consider . Let denote theRW-cell in which node is at time slot , anddenote the displacement of node at time slot , i.e.,

w.p.w.p.w.p.

andw.p.w.p.w.p. .

It is easy to see that

Further, let denote the relative position ofnode from node , i.e.,


wherew.p.w.p.w.p.w.p.w.p.

andw.p.w.p.w.p.w.p.w.p.

So is the consequence of random walk

with initial position

Note that node and node are in the same RW-cell if, and in neighboring RW-cells

if

Similar to the argument in Lemma 25, we can conclude that for

which implies that

APPENDIX HPROOF OF INEQUALITIES (31), (32), AND (33)

Proof of Inequality (31): Since im-plies . Inequality (31) can be obtained from theChernoff bound and union bound.

Proof of Inequality (32): Consider the broadcasting. Notethat when occurs, node is selected to broadcast with proba-bility at least at each time slot. Let denote theevent that node is selected to broadcast in time slot . From theChernoff bound, we have

(61)

So node broadcasts coded packets with a highprobability. Each coded packet is broadcast to relaynodes.

According to Step (2) (ii) of Joint Coding–Scheduling Al-gorithm I, each relay node keeps at most one packet for eachsource. Consider duplicate packet . It could be dropped

if node is in the same cell as node and node is selectedto broadcast. Thus, the probability that is dropped is atmost

(62)

due to the following two facts.(1) Let denote the event that node is in the same

cell as node in at least one of consecutive time slots.Similar to (30), it can be shown that

(63)

under the delay constraint given in the theorem.(2) When occurs, node is selected to broadcast with

probability at most at each time slot.Now suppose source broadcasts coded packets, so

duplicate copies are generated. Let denote thenumber of duplicate packets of node dropped in the broad-casting. From the Markov inequality and inequality (62), wehave

which implies

(64)

since otherwise, more than duplicate packets wouldbe dropped. Inequality (32) follows from inequalities (64) and(61).

Proof of Inequality (33): We group every timeslots into big time slots, named as b-time slot and indexed by ,and then divide every b-time slot into three equal parts, indexedby , , and as in Fig. 14. We first calculate the proba-bility that coded packet is delivered in . Letdenote the event that at least one copy of packet becomesdeliverable in . If is in at least relay nodes, wecan obtain

(65)

due to the following facts.


Fig. 14. Division of a super-time slot.

(1) Given , from Lemma25 provided in Appendix F we know that with proba-bility at least , two nodes are in the same RW-cellfor at least time slots.

(2) Given two nodes are in the same RW-cell, the probabilitythat they are in the same cell is .

Next note that the duration of and are of a larger orderthan the mixing time of the random walk. From the definition ofthe mixing time, we have that at any time slot belonging to ,the nodes are almost uniformly distributed in the unit square. Let

denote the event that coded packet is deliveredto its destination in . Following the argument used to showinequality (56), we have

(66)

Now let denote the positions of the nodes at time slot ,and

for . Also, let denote the event thatis delivered in the receiving. It is easy to see that

occurs if occurs for some .Note that are mutually independent given ,so from inequality (66), we have

Further, since , we can conclude that

(67)

We next bound the number of distinct coded packets deliver-able in . Similar to inequality (65), we have

Note that no two duplicate packets from node are in one relaynode, so are mutually independent. From theChernoff bound, we have

Let denote the event that node obtains no more thancoded packets at each b-time slot in the

receiving. From the union bound, we have that for sufficientlylarge

(68)

Now let denote the number of distinct codedpackets delivered to the destination of node given ,and denote an matrix where the entryis the position of node at the th time slot of b-time slot . Itis easy to see that the value of is determined by

, i.e., there exists a function such that

From the definition of , function satisfies the fol-lowing condition:

(69)

It is easy to see that are mutually independent given. Then invoking Azuma–Hoeffding inequality pro-

vided in Appendix A, we can conclude that

(70)

holds for any and . Inequality (33) follows from inequali-ties (67), (68), and (70).

APPENDIX IPROOF OF INEQUALITIES (40) AND (41)

Proof of Inequality (40): Assume that occurs, then ateach time slot, node is selected with probability .Note that there are cells on the orbit of node , and node

is uniformly randomly positioned in one of the cells. Thus, thenumber of coded packets broadcast by node is equal to thenumber of nonempty bins of following balls-and-bins problem.

Balls-and-Bins Problem: Suppose we have binsand one trash can. At each time slot, we drop a ball. Each binreceives the ball with probability , and the trashcan receives the ball with probability . Repeatthis times, i.e., balls are dropped.

From Lemma 22, we have

We say two nodes are competitive with each other if the orbitsof their destinations are contained in the same rectangle, so eachnode has competitive nodes. Suppose that node is anH-node and node is a V-node. Let denote the numberof node ’s competitive nodes in the V-rectangle containing theorbit of node at time slot . Since nodes are uniformly, ran-domly positioned on their orbits, from the Chernoff bound, wehave

(71)


Now consider B-duplicate packet and assume that node, a competitive of node , is in the V-rectangle containing the

orbit of node . Then might be dropped when it is in thesame cell as node , and node is selected to broadcast. Theprobability of this event is at most

(72)

From (71), (72), and the union bound, we can conclude that theprobability that is dropped at time slot is at most

(73)

which implies that the probability of dropped in thebroadcasting is at most

Inequality (40) follows from above inequality and the Markovinequality.

Proof of Inequality (41): Consider an H-node . Letdenote the number of B-duplicate packets which are containedin the V-rectangle where broadcast, and are destined tothe same H-rectangle as node . Note the following facts.

(1) Each node has competitive nodes.(2) Each H-node broadcasts at most coded packets.

The probability that a coded packet broadcast in a spe-cific V-rectangle is at most

(3) Each broadcast generates duplicate copies.Thus, from the Chernoff bound, we have that

which implies that for sufficiently large

(74)

Let denote the event that a B-duplicate packet is broadcastat time slot in the transporting. If is successfully dupli-cated, i.e., there are at least B-duplicate copies of ,we have

Further, let denote the event that at least one copy ofis broadcast in the transporting. Then for sufficiently large , wecan obtain that

Let denote the number of distinct coded packets of nodebroadcast in the transporting, i.e.,

Since different coded packets of node are broadcast in dif-ferent V-rectangles, are mutually independent. Fromthe Chernoff bound, we have

(75)

In the transporting, a T-duplicate copy will be dropped if thenode carrying it obtains another packet destined to the samedestination. Consider a T-duplicate packet carried bynode . Note following facts.

(1) Coded packets of node are broadcast in at mostV-rectangles.

(2) Each rectangle contains at most B-duplicatecopies from node .

Thus, the probability of dropped at time slot is at most

The node mobility is independent across time, so the probabilityof dropped in the transporting is at most

Let denote the number of duplicate packets dropped in thetransporting. Note that T-duplicate packets are gen-erated, and each of them has probability to be dropped.Using the Markov inequality, we have

which implies

(76)

since otherwise, more than duplicate copies aredropped. Inequality (41) follows form inequalities (74)–(76).

ACKNOWLEDGMENT

The authors gratefully acknowledge the useful discussionswith Prof. Geir E. Dullerud and Prof. Bruce Hajek, Univer-sity of Illinois at Urbana-Champaign. They also wish to thankProf. Michael J. Neely, University of Southern California, forhis comments on the earlier version of this paper.

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