Throughput-Delay Tradeoffs in Large-Scale MANETs with Network Coding

Transcription

1 Throughput-Delay Tradeoffs i Large-Scale MANETs with Network Codig Chi Zhag ad Yuguag Fag Departmet of Electrical ad Computer Egieerig Uiversity of Florida, Gaiesville, FL {zhagchi@, fag@ece.}ufl.edu Xiaoya Zhu Natioal Key Laboratory of Itegrated Services Networks Xidia Uiversity, Xi a, Chia xyzhu@mail.xidia.edu.c Abstract This paper characterizes the throughput-delay tradeoffs i mobile ad hoc etworks (MANETs) with etwork codig, ad compares results i the situatio where oly replicatio ad forwardig are allowed i each ode. The schemes/protocols achievig those tradeoffs i a effective ad decetralized way are proposed ad the optimality of those tradeoffs is established. The scearios i which etwork codig ca provide sigificat improvemet o etwork performace are idetified uder differet ode mobility patters (fast ad slow mobility). The isights o whe ad how iformatio mixig is beeficial for MANETs with multiple uicast ad multicast sessios are provided. As far as we kow, this is the first work characterizig scalig laws of throughput ad delay of MANETs with etwork codig. I. INTRODUCTION Oe distict characteristic of wireless mobile ad hoc etworks (MANETs) is that, besides trasportig data through multi-hop coected paths betwee the source ad destiatio, packets ca also be delivered through the physical mobility of relay odes which is called store-carry-ad-forward paradigm i the literature []. Grossglauser ad Tse [2] have show that sigificat gais i per-ode throughput ca be obtaied by exploitig this paradigm. I particular, they proposed a 2-hop relayig scheme, ad showed that it ca achieve a costat per-ode throughput. The scheme overcomes the throughput boud of O(/ log ) origially established by Gupta ad Kumar [3] for a static wireless etwork, where is the umber of odes for defiitios of the stadard asymptotic otatio used throughout the paper). Although heavy use of relayig through ode mobility allows for higher throughput, it also bears egative side-effects: icreased delay. It has bee show i [4], [5] that the 2-hop relayig scheme i [2] yields a extremely large average delay of Ω(). Sice both throughput ad delay are importat etwork performace metrics from the perspective of a applicatio, sigificat effort i the last few years has bee devoted to uderstad the throughput-delay relatioship i MANETs (refer to Sectio II-A ad the refereces therei) i the etworkig research commuity. A iterestig work by Neely ad Modiao [5] suggested to utilize redudat packets trasmissio through multiple opportuistic paths (which are composed of multiple opportuistic liks) of a MANET to balace the coflictig requiremets o throughput ad delay. The basic idea is that the time required for a packet to reach the destiatio (i.e., ed-to-ed delay) ca be reduced by repeatedly trasmittig this packet to may relay odes of the etwork, ad thus improvig the chaces that some user This work was supported i part by the U.S. Natioal Sciece Foudatio uder grat CNS , CNS , CNS ad DBI The work of Fag ad Zhu is also partially supported by the project uder B08038 with Xidia Uiversity. holdig a origial or duplicate versio of the packet reaches the destiatio ode. Clearly, the cost of this approach is the decreased throughput sice duplicate packets waste scarce opportuities of wireless trasmissios. I particular, with i.i.d. mobility, it was show that for per-ode throughput T () =O(), the relayig strategies with replicatio could yield ed-to-ed delay D() scalig as Θ( T ()) [5]. Previous studies o the scalig laws of MANETs, as discussed above, are all based o the implicit assumptio that each ode ca oly perform traditioal operatios o packets, such as storage, replicatio ad forwardig. Recetly, etwork codig, first itroduced by Ahlswede et al. [6] i 2000, has bee widely recogized as a promisig primitive operatio besides simple replicatig ad forwardig icomig packets [7]. Usig the paradigm of etwork codig, whe a ode is scheduled to trasmit, it may trasmit a mixed packet as a result of algebraic operatios o its icomig packets to maximize the usefuless of this trasmissio to all receivers i its trasmissio rage. It is worth otig that a particular useful form of etwork codig called Radom Liear Codig (RLC) was proposed i the literature [8], [9] to idepedetly ad radomly mix icomig packets at each ode with liear operatios, which allows the odes of the etwork to achieve the optimal performace i a decetralized fashio. Ituitively, whe RLC istead of replicatio is used to miimize the ed-to-ed delay, etwork cogestio ca be alleviated ad the requiremet o buffer size ca be relaxed. Therefore, a better throughput-delay tradeoff is expected to be obtaied. Sice etwork codig was ot take ito cosideratio i Grossglauser ad Tse s origial work [2] ad the related work [4], [5], [0] that followed, a iterestig questio raised aturally is how much beefit etwork codig ca provide to the etwork performace of MANETs compared to whe oly simple replicatio ad forwardig are allowed for relay odes. Aswerig this questio will help us better uderstad ot oly the beefits ad limitatios of etwork codig i wireless etworks but also the fudametal tradeoffs determiig MANET s performace. I this paper we coduct a thorough study o the scalig laws goverig MANETs. We characterize the throughputdelay tradeoffs with respect to differet ode mobility patters. We idetify scearios i which etwork codig ca provide sigificat improvemet o etwork performace. Note that our work differetiates MANETs from static wireless etworks by the roles etwork codig plays, because previous work showed that etwork codig could oly provide costat improvemet o the throughput of static wireless etworks (cf. Sectio II-C ad the refereces therei). We also provide isights o whe ad how iformatio mixig is beeficial ad propose algorithms to show that these beefits ca be achieved i a effective ad decetralized fashio /09/$ IEEE 99

2 II. BACKGROUND AND RELATED WORK A. Scalig Laws of MANETs without Network Codig Semial work of Gupta ad Kumar [3] iitiated the ivestigatio o how the throughput of wireless etworks scales with, the umber of odes. Uder the assumptio that odes with commo trasmissio rage are radomly distributed, it is show that per-ode throughput for static wireless etworks scales as Θ(/ log ). Note that [3] implicitly used a fluid model for establishig throughput scalig. Later work by Kulkari ad Viswaath [] cosolidated the result of [3] with a explicit costat packet size model. I [2], with percolatio theory, Fraceschetti et al. showed that the Θ(/ ) per-ode throughput is achievable if each ode ca adjust its trasmissio rage (through power cotrol), however, the throughput vaishig problem for large-scale ( ) static wireless etworks still remais. I [2], Grossglauser ad Tse showed that the mobility of the odes i a MANET ca be exploited to overcome this problem. The 2-hop relayig scheme they proposed achieves a costat per-ode throughput at the cost of a large delay o the order of [4], [5]. This result reveals the possibility of tradig larger delay for higher throughput or lower throughput for smaller delay i MANETs. Sice the, a flurry of research activities have tried to characterize the throughput-delay relatioship with respect to ode mobility, e.g., [4], [5], [3] [20]. I geeral, there are two ways to trade throughput for delay i the literature. Kleirock et al. [2] may be the first to fid that delay ca be reduced by icreasig the trasmissio radius of each relay ode, at the expese of reducig the umber of simultaeous trasmissios the etwork ca support, which leads to a lower throughput. Similar trasmissio radius scalig techiques have appeared i [4], [4] [20]. Aother approach, which improves delay via redudat packet trasfers is cosidered i [5], [22]. I this paper, we follow this approach, adoptig redudat strategy ad comparig it with etwork codig for the followig reasos: First of all, the assumptio that trasmissio rages ca scale with, the umber of odes, is impractical for large-scale MANETs. To obtai the scalig law of MANETs, we usually require tedig to ifiity, which is equivalet to assumig A for exteded etwork model, where A is the area of the etwork (cf. Sectio III-A). I geeral, wireless device is power limited, rederig it impossible to require the trasmissio rage reachig the order of A. Secod, tradeoffs theoretically aalyzed usig the first meas metioed above are maily based o fluid model, i which the packets are allowed to be arbitrarily small as (e.g., [4], [4] [7], [9], [20]). O the other had, tradeoffs obtaied through the secod approach assume costat packet size model, where the packet size remais costat, i.e., does ot scale dow with (e.g., [5]). We prefer the costat packet size model sice i reality, packet size does ot chage whe more odes are added to the etwork. Furthermore, fluid model caot be applied to scearios with etwork codig, sice every coded packet icludes a code vector of at least costat size for successful decodig. Note that with the additioal costrait that the packet size remais costat, the throughput-delay tradeoff ca be o better tha that i the fluid model, ad the aalysis of costat packet size model is much harder tha that of fluid model [8]. Throughout the paper, our results o scalig laws of MANET with or without etwork codig are all based o costat packet size model (cf. Sectio III-A) for the rigor of theoretical aalysis. Fially, i this paper we are iterested i examiig pure gais itroduced by etwork codig i MANETs. Replicatio strategies ca be replaced by etwork codig, which provides a good chace for compariso. Trasmissio radius scalig techiques, however, are orthogoal to etwork codig, ad should be studied separately. We would like to poit out that all the results discussed above are based o the implicit assumptio that oly storagead-forwardig (without etwork codig) is allowed i each ode, while i this paper we seek to uderstad whether etwork codig ideed affects the throughput-delay tradeoffs i MANETs. B. Network Codig Applicatios i Wireless Networks At the very begiig, research o etwork codig maily focused o multicast sceario. I their pioeerig theoretical work, Alswede et al. [6] showed that the mi-cut throughput of a multicast sessio o a directed graph ca be achieved, provided that oe allows etwork codig, i.e., ecodig at the itermediate odes of the etwork. Coversely, it is geerally ot possible to achieve this commuicatio rate if oe allows oly routig or copyig packets at the itermediate odes of the etwork. Shortly afterwards, Li, Yeug, ad Cai [23] showed that it is sufficiet for the ecodig fuctios at the iterior odes to be liear. Subsequet work by Jaggi et al. [8] ad Ho et al. [9] showed that the liear ecodig fuctios ca be desiged radomly ad idepedetly at each ode, which leads to a particular useful form of etwork codig, the RLC. Sice RLC operates i a decetralized fashio [24], [25] which is extremely suitable for MANETs where cetralized cotrol is almost impossible or costly, we cocetrate o RLC throughout the paper. Performace of RLC for local ad global broadcast i wireless etworks has bee extesively studied i the literature, e.g., [7], [26] [30]. For practically more importat case of multiple uicast, we ca oly ascertai that it is totally differet from multicast cases. For example, Li ad Li cojectured i [3] that for udirected etwork with multiple uicast sessios, etwork codig does ot help much o throughput. A deep uderstadig o achievable throughput for multiple uicast sessios i a etwork is still a ope problem. I geeral, it is ot clear whether etwork codig should be performed, ad if it should, what the strategy must be [7]. Oe of our mai cotributios i this paper is that we aalytically address this problem ad show that, although RLC still caot improve the order of throughput i MANETs, it chages the achievable throughput-delay tradeoffs sigificatly, which we believe will help improve our uderstadig of the theoretical limits o the beefits of etwork codig ad o how to achieve them for MANETs with multiple uicast sessios. The idea that, whe RLC is allowed i itermediate odes, compared to replicatio strategies [5], [22], larger throughput ca be achieved with the same delay ad smaller sizes of ode buffers, was perhaps first explicitly developed i the work [32] by Zhag et al., where a simulatio-based study of the beefit of RLC i oe uicast commuicatio is also preseted. The recet work by Li ad Li [33] gives a rigorous aalysis of this idea based o ordiary differetial equatios. To our kowledge, [32] ad [33] are the closest to our work i terms of uderstadig the relatioships betwee throughput ad delay with etwork codig. However, our work has the followig advatages: First of all, istead of explicitly modelig odes spatial distributios as i this paper, the mobility of odes 200

3 i [32] ad [33] is modeled with meetig times of ay pair of odes, to simplify the aalysis. The problem is that, the most importat feature of wireless trasmissio, i.e., iterferece, is ot icluded i their modelig. It is evertheless still reasoable for [32] ad [33], sice the authors are maily iterested i delay tolerat etworks, where odes are assumed to be sparsely distributed ad iterferece from simultaeous trasmissios ca be igored. However, it is obviously ot suitable for the study of geeral MANETs. Secod, the traffic patter cosidered i our paper is more practical. The umber of uicast sessios supported i this paper is Θ(), while oly oe uicast or broadcast sessio is assumed i [32] ad [33]. Next, oly epidemic routig ad its replacemet of etwork codig are cosidered i [32] ad [33], while i our work, several alteratives are cosidered ad differet algorithms are developed, which achieve throughputdelay pairs o differet orders of. Therefore, we obtai a complete characterizatio of tradeoffs i MANETs. Most importatly, explicit expressios of etwork performace or tradeoffs are obtaied i our paper for the first time, which provides isights o the degree of scalability of MANETs with etwork codig. C. Scalig Laws of Wireless Networks with Network Codig Scalig laws goverig wireless etworks with etwork codig have oly bee ivestigated i the limited scearios i the literature recetly. The delay gais ad reliability beefit (measured i the reduced umber of trasmissios) of etwork codig i ureliable wireless etworks were characterized i [34], [35] ad [36], respectively. However, these results are for oe multicast sessio with oe-hop trasmissio or stable etwork topology. For multiple uicast sceario, Liu et al. [37], [38] ad Keshavarz-Haddadt et al. [39] showed that for static wireless etworks, etwork codig ad broadcastig at most provide a costat-factored improvemet i the throughput, compared to Gupta ad Kumar s Θ(/ log ) per-ode throughput [3]. As far as we kow, the scalig laws for throughput ad delay are still uexploited for MANETs i the literature. More importatly, our results show that, etwork codig ca provide sigificat improvemet o etwork performace whe mobility is utilized, which is impossible i static wireless etworks [37] [39]. We believe it reveals the itrisic differece betwee MANETs ad static wireless etworks. III. MANET MODELS AND DEFINITIONS A. Network Models Radom etwork model for MANETs: Cosider a ad hoc etwork where odes are distributed uiformly at radom i a square area of A. The square is assumed to be a torus 2, i.e., the top ad bottom edges are assumed to touch each other ad similarly the left ad right edges also are assumed to touch other. We cosider a multiple () uicast sceario i which each ode i {, 2,,} is a source ode for oe uicast sessio, ad a destiatio ode for aother uicast sessio. Suppose that the source ode i has data iteded for destiatio ode d(i). We assume that each source ode has a ifiite stream of packets to sed to its destiatio. 2 We assume the torus to avoid edge effects, which otherwise complicates the aalysis. We ote, however, that the results i the paper hold for square, disk or ay other shapes of practical iterests. s = A / m Cell Coordiates Fig.. (,) (, 2) ( 2,) ( 2,2) Fast Mobility (i-, j) Slow Mobility (i, j-) (i+, j) (i, j) (i, j+) ( m,) ( m,2) (, ) m (2, ) m m m (, ) Fast ad slow mobility models for MANETs. The source-destiatio (S-D) associatio does ot chage with time, although the odes themselves move. Mobility models: The torus is divided ito m =Θ() square cells of area A /m each, resultig i a two-dimesioal m m discrete torus 3, see Fig. for a illustratio. The iitial positio of each ode is equally likely to be ay of the m possible cells idepedet of others. We assume the time is slotted ad we study the followig mobility models i this paper: Fast mobility model (i.i.d. mobility model): At each time slot, odes radomly choose a ew cell locatio idepedetly ad idetically (i.i.d.) distributed over all cells i the etwork. This model captures the situatio whe mobile user moves so quickly that its positio is almost idepedet from time to time. With this assumptio, the etwork topology dramatically chages i every time slot, so that the etwork behavior caot be predicted ad fixed routig algorithms caot be used. This mobility model is also used i [5], [4], [6], [9], [20]. Slow mobility model (radom walk model): Let a ode be i cell (i, j) {, m} 2 at time slot t, the, at time slot t +, the ode is equally likely to be i the same cell (i, j) or ay of the four adjacet cells {(i,j), (i +,j), (i, j ), (i, j +)}, where additio ad subtractio are modulo m. So each ode i fact idepedetly performs a simple radom walk o the two-dimesioal m m discrete torus. Note that this model implicitly sets a upper boud o the velocity of odes as 2A /m. Therefore, it is a suitable model for capturig real motio of odes with slow mobility. Similar mobility model is also adopted i [4], [7] [9]. Model for successful trasmissio: For characterizig the coditio for a successful trasmissio, we adopt the protocol model as defied i [3]. We assume that all odes use a commo rage r c for their trasmissios, ad a trasmissio from ode i to ode j is successful if ad oly if d ij r c ad d kj ( + Δ)r c for ay other simultaeous trasmitter, say ode k. Here, d ij is the distace betwee odes i ad j, ad Δ is a positive costat idepedet of. Durig a successful trasmissio, odes sed data at a costat rate of W bits per secod. I the other commoly used model of successful trasmissio, amely, the physical model, a trasmissio is successful if the SINR is greater tha some costat. It is well kow [3], [4] that with a fadig factor α > 2, the protocol model is equivalet to the physical model. Therefore, we prefer the use of the protocol model i this paper for a cleaer presetatio of the key ideas. Cocurretly trasmittig cells: Now we defie the trasmissio rage ad schedule. We choose r i such a way 3 For simplicity, assume m is a iteger. A 20

4 K = 4 Group Number s = A / m Trasmitter ( +Δ) r c Receiver r = Fig. 2. Cell trasmissio schedulig. Here is a illustratio of the cells beig divided ito K 2 groups for the case of K =4, i.e., 6 groups. All the blue cells which are i group trasmit i the same timeslot. I the ext timeslot all the cells i group 2 trasmit ad so o. that ay ode i a cell ca always directly trasmit to ay other ode i the same cell usig the smallest commo rage of trasmissio. Obviously, r c = 2s = 2A /m = Θ( A /). Time is slotted for packetized trasmissio. We assume oly O() packets ca be trasmitted per cell per timeslot, i.e., our aalysis is explicitly based o costat packet size model (see [8] ad Sectio II-A for detailed discussio). We adopt the cell schedulig scheme show i Fig. 2, which has the followig Propositio []. Propositio : Uder the Protocol model, there exists a iterferece-free schedule such that each cell becomes active regularly oce i K 2 timeslots ad it does ot iterfere with ay other simultaeously trasmittig cells. Here K depeds oly o Δ, ad is idepedet of. Exteded etwork model: We are particularly iterested i asymptotic properties of MANETs, which hold with high probability 4 for large-scale MANETs. Therefore, we eed ofte take limits as. Whe the regio area A is fixed, it correspods to a dese etwork model [3], [40], because the desity of the etwork d = /A also teds to ifiity as. Aother widely used model is the exteded etwork model [4], [42], i which both the umber of odes ad the area of the regio A go to ifiity while d is kept costat. Both models are widely used i the literature ad we will focus o the latter oe. I the exteded etwork model, A = /d = Θ(), ad correspodigly r c =Θ( A /) = Θ(). Thisis more practical, sice power costrait o wireless devices does ot chage whe more odes are added to the etwork. We ote that, however, results obtaied i this paper ca be easily exteded to dese etwork model. B. Network Performace Metrics Defiitio of throughput: A throughput λ>0 is said to be feasible/achievable if every ode ca sed at a rate of at least λ bits per secod to its chose destiatio. We deote by T (), the maximum feasible throughput w.h.p. Give a scheme Π, let M Π (i, t) be the umber of packets from source ode i that destiatio ode d(i) receives i t timeslots uder scheme Π, for i. Note that this could be a radom quatity for a give realizatio of the etwork. Defie the log term throughput of S-D pair i, deoted by λ i Π (), tobe c 2s λ i Π() = lim if t t M Π(i, t). Scheme Π is said to have throughput T Π () if lim P ( λ i Π() T Π () for all i ) =. 4 We say that a evet occurs with high probability (w.h.p.) if its probability teds to as. A We allow radomess i schemes ad, as a result, radom quatities above are i the joit probability space icludig both the radom etwork of size ad the scheme Π. Note that whe etwork codig is utilized i scheme Π, M Π (i, t) is the umber of successfully decoded packets received by the destiatio d(i) of S-D pair i i t timeslots uder scheme Π. Defiitio of delay : The delay of a packet is the time it takes the packet to reach the destiatio after it leaves the source. We do ot take queueig delay at the source ito accout, sice our iterest is i the etwork delay. Let DΠ i (j) deote the delay of packet j of S-D pair i uder scheme Π, the the sample mea of delay for S-D pair i is D i Π = lim sup k k k DΠ(j). i j= The average delay over all S-D pairs for a particular realizatio of the radom etwork is the D Π = i= D Π. i The delay for a scheme Π is the expectatio of the average delay over all S-D pairs ad all radom etwork cofiguratios, i.e., D Π () =E [ D Π ] = i= [ ] E D i Π. Whe etwork codig is utilized, we cosider the delay of gettig origial packets. Whe a origial packet m i belogs to the geeratio M, the delay of m i uder scheme Π is the time from the first packet belogig to M leaves the source to the origial packet m i has bee decoded i the destiatio. IV. THROUGHPUT-DELAY TRADEOFFS WITHOUT NETWORK CODING: SCHEMES AND RESULTS I this sectio, we give a brief overview of the redudacybased schemes as preseted i [5] ad establish the throughput-delay tradeoffs i MANETs without etwork codig. Some of the discussios preseted here directly build o results already established i [5]. They are icluded here for completeess ad compariso purposes. We first describe three relay schemes with differet redudacy proposed i [5] from a uified poit of view. Three Redudacy-Based Schemes Proposed i [5]: We ca cotrol the trasmissio redudacy of each packet with two methods: the umber of hops each packet will take from source to destiatio, ad the total umber of copies (replicas) of each origial packet i the etwork. The three schemes, amely, 2-hop relay without replicas, 2-hop relay with k replicas, ad multi-hop relay with k 2 replicas represet differet combiatios of the two methods. Each scheme has two parts: () schedulig of active cells; (2) schedulig of trasmissio i a active cell. The three schemes have the same cell schedulig policy (Part ()) as follows: Each cell becomes active oce i every K 2 timeslots as discussed i Propositio. I a active cell, trasmissio is always betwee two odes withi the same cell. I every active cell with at least two odes, itra-cell trasmissio schedulig (Part (2)) is eeded. For 2-hop relay schemes, each packet at most takes two hops from source to destiatio. The differece is that, for 2-hop relay without replicas, packets are ot duplicated ad are held by at most oe ode (source or relay) at ay timeslot, while for 2-hop relay with k replicas, the source will sed k replicas to distict odes as relays. 202

5 Whe destiatio odes receive packets from relays, they will first tell relays which packet they are lookig for before the trasmissio begis (usig the hadshake). Multi-hop relay with k 2 replicas is just aother type of floodig scheme, which trasmits k 2 replicas of each packet, ad places o costraits o the umber of hops. At every timeslot i each active cell, the oldest packet will be selected to sed to all odes i the cell. Now, we aalyze the performace of the schemes described above. First of all, we give the lower bouds o delays uder fast mobility model as follows [5]. Theorem : Uder fast mobility model, D() = Ω(log ) for ay scheme ad D() = Ω( ) for ay 2-hop relay scheme. Remark : To achieve the optimal throughput with the miimal delay give above, parameters like k ad k 2 i the proposed schemes should be carefully chose uder differet mobility models. It has bee show i [5] that uder fast mobility model, k = Θ( ) is eough for 2-hop relay scheme to achieve the miimal delay Θ( ). Further icreasig redudacy k will oly reduce throughput without decreasig delay. Followig the same argumet, we have the optimal k 2 = Θ(log ) for floodig scheme. These lead to the followig Theorem. Theorem 2: Assumig ifiite buffer space at each ode, throughput-delay tradeoffs achieved by the three redudacybased schemes proposed i [5] for MANETs uder fast mobility model ca be summarized i the followig table. scheme throughput delay 2-hop relay without replicas Θ() Θ() 2-hop relay with k replicas Θ(/ ) Θ( ) multi-hop relay with k 2 replicas ( Θ log ) Θ(log ) Proof: The proof of Theorems ad 2 is similar to the proof of Theorems 3, 6, 7 ad 8 i [5], with mior differeces caused by our use of the protocol model ad cell schedulig scheme, igorig queueig delays at source odes. Due to space costraits, we do ot repeat the proof here. Next, we cosider the throughput-delay tradeoffs uder slow mobility model. We first show that the tradeoff with slow mobility is dramatically differet from the oe with fast mobility by the followig theorem. Theorem 3: Uder slow mobility model, D() =Ω( ) for ay scheme ad D() =Ω( log ) for ay 2-hop relay scheme. Proof: From radom walk model, ode speed is upper bouded by 2A /m = O() ad the trasmissio rage r c = Θ(). Therefore, iformatio propagatio speed will be o larger tha Θ() per timeslot. It ca be show that the distace betwee the iitial positios of S-D pair is Ω( A )=Ω( ) w.h.p. []. Hece, the expected delay is at least Ω( ) timeslots. For 2-hop relay cases, see our techical report [43] for a complete proof. Theorem 4: Assumig ifiite buffer space at each ode, throughput-delay tradeoffs achieved by the three redudacybased schemes proposed i [5] for MANETs uder slow mobility model ca be summarized i the followig table. scheme throughput delay 2-hop relay without replicas Θ() ( ) Θ( log ) 2-hop relay with k replicas Θ log Θ( log ) multi-hop relay with k 2 replicas ( Θ ) Θ( ) Proof: The proof for the performace of 2-hop relay without replicas ca be foud i [4], [7]. The proof for the other two schemes are ot reported i the literature. We complete the proof of this theorem i our techical report [43]. Note that the performace above is achieved with k =Θ( log ) ad k 2 =Θ( ), respectively. V. THROUGHPUT-DELAY TRADEOFFS WITH NETWORK CODING: SCHEMES AND RESULTS We first review RLC used i our etwork codig based schemes. This bears exactly the same setup as i [25]. The we describe the schemes developed for aalyzig tradeoffs i MANETs with etwork codig, ad idetify scearios i which RLC improves etwork performace of MANETs. A. Network Codig Operatio Radom liear codig (RLC) is applied to a fiite set of k origial packets (i.e., M = {m,m 2,,m k }), that is called a geeratio. Each packet is viewed as a r-dimesioal vector over a fiite field, F q of size q, i.e., m i F r q, i =, 2,,k. If the packet size is m bits, this ca be doe by viewig each packet as a r = m/ log 2 (q) -dimesioal vector over F q (istead of viewig each packet as a m-dimesioal vector over biary field). Typically, F 2 8 (i.e., F 256 ) is used. All the additios ad multiplicatios i the followig descriptio are assumedtobeoverf q. We assume that all the k packets i M are liearly idepedet. Durig the executio of a RLC based relay scheme, the destiatio ode of M collects liear combiatios of the packets i M. Oce there are k idepedet liear combiatios at a ode, it ca recover all the origial packets i M successfully. Now, cosider a certai timeslot t. LetS v (t) ad S u (t) deote the set of all the coded packets (each coded packet is a liear combiatio of the packets i M) at ode v ad u, respectively, at the begiig of the timeslot t. More precisely, if coded packet f l S v (t), where l =, 2,, S v (t), the f l F r q has the form f l = k i= α li m i, α li F q.the scheme esures that a li s are kow to ode v by appedig each packet f l with a code vector, which will be explaied a little later. Let S v (t) ad S u (t) deote the subspaces spaed by the coded packets i S v (t) ad S u (t), respectively. If S v (t) S u (t), we say ode v has useful iformatio about M for u. I timeslot t, if ode v is scheduled by the scheme to trasmit a packet related to M to ode u, v first checks if it has useful iformatio for u. Ifso,v trasmits a radom coded packet with payload f ew F r q to u, where f ew = f l S v(t) β l f l, β l F q ad P(β l = β) = q, β F q. It is easy to check that f ew is still a liear combiatio of the k origial packets, ad ca be writte as f ew = k i= θ i m i where θ i = f β l S v(t) l α li F q. For decodig purposes, the vector (θ,θ 2,,θ k ) F r q, called code vector, will be appeded to f ew, ad set as overhead. This overhead clearly requires a paddig of additioal k log 2 (q) bits. If the packet size m log 2 (q), which would be the case uder our costat packet size model, the the overhead required by the RLC based scheme ca be igored i our aalysis. 5 5 More precisely, the costat packet size model for origial packets meas that the packet size scales as Θ(log ) bits, sice it eeds to carry the ID of the destiatio ode with Θ(log ) bits. For a fair compariso, we require that k = O(log ) for the coded packets throughout the paper. Therefore the overhead itroduced by the code vector will ot chage the order of our results o T () ad D() for RLC-based schemes. 203

6 We say that v seds a iovative coded packet f ew to u, if f ew ca icrease the dimesio of the subspace S u (t), i.e., dim(s u (t) ). Note that dim(s u (t) ) k i geeral ad if dim(s u (t) )=k, ode u ca recover all the k origial packets at oce. We ow recall the followig key result about RLC, which says that f ew will be a iovative coded packet for u with probability o less tha q. Propositio 2: (Lemma 2. i [25]) Let S u (t) + = S u (t) {f ew } be the subspace spaed by the code vectors i u at the ed of timeslot t, i.e., after receivig a coded packet f ew from v accordig to the RLC based scheme described as above. The, P ( dim(s u (t) + ) > dim(s u (t) ) S v (t) S u (t) ) q. B. RLC-Based Relay Schemes I this subsectio, we describe RLC-based relay schemes with differet routig strategies, which will be used later to exploit throughput-delay tradeoffs i MANETs. We first itroduce the cocept of big geeratio. I what follows, whe we say that the source ode groups k = ω(log ) origial packets ito oe big geeratio, we i fact separate these k packets ito k/θ(log ) geeratios, each with Θ(log ) packets. Whe the destiatio ode tries to decode oe origial packet, it first eeds to collect Θ(k) coded packets from the big geeratio (with Θ(log ) coded packets from each geeratio). Therefore the overhead itroduced by RLC is igorable i our aalysis (cf. footote 5). Schemes : 2-hop Relay with RLC () k origial packets i each source ode will be grouped ito oe (big) geeratio. Each source will sed m =(+ɛ)k coded packets for each (big) geeratio, where ɛ is a costat. (2) Coded packets for each geeratio will have the same timestamp t p.thevalueoft p is the time the first coded packet of that geeratio leaves the source. All coded packets of a geeratio will be deleted from the relay buffer at the timeslot t if t t p >th p, where the threshold th p depeds o D() of the scheme ad will be sufficietly larger tha D(). (3) Each cell becomes active oce i every K 2 timeslots as discussed i Propositio. I a active cell, trasmissio is always betwee odes withi the same cell. (4) For a active cell with at least two odes, a radom trasmitter-receiver pair is selected, with uiform probability over all possible ode pairs i the cell. With probability /2, the trasmitter is scheduled to operate i either Source-to- Relay or Relay-to-Destiatio mode, described as follows: Source-to-Relay Mode: The trasmitter seds a coded packet of its curret geeratio, ad does so upo every trasmissio opportuity while it is i source-to-relay mode util m coded packets have bee delivered to distict odes. If all other odes i the cell already have oe coded packet for that geeratio, the source will begi to trasmit coded packets from the ext geeratio. Every ode stores a sigle packet per S-D pair per geeratio. Whe the ode receives a ew packet, a relay liearly combies the icomig packet with the stored oe, ad replaces the stored packet with the result. Note that the odes operate i broadcast mode, i.e., every ode will hear every trasmissio i its rage, ad update the packet storage as described above. Relay-to-Destiatio Mode: If the desigated trasmitter has a coded packet i its relay buffer for the destiatio ode, ad the rak of coded packets of that geeratio i the receiver is smaller tha k, the coded packet is trasmitted to the desigated receiver. Remark 2: Sice m>k, we eed a mechaism to stop uecessary relay of coded packets of a geeratio whe it is already decoded i the destiatio. Here we use a proactive stoppig mechaism, i.e., the timestamp of each geeratio, sice we ca boud the delay of the scheme. I the aalysis part preseted later, we will show that k =Θ(), ad D() for this scheme is also Θ() for fast ad slow mobility models. Therefore, th p should be larger tha Θ(). More complicated reactive stoppig mechaisms (cf. [33] ad the refereces therei) ca be adopted to ehace the efficiecy of the scheme i practice. However, we follow the simplest desig for aalytical tractability of the scheme. Schemes 2: Multi-hop Relay with RLC () k origial packets i each source ode will be grouped ito oe (big) geeratio. Each source will sed m =(+ɛ)k coded packets for each geeratio, where ɛ is a costat. Two timestamps for each geeratio are used. Oe is called the geeratig time t g, based o the time for k origial packets to be grouped ito a geeratio i the source. Aother is called trasmissio time t p, based o the time the first coded packet of that geeratio is trasmitted by the source. (2) Each cell becomes active oce i every K 2 timeslots as discussed i Propositio. I a active cell, trasmissio is always betwee odes withi the same cell. (3) For a active cell with at least two odes, perform the followig: amog all packets cotaied i at least oe ode of the cell ad which have useful iformatio for some other ode i the same cell, choose the packet with the smallest geeratig time t g. If there are ties, choose the packet from the S-D pair i which maximizes (t g + i) mod. Trasmit this packet to all other odes i the cell. If the selected packet is i the source, the the source will trasmit the liear combiatio of its k origial packets of the same geeratio, istead of a particular packet belogig to that geeratio. (4) Every ode stores a sigle packet per S-D pair per geeratio. Whe the ode receives a ew packet, a relay liearly combies the icomig packet with the stored oe, ad replaces the stored packet with the result. (5) All coded packets of a geeratio will be deleted from the relay buffer at the timeslot t if t t p >th p, where the threshold th p depeds o D() of the scheme ad should be sufficietly larger tha D(). Remark 3: The geeratig timestamp t g is used to costruct a floodig scheme for oe particular S-D pair where all S-D pairs are active ad share the etwork resource. It is easy to see that the packets from the oldest geeratio that has ot bee delivered to all odes will domiate the trasmissios over the whole etwork very quickly. The log-term fairess betwee all S-D pairs is guarateed sice i the case of ties, packets from S-D pair i are give top priority i every timeslots. Also ote that, sice at oe particular timeslot, oly oe geeratio from oe S-D pair domiates the whole etwork, the umber of packets each relay eeds to store i step (4) is, i.e., just for oe geeratio w.h.p. Aother timestamp t p used here has the same fuctioality as the previous scheme. The threshold th p should be larger tha D(), scalig as Θ(log ) ad Θ( ), respectively, for fast ad slow mobility models. C. Mai Results for RLC-Based Schemes I this subsectio, we summarize the performace of the above schemes uder differet mobility models. Here, we 204

7 Θ( ) Sessio Sessio 2 Θ( ) (a) 2-hop relay with RLC codig time Θ( ) Θ ( ) Θ ( ) Θ( ) Sessio Sessio 2 Θ( ) (b) Multi-hop relay with RLC codig Fig. 3. Timetables for differet RLC-based schemes uder slow mobility model. focus o the ituitio ad explaatio of these results. Proofs of theses results will be give i the ext Sectio. Theorem 5: Whe 2-hop relay with RLC scheme is used ad k =Θ(), wehavet () = Θ() ad D() =Θ() for fast ad slow mobility models. Remark 4: Compare to Theorems 2 ad 4, it is easy to see that, RLC provides delay improvemet Θ(log ) uder slow mobility model. No gai is foud uder fast mobility model. It is ot surprisig, sice 2-hop relay with RLC scheme is used to replace 2-hop relay without replicas, ad we kow that i the latter, there is o duplicated packets i order to maximize the throughput. Thus we caot expect ay gais whe etwork codig is used. The gai Θ(log ) of delay uder slow mobility model comes from the lower iformatio propagatio speed, ad the mixig of packets icrease this speed by guarateeig that every packet the destiatio received from relay odes will cotribute some iformatio for the decodig of the packet from the same geeratio. For fast mobility model, this beefit vaishes sice the iformatio propagatio speed is high eough, ad the delay for waitig k coded packets for decodig domiates the whole delay. Theorem 6: Whe multi-hop relay with RLC scheme is used, uder fast mobility model with k = Θ(log ), wehave T () = Θ(/) ad D() = Θ(log ). Uder slow mobility model with k = Θ( ), we have T () = Θ(/) ad D() =Θ( ). Remark 5: Uder fast ad slow mobility models, multihop RLC-based schemes always provide sigificat gais compared to floodig schemes. We ca see that the RLC-based scheme ca achieve miimal delay, with a improved delaycostraied throughput. The ituitio is that, whe floodig is used, there exist eough opportuities to ehace performace by replacig replicas with more itelliget codig. Fig. 3 compares timetables of 2-hop ad multi-hop RLCbased relay schemes. It ca be foud that i 2-hop relay schemes, multiple sessios operate i a parallel fashio, while i multi-hop relay schemes, they operate i a sequetial fashio. Therefore, at each timeslot, for 2-hop relay schemes, traffic patter is still multiple uicasts. Recall our discussio i Sectio II-B, for multiple uicasts, we seldom fid gais from etwork codig. While for multi-hop relay schemes, at each timeslot, traffic patter looks more like oe broadcast sessio, where gais from etwork codig are aturally expected. Remark 6: Also otice that, multi-hop relay schemes ca be divided ito multiple phases, ad i each phase, relayig for oe geeratio from oe S-D pair will domiate the etwork, which is i fact a type of iformatio floodig i this phase (refer to Fig. 3(b) for illustratio). The result is that i each phase, packets from oe geeratio will be broadcasted to the whole etwork, ad if the other odes are receivers, they ca all decode the origial packets i that geeratio at the ed of that phase. So it guaratees that multi-hop relay with RLC codig ca support all-to-all traffic patter ( broadcast sessios) with the same performace. Note that this also meas that the same etwork performace ca be achieved for ay time multicast sessios (sice receivers i this case are just a subset of receivers i the broadcast case). From Theorem 6, we ca easily obtai the followig corollary o the performace of multiple broadcasts ad multicasts with etwork codig. Corollary : For all-to-all commuicatios or ay multicasts with sources, whe multi-hop relay with RLC scheme is used, uder fast mobility model with k = Θ(log ), we have T () = Θ(/) ad D() = Θ(log ). Uder slow mobility model with k =Θ( ), wehavet () = Θ(/) ad D() =Θ( ). I [30], Fragouli et al. desiged a RLC-based scheme based o results from [25]. For all-to-all commuicatios, they showed that their scheme achieves T () = Θ(/) ad D() = Θ() uder fast mobility model. Obviously, their scheme obtaied the same throughput as ours at the cost of much larger delay. The basic idea of their scheme is that, k packets from k differet sources will be grouped ito oe geeratio, ad the relayig scheme is essetially the same as ours. The compariso here raises a iterestig questio why i our RLC-based schemes we oly mix packets from the same source? The reasos are the followig: first of all, as show i the above compariso, eve for all-to-all commuicatio scearios, mixig packets from differet sources is ot a good choice. Secod, for multiple uicast scearios, we mix packets from differet sources ad these packets have differet destiatios. Whe oe destiatio decodes a packet desigated for aother destiatio, this packet is i fact a duplicate at the first destiatio which will reduce the throughput. I our multi-hop relay with RLC, we also itroduce redudacy for the same reaso. However, the redudacy here is explicitly desiged for decreasig the delay. While for the former case, it is purely a waste of etwork resource i multiple uicast scearios. Fially, groupig packets from differet sources requires coordiatios. We are ot sure about the cost for performig this coordiatio task, ad we are iterested i desigig fully decetralized schemes, i which the operatios from differet odes should be decoupled as much as possible. VI. THROUGHPUT-DELAY TRADEOFFS WITH NETWORK CODING: ANALYSIS I this sectio, we give outlies of proofs for the results o RLC-based relayig schemes discussed i the previous sectio. Ituitios behid these proofs are also provided. A. Prelimiaries To facilitate the theoretical aalysis, we eed first ivestigate two critical delays for fast ad slow mobility models: miimal delays for 2-hop relays ad for floodig. Here, 2-hop relay represets ay scheme with cotrolled redudacy o the umber of hops (i the 2-hop relay case, the umber of hops for each packet is 2, ad other schemes with costat hop costrais will yield the critical delays o the same order of ), ad floodig represets all schemes that remove this costrait totally. Cosider the followig situatio: iitially, oly oe ode s color is red, which we call the source. All other odes are blue. Wheever a source ode ecouters a blue ode (i the same cell), the latter is colored red. The time for Θ() odes to become red is called miimal 2-hop delay. Ifwe chage the rule slightly: wheever a red ode ecouters a blue ode, the latter is colored red, the the correspodig time is amed as miimal floodig delay. Obviously, these two critical delays reflect the itrisic properties of how mobility will facilitate iformatio propagatio. These two quatities 205

8 are scheme-idepedet, i.e., they hold for ay scheme with or without replicas ad with or without etwork codig. For fast mobility model, the values of these two critical delays are available i the literature [5], ad are icluded here for completeess. Lemma : (Theorem 3 ad Lemma 3 i [5]) The miimal 2-hop delay ad the miimal floodig delay uder fast mobility model are Θ() ad Θ(log ), respectively. Next, we preset the results for slow mobility model. Lemma 2: The miimal 2-hop delay uder slow mobility model is Θ(). Proof: Uder slow mobility model, the joit positio of two odes due to idepedet radom walks ca be viewed as a differece radom walk relative to the positio of oe ode. The the iter-meetig times are just the iter-visit times of cell (, ) for the differece radom walk o a torus. Let τ be the radom variable represetig the iter-meetig time defied as above. El Gamal et al. prove the followig Lemma i [7]. Lemma 3: E [τ] = ad E [ τ 2] =Θ( 2 log ). Let N be the umber of distict odes the red ode has met i timeslots. Based o above results, we ca obtai that E[N] =( ε), where 0 <ε< is a costat, ad σ N = O( log ). By Chebyshev iequality, for ay 0 <κ<, P {N ( κ)e[n]} σ ( ) N log κ 2 E[N] 2 = O 0, which meas that N =Θ() w.h.p. Lemma 4: The miimal floodig delay uder slow mobility model is Θ( ). Proof: Note that, Theorem 3 already shows that o scheme will obtai a delay better tha Θ( ) uder slow mobility model. We eed just to show that this is achievable usig floodig. We cite the followig importat result about rumor spreadig o torus: Theorem 3 i [44] states that followig the floodig rule metioed above, at timeslot t, there exists a sub-torus of size t t, where for each cell i this sub-torus, there exists at least oe red ode. Therefore, i Θ( ) timeslots, we ca cover the whole torus of size w.h.p. The followig lemma is useful i delay aalysis, sice it cofirms that the effect of trasmissio schedulig oly cotributes a costat factor, which ca be igored i asymptotic aalysis. Therefore, the time for two desired odes to meet will domiate the delay of the scheme. Lemma 5: I the schemes metioed above, every ode will be scheduled to trasmit or receive a packet with a costat, o-vaishig probability that is idepedet of. Proof: This result ca be obtaied from Propositio. It oly depeds o the steady state ode locatio distributio. Note that fast ad slow mobility models have the same ode locatio distributios i the steady state. Therefore, this result applies to both mobility models. B. Proof for Mai Results Proof outlie for 2-hop relay with RLC (Theorem 5): We first prove the case for fast mobility model. Obviously, if we ca prove that i N =Θ() timeslots, the destiatio ca receive Θ() coded packets, the based o Propositio 2 the destiatio has eough coded packets to recover k =Θ() origial packets w.h.p. Therefore, delay is upper bouded by O() ad the throughput is T () =O(). From the descriptio of the scheme, we kow the source will sed m =Θ() coded packets to the etwork. However, the destiatio may get k packets, which is fewer tha m packets for the followig reasos: () The source ode oly delivers coded packets to m <m differet odes actig as relays i N timeslots; (2) The Destiatio ode oly meets m 2 < m relay odes that have useful iformatio about the geeratio the destiatio wats to decode i N timeslots; (3) Furthermore, k <m 2 sice whe the destiatio meets a relay, it will ot always be scheduled to receive a packet from the relay. However, based o Lemma 5, the differece betwee k ad m 2 caused by the above reasos is upper-bouded by a costat factor ad we ca assume that Θ(k )=Θ(m 2 ). Obviously, if there are Θ() odes i the etwork which have useful iformatio for the destiatio, the probability for the destiatio to meet a relay i each timeslot is a costat value, i.e., will ot scale dow with. Therefore, P{k =Θ() m =Θ()} =. Thus, we eed to prove that m =Θ(), which is established i Lemma ). Due to space costraits, we do ot repeat the proof here. The proof for the slow mobility case is similar to the argumet above, ad the differeces are give as follows: Recall that Lemma 2 already shows that after N =Θ() timeslots, m =Θ(). From [45], we kow that the mixig time of a simple radom walk o a torus is also Θ(). Therefore, there exist a costat ε such that after N 2 = ε timeslots, these m odes with coded packets are uiformly distributed i the torus w.h.p. which meas that each ode i the etwork has coded packets with a costat probability. Istead of collectig coded packets as soo as possible, the destiatio odes begi to collect packets after N + N 2 timeslots. It ca be proved that the timeslots N 3 required to collect coded packets is still Θ(). Therefore the total delay N = N + N 2 + N 3 =Θ() w.h.p. I both mobility models, each source ode seds m =Θ() coded packets for each big geeratio, ad each big geeratio has Θ() origial packets, the each coded packet cotais Θ() iformatio of origial packets. Because every coded packet is trasmitted twice, we have T () = Θ(). Proof for multi-hop relay with RLC scheme (Theorem 6): The cetral problem here is still the followig: ca destiatio ode get Θ(k) coded packets withi Θ(k) timeslots? If it is the case, based o Propositio 2 the destiatio has eough coded packets to recover k origial packets w.h.p. The the delay is upper bouded by O(k). Replacig k with Θ(log ) ad Θ( ) for fast ad slow mobility cases, respectively, we obtai the results o delays. Sice we get k origial packets i Θ(k) timeslots, the throughput for the phase whe the trasmissios of this S-D pair domiate the etwork is Θ(). For fairess embedded i the scheme, this situatio happes oce for Ω(/) phases. Therefore, logterm throughput T () =O(), which completes the proof. Next, we cocetrate o a equivalet problem: how may timeslots do we eed i order to receive at least Θ(k) coded packets at the destiatio? We deote it as N. Obviously, E[N] E[S ]+E[S 2 ], where S ad S 2 respectively represet the timeslots required for Θ() odes i the etwork to have oe coded packet for that geeratio, ad the time required for the destiatio to receive Θ(k) packets give that other Θ() odes hold coded packets. Note that Lemma ad Lemma 4 establish that E[S ]= Θ(log ) ad E[S ] = Θ( ) for fast ad slow mobility models, respectively. They both agree with Θ(k) i respective schemes. I fact, it does ot happe by coicidece but by 206