The Fundamental Capacity-Delay Tradeoff in Large Mobile Ad Hoc Networks

Transcription

1 The Fudametal Capacity-Delay Tradeoff i Large Mobile Ad Hoc Networks Xiaoju Li ad Ness B. Shroff School of Electrical ad Computer Egieerig, Purdue Uiversity West Lafayette, IN 47907, U.S.A. {lix, shroff}@ec.purdue.edu Abstract There has bee recet iterest withi the etworkig research commuity to uderstad how mobility ca improve the capacity of mobile ad hoc etworks. Of particular iterest is the achievable capacity uder delay costraits. I this paper, we establish the followig upper boud o the optimal capacity-delay tradeoff i mobile ad hoc etworks for a i.i.d. mobility model. For a mobile ad hoc etwork with odes, if the per-bit-averaged mea delay is bouded by D, the the per-ode capacity λ is upper bouded by λ 3 the coditio uder which the upper boud is tight, we are able to idetify the optimal values of several key schedulig parameters. We the develop a ew scheme that ca achieve a capacity-delay tradeoff close to the upper boud up to a logarithmic factor. Our ew scheme achieves a larger per-ode capacity tha the schemes reported i previous works. I particular, whe the delay is bouded by a costat, our scheme achieves a per-ode capacity of Θ( /3 / log ). This idicates that, for the i.i.d. mobility model, mobility results i a larger capacity tha that of static etworks eve with costat delays. Fially, the isight draw from the upper boud allows us to idetify limitig factors i existig schemes. These results preset a relatively complete picture of the achievable capacity-delay tradeoffs uder differet settigs. I. INTRODUCTION O( D log3 ). By studyig Sice the semial paper by Gupta ad Kumar [], there has bee tremedous iterest i the etworkig research commuity to uderstad the fudametal achievable capacity i wireless ad hoc etworks. For a static etwork (where odes do ot move), Gupta ad Kumar show that the per-ode capacity decreases as O(/ log ) as the umber of odes icreases []. This work has bee partially supported by the NSF grats ANI , EIA , ad the Idiaa st Cetury Fud for Wireless Networks. We use the followig otatio throughout: f() f() = o(g()) lim g() = 0, The capacity of wireless ad hoc etworks ca be improved whe mobility is take ito accout. Whe the odes are mobile, Grossglauser ad Tse show that perode capacity of Θ() is achievable [], which is much better tha that of static etworks. This capacity improvemet is achieved at the cost of excessive packet delays. I fact, it has bee poited out i [] that the packet delay of the proposed scheme could be ubouded. There have bee several recet studies that attempt to address the relatioship betwee the achievable capacity ad the packet delay i mobile ad hoc etworks. I the work by Neely ad Modiao [3], it was show that the maximum achievable per-ode capacity of a mobile ad hoc etwork is bouded by O(). Uder a i.i.d. mobility model, the authors of [3] preset a scheme that ca achieve Θ() per-ode capacity ad icur Θ() delay, provided that the load is strictly less tha the capacity. Further, they show that it is possible to reduce packet delay if oe is willig to sacrifice capacity. I [3], the authors formulate ad prove a fudametal tradeoff betwee the capacity ad delay. Let the average ed-toed delay be bouded by D. For D betwee Θ() ad Θ(), [3] shows that the maximum per-ode capacity λ is upper bouded by λ O( D ). () The authors of [3] develop schemes that ca achieve Θ(), Θ(/ ), ad Θ(/( log )) per-ode capacity, whe the delay costrait is o the order of Θ(), Θ( ), ad Θ(log ), respectively. Iequality () leads to the pessimistic coclusio that a mobile ad hoc etwork ca sustai at most O(/) f() = O(g()) f() lim sup g() <, f() = ω(g()) g() = o(f()), f() = Θ(g()) f() = O(g()) ad g() = O(f()).

2 per-ode capacity with a costat delay boud. This capacity is eve worse tha that of static etworks. It turs out that this pessimistic coclusio is due to certai restrictive assumptios that are implicit i the work i [3] (we will elaborate o these assumptios i Sectio VI). I fact, Toumpis ad Goldsmith [4] preset a scheme that ca achieve a per-ode capacity of Θ( (d )/ / log 5/ ) whe the delay is bouded by O( d ). The result of [4] has icorporated the effect of fadig. If we remove fadig, the per-ode capacity will be of the order Θ( (d )/ / log 3/ ). Igorig the logarithmic term, we fid that i [4] the followig capacity-delay tradeoff is achievable: λ = Θ( D ). () This is better tha (). I particular, the authors of [4] preset a scheme that ca achieve Θ(/( log 3/ )) per-ode capacity with a costat delay boud. (The capacity will be Θ(/( log )) with o fadig.) This capacity is ow comparable to that of the static ad hoc etworks. A ope questio that still remais is: what is the optimal capacity-delay tradeoff i mobile ad hoc etworks? Iequality () is clearly ot optimal. The methodology of [4] is costructive i ature. Hece, iequality () is oly a lower boud. The search for the optimal capacitydelay tradeoff is importat for two reasos. First, it will allow us to see where the fudametal limits (i.e., upper bouds) are, ad how far existig schemes could possibly be improved. Secodly, as has happeed i previous works [], [3], a careful study of the upper boud is usually able to reveal the delicate tradeoffs iheret to the problem. A complete uderstadig of these tradeoffs will help us idetify the possible poits of iefficiecy i existig schemes ad provide directios for further improvemet. The ultimate goal is to fid a scheme that ca achieve the optimal capacity-delay tradeoff. This paper accomplishes these two goals. Uder the i.i.d. mobility model studied i [3], we will first establish a upper boud o the optimal capacity-delay tradeoff i mobile ad hoc etworks. We will show that, if the per-bit-averaged mea delay is bouded by D, the the per-ode capacity λ is upper bouded by λ 3 O( D log3 ). (3) I Fig., we draw this upper boud alogside the capacity-delay tradeoffs achieved by the schemes i [3] ad [4]. There is obviously a gap betwee the upper boud ad what ca be achieved by existig schemes. Capacity O() O( 3 ) O( ) O( ) O() O( d ) O() Delay Fig.. The achievable capacity-delay tradeoffs of existig schemes compared with the upper boud (igorig the logarithmic terms). The top lie correspods to our upper boud (achievable by the scheme outlied i Sectio V up to a logarithmic factor). The middle lie is achieved by the scheme i [4], ad the bottom oe is achieved by the scheme i [3]. Further, i the process of provig the upper boud, we are able to idetify the optimal choices for several key parameters of the schedulig policy. We the develop a ew scheme that achieves the upper boud o the capacity-delay tradeoff upto a logarithmic factor, which suggests that our upper boud is fairly tight. Our ew scheme achieves a larger per-ode capacity tha the oes i [3] ad [4]. I particular, our scheme ca achieve Θ( /3 /log ) per-ode capacity with costat delay. Ulike previous works, this result shows that, eve for a costat delay boud, the per-ode capacity of mobile ad hoc etworks ca be larger tha that of the static etworks! Fially, the isight draw from the upper boud allows us to idetify the limitig factors of the schemes i [3] ad [4]. The rest of the paper is orgaized as follows. I Sectio II, we outlie the etwork ad mobility model. I Sectio III, we prove several key properties that capture various tradeoffs iheret i mobile ad hoc etworks. We establish the upper boud o the optimal capacity-delay tradeoff i Sectio IV ad preset a scheme i Sectio V that achieves a capacity-delay tradeoff close to the upper boud. I Sectio VI, we discuss the existig schemes described i [3] ad [4]. The we coclude. II. NETWORK AND MOBILITY MODEL We cosider a mobile ad hoc etwork with odes movig withi a uit square. We assume that time is divided ito slots of uit legth. We assume the Note that chagig the shape of the area from a square to a circle or other topologies will ot affect our mai results.

3 followig i.i.d. mobility model proposed i [3]. At each time slot, the positios of each ode are i.i.d. ad uiformly distributed withi the uit square. Betwee time slots, the distributios of the positios of the odes are idepedet. Although the assumptio o a i.i.d. mobility model is somewhat restrictive, its mathematical tractability allows us to gai importat isights ito the structure of the problem. We will commet o some extesios to the i.i.d. mobility model i the coclusio. For simplicity, we assume the followig traffic model similar to the models i [3], [4]. We assume that the umber of odes is eve ad the odes ca be labeled i such a way that ode i commuicates with ode i, ad ode i commuicates with ode i, i =,..., /. The commuicatio betwee ay sourcedestiatio pairs ca go through multiple other odes as relays. That is, the source ca either sed a message directly to the destiatio; or, it ca sed the message to oe or more relay odes; the relay odes ca further forward the message to other relay odes (possibly after movig to aother positio); ad fially some relay ode forwards the message to the destiatio. We assume the followig Protocol Model from [] that govers direct radio trasmissios betwee odes. Let W be the badwidth of the system. Let X i deote the positio of ode i, i =,...,. Let X i X j be the Euclidea distace betwee odes i ad j. At each time slot, ode i ca commuicate directly with aother ode j at W bits per secod if ad oly if the followig iterferece costrait is satisfied []: X j X k ( + ) X i X j for every other ode k i, j that is simultaeously trasmittig. Here, is some positive umber. Note that a alterative model for direct radio trasmissio is the Physical Model [], [4]. I the Physical Model, a ode ca commuicate with aother ode if the sigal-toiterferece ratio is above a give threshold. It has bee show that, uder certai coditios, the Physical Model ca be reduced to the Protocol Model with a appropriate choice of []. Hece, we will ot cosider the Physical Model ay further i this paper. We also assume that o odes ca trasmit ad receive over the same frequecy at the same time. We further assume the followig separatio of time scale, i.e., radio trasmissio ca be scheduled at a time scale much faster tha that of ode mobility. This is usually a reasoable assumptio i real etworks. Hece, a message may be divided ito multiple bits ad each bit ca be forwarded multiple hops separately withi a sigle time slot. We assume a uiform traffic patter, that is, all source odes commuicate with their destiatio odes at the same rate λ. let D be the mea delay averaged over all messages ad all source-destiatio pairs. Both λ ad D will deped o how the trasmissios betwee mobile odes are scheduled. We are iterested i capturig the fudametal tradeoff betwee the achievable capacity λ ad the delay D. That is, over all possible ways of schedulig the radio trasmissios, what is the maximum per-ode capacity λ give certai costrait o the delay D. III. PROPERTIES OF THE SCHEDULING POLICIES I this sectio, we will prove several key results that capture the various tradeoffs iheret i mobile ad hoc etworks. We will first defie the class of schedulig policies that we will cosider. Because we are iterested i the fudametal achievable capacity for a give delay, we will assume that there exists a scheduler that has all the iformatio about the curret ad past status of the etwork, ad ca schedule ay radio trasmissio i the curret ad future time slots. At each time slot t, for each bit b that has ot bee delivered to its destiatio yet, the scheduler eeds to perform the followig two fuctios: Capture: The scheduler eeds to decide whether to deliver the bit b to the destiatio withi the curret time slot. If yes, the scheduler the eeds to choose oe relay ode (possibly the source) that has a copy of the bit b at the begiig of the time slot t, ad schedule radio trasmissios to forward this bit to the destiatio withi the same time slot, usig possibly multi-hop trasmissios. Whe this happes successfully, we say that the chose relay ode has successfully captured the destiatio of bit b. It is importat to forward the bit to the destiatio withi a sigle time slot. Otherwise, sice the chose relay ode may move far away from the destiatio i the ext time slot, the odes that received the bit b i the curret time slot will oly cout as ew relay odes for the bit b, ad they have to capture agai i the ext time slot. Duplicatio: If capture does ot occur for bit b, the scheduler eeds to decide whether to duplicate bit b to other odes that do ot have the bit at the begiig of the time slot t. The scheduler also eeds to decide which odes to relay from ad relay to, ad how to schedule radio trasmissios to forward the bit to these ew relay odes.

4 I this paper, we will cosider the class of causal schedulig policies that perform the above two fuctios at each time slot. The causality assumptio essetially requires that, whe the scheduler makes the capture decisio ad the duplicatio decisio, it ca oly use iformatio about the curret ad the past status of the etwork. I particular, at ay time slot t, the scheduler caot use iformatio about the future positios of the odes at ay time slot s > t. This class of schedulig policies is clearly very geeral, ad ecompasses early ay practical schedulig scheme we ca thik of. (Note that eve predictive schedulig schemes have to rely o curret ad past iformatio oly.) Some remarks o the capture process is i order. Although we do allow for other less ituitive alteratives, i a typical schedulig policy a successful capture usually occurs whe some relay odes are withi a area close to the destiatio ode, so that fewer resources will be eeded to forward the iformatio to the destiatio. For example, a relay ode could eter a disk of a certai radius aroud the destiatio, or a relay ode could eter the same cell as the destiatio. We call such a area a capture eighborhood. The relay odes that has the bit b at the begiig of the time slot t are called mobile relays for bit b. The mobile relay that is chose to forward the bit b to the destiatio is called the last mobile relay for bit b. The followig examples are illustrative of the possible schedulig policies withi this broad class. The schemes i previous works [3], [4] are all special cases or variats of these examples. Example A: The umber of mobile relays R is fixed ad the capture eighborhood is chose to be a disk with a fixed radius ρ aroud the destiatio. Oce a bit b eters the system, it is immediately broadcast to the earest R eighborig odes. Whe ay of the R mobile relays (icludig the source ode) move withi distace ρ from the destiatio, the bit b is the forwarded from the earest mobile relay to the destiatio. Example B: The uit area is divided ito a umber of cells. Oce a bit b eters the system, it is immediately broadcast to all other odes i the same cell. The umber of mobile relays for the bit b the stay uchaged. Note that the actual umber of mobile relays depeds o the umber of odes that reside i the same cell as the source (at the time slot whe the bit b eters the system), ad is thus a radom variable. Whe oe of the mobile relays moves ito the same cell as the destiatio, the bit b is the forwarded from the earest mobile relay to the destiatio. Example C: I the above two schemes, o duplicatio for bit b is carried out except at the first time slot whe the bit eters the system. A more sophisticated strategy is to use a opportuistic duplicatio scheme such as the example below. The uit area is divided ito a umber of cells. After a bit b eters the system, at each time slot t, if oe of the mobile relays moves ito the same cell as the destiatio, bit b is the forwarded from the earest mobile relay to the destiatio. Otherwise, the source ode (or, alteratively, the curret mobile relays) broadcasts the bit to all other odes that reside at the same cell. Hece, duplicatio may occur at each time slot util bit b is delivered to its destiatio. I the sequel, we will prove several key iequalities that capture the various tradeoffs iheret i the class of schedulig policies outlied above. Ituitively, the larger the umber of mobile relays ad the larger the capture eighborhood, the smaller the delay. O the other had, i order to improve capacity, we eed to cosume fewer radio resources, which implies a smaller umber of mobile relays ad a shorter distace from the last mobile relay to the destiatio. As we will see later, these tradeoffs will determie the fudametal relatioship betwee achievable capacity ad delay i mobile ad hoc etworks. A. Notatios Fix ay schedulig policy. Defie the followig radom variables for each bit b that eeds to be commuicated from its source ode to its destiatio ode. Let t 0 (b) deote the time slot whe bit b first eters the system. Let s b t 0 (b) be the time slot whe the scheduler decides that a successful capture for bit b occurs. The scheduler also eeds to choose oe mobile relay that has a copy of bit b at the begiig of the time slot s b to forward bit b towards its destiatio withi the same time slot s b. Let R b be the umber of mobile odes holdig the bit b at the time of capture, let D b s b t 0 (b) be the umber of time slots from the time bit b eters the system to the time of capture, ad let l b deote the distace from the chose last mobile relay to the destiatio of bit b. The quatities R b, D b, ad l b are essetial for the tradeoffs that follow 3. Note 3 Usig the otio of filtratios ad stoppig times [5, p3], we ca rigorously defie these quatities as radom variables o the same probability space where the radom mobility of the mobile odes is defied. The expectatio i the followig tradeoffs are the take with respect to this probability space. See [6] for details.

5 that D b icludes possible queueig delay at the source ode or at the relay odes. B. Tradeoff I : D b versus R b ad l b Propositio : Uder the i.i.d. mobility model, the followig iequality holds for ay causal schedulig policy whe 3, c log E[D b ] (E[l b ] + ) E[R b ] for all bits b, (4) where c is a positive costat. The proof is available i [6]. This ew result is the corerstoe for derivig the optimal capacity-delay tradeoff i mobile ad hoc etworks. It captures the followig tradeoff: the smaller the umber R b of mobile relays the bit b is duplicated to, ad the shorter the targeted distace l b from the last mobile relay to the destiatio, the loger it takes to capture the destiatio. This seemigly odd relatioship is actually motivated by some simple examples. Cosider Example A at the begiig of this sectio. Whe R b ad the area of the capture eighborhood A b are costats, the ( A b ) Rb is the probability that ay oe out of the R b odes ca capture the destiatio i oe time slot. It is easy to show that, the average umber of time slots eeded before a successful capture occurs, is, E[D b ] =. ( A b ) Rb A b R b If, as i Example B, R b ad possibly A b are radom but fixed after the first time slot t 0 (b), the E[D b R b, A b ] A b R b. By Hőlder s Iequality [5, p5], E [ ] E[R b ]E[ ]. Ab A b R b Hece, E[D b ] E[ ] E [ ] A b R b Ab E[R b ] E [ A b ]E[R b ], where i the last step we have applied Jese s Iequality [5, p4]. Note that o average l b is o the order of A b. Hece, E[D b ] c E [l b ]E[R b ] for all bits b, (5) where c is a positive costat. It may appear that, whe a opportuistic duplicatio scheme such as the oe i Example C is employed, such a scheme might achieve a better tradeoff tha (5) by startig off with fewer mobile relays ad a smaller capture eighborhood, if the ode positios at the early time slots after the bit s arrival turs out to be favorable. However, Propositio shows that o schedulig policy ca improve the tradeoff by more tha a log factor. C. Tradeoff II : Multihop Oce a successful capture occurs, the chose mobile relay (i.e., the last mobile relay) will start trasmittig the bit to the destiatio withi a sigle time slot, usig possibly other odes as relays. We will refer to these latter relay odes as static relays. The static relays are oly used for forwardig the bit to the destiatio after a successful capture occurs. Let h b be the umber of hops it takes from the last mobile relay to the destiatio. Let Sb h deote the trasmissio rage of each hop h =,.., h b. The followig relatioship is trivial. Propositio : The sum of the trasmissio rages of the h b hops must be o smaller tha the straight-lie distace from the last mobile relay to the destiatio, i.e., h b h= D. Tradeoff III : Radio Resources S h b l b. (6) It cosumes radio resources to duplicate each bit to mobile relays ad to forward the bit to the destiatio. Propositio 3 below captures the followig tradeoff: the larger the umber of mobile relays R b ad the further the multi-hop trasmissios towards the destiatio have to traverse, the smaller the achievable capacity. Cosider a large eough time iterval T. The total umber of bits commuicated ed-to-ed betwee all source-destiatio pairs is. Propositio 3: Assume that there exist positive umbers c ad N 0 such that D b c for N 0. If the positios of the odes withi a time slot are i.i.d. ad uiformly distributed withi the uit square, the there exist positive umbers N ad c 3 that oly deped o c, N 0 ad, such that the followig iequality holds for ay causal schedulig policy whe N, 4 E[R b ] h b π +E[ 4 (Sh b ) ] c 3 W T log. h= (7) The assumptio that D b c for large is ot as restrictive as it appears. It has bee show i [3] that the maximum achievable per-ode capacity is Θ() ad this

6 capacity ca be achieved with Θ() delay. Hece, we are most iterested i the case whe the delay is ot much larger tha the order O(). Further, Propositio 3 oly requires that the statioary distributio of the positios of the odes withi a time slot is i.i.d. It does ot require the distributio betwee time slots to be idepedet. We briefly outlie the motivatio behid the iequality (7). The details of the proof are quite techical ad available i [6]. Cosider odes i, j that directly trasmit to odes k ad l, respectively, at the same time. The, accordig to the iterferece costrait: Hece, Similarly, Therefore, X j X k ( + ) X i X k ] X i X l ( + ) X j X l ]. X j X i X j X k X i X k X i X k. X i X j X j X l. X i X j ( X i X k + X j X l ). times the trasmissio rage That is, disks of radius cetered at the trasmitter are disjoit from each other 4. This property ca be geeralized to broadcast as well. We oly eed to defie the trasmissio rage of a broadcast as the distace from the trasmitter to the furthest ode that ca successfully receive the bit. The above property motivates us to measure the radio resources each trasmissio cosumes by the areas of these disjoit disks []. For uicast trasmissios from the last mobile relay to the destiatio, the area cosumed by each hop is π 4 (Sh b ). For duplicatio to other odes, broadcast is more beeficial sice it cosumes fewer resources. Assume that each trasmitter chooses the trasmissio rage of the broadcast idepedetly of the positios of its eighborig odes. If the trasmissio rage is s, the o average o greater tha πs odes ca receive the broadcast, ad a disk of radius π s (i.e., area 4 s ) cetered at the trasmitter will be disjoit from other disks. Therefore, we ca use E[R b] as a lower boud o the expected area cosumed by duplicatig the bit to R b mobile relays (excludig the source ode). This lower boud will hold eve if the duplicatio process is 4 A similar observatio is used i [] except that they take a receiver poit of view. 4 carried out over multiple time slots, because the average umber of ew mobile relays each broadcast ca cover is at most proportioal to the area cosumed by the broadcast. Therefore, ispired by [], the amout of radio resources cosumed must satisfy E[R b ] h b π + E[ 4 4 (Sh b ) ] c 3W T, h= (8) where c 3 is a positive costat. However, E[R b] 4 may fail to be a lower boud o the expected area cosumed by duplicatig to R b mobile relays if the followig opportuistic broadcast scheme is used. The source may choose to broadcast oly whe there are a larger umber of odes close by. If the source ca afford to wait for these good opportuities, a opportuistic broadcast scheme may cosume less radio resources tha a o-opportuistic scheme to duplicate the bit to the same umber of mobile relays. Noetheless, we ca show that, whe is large, the probability that the umber of odes i a eighborhood is orders of magitude larger tha its average value will be very small. We ca the prove Propositio 3, which implies that o schedulig policies ca improve the tradeoff by more tha a log factor. For details, please refer to [6]. E. Tradeoff IV : Half Duplex Fially, sice we assume that o ode ca trasmit ad receive over the same frequecy at the same time (a practically ecessary assumptio for most wireless devices), the followig property ca be show as i []. Propositio 4: The followig iequality holds, h b h= W T. (9) IV. THE UPPER BOUND ON THE CAPACITY-DELAY TRADEOFF Our first mai result is to derive, from the above four tradeoffs, the upper boud o the optimal capacitydelay tradeoff of mobile ad hoc etworks uder the i.i.d. mobility model. Sice the maximum achievable per-ode capacity is Θ() ad this capacity ca be achieved with Θ() delay by the scheme of [3], we are oly iterested i the case whe the mea delay is o(). Propositio 5: Let D be the mea delay averaged over all bits ad all source-destiatio pairs, ad let λ be the throughput of each source-destiatio pair. If

7 D = O( d ), 0 d <, the followig upper boud holds for ay causal schedulig policy, λ 3 O( D log3 ). Proof: Usig the Cauchy-Schwartz iequality, we have ( ) h ( b ) ( h b ) h b Sb h (Sb h ) h= h= W T h= h b (Sb h ), (0) h= where i the last step we have used Tradeoff IV (9). Equality holds i (0) whe iequality (9) is tight ad whe Sb h is equal for all b ad h. We thus have, ( h b ) E[ (Sb h h b ) ] W T E[ Sb h ] h= W T W T h= ( ) h b E[ Sb h ] () h= ( E[l b ]), () where i the last two steps we have used Jese s Iequality ad the Tradeoff II (6), respectively. Iequality () is tight whe h b is almost surely a costat, Sb h h= ad () is tight whe (6) is tight. From Tradeoff I (4), we have Let E[R b ] D = c log (E[l b ] + ) E[D b ]. (3) E[D b ] = E[D b ]. Usig Jese s Iequality ad Hőlder s Iequality, we have, P (E[l b]+ P ) (E[l b]+ ) E[R b ] (E[l b]+ ) E[D b] c log E[D b ]. (4) Equality holds whe E[l b ] is the same for all b ad E[D b ] = D for all b. Substitutig (4) i (3), we have ( ) 3 ( ).(5) D (E[l b ] + ) Substitutig () ad (5) ito Iequality (7), we have 4c 3 W T log E[R b ] + πe[ h b h= ( ) 3 c log ( D (E[l b ] + ) + π W T ( E[l b ]) λt. There are two cases that we eed to cosider. Case : If E[l b ] λt, the 4c 3 W T log = c log 4c log (S h b ) ] ) ( ) 3 D ( ) λt λt λt 4 D λt. Whe D = O( d ), d <, the first term domiates whe is large. Hece, for large eough, 4c 3 W T log Case : If λt 4 8c log D λ 3c c 3 W E[l b ] λt, the 4c 3 W T log c log ( D ( ) 3 E[l b ] D log 4. (6) ) + π W T ( E[l b ]) λt (7)

8 Therefore, either π ( ) c log W T λt 4 D (8) = π λ 3 T λt. c log DW (9) λ O( D log ), (0) or, if λ = ω( D log ), the the first term i (9) domiates whe is large. I the latter case, for large eough, 4c 3 W T log π λ 3 T c log DW λ 3 3c c 3 W 3 D log 3 4. () Fially, we compare the three iequalities we have obtaied, i.e., (6), (0) ad (). Sice D = o( d ), d <, iequality () will evetually be the loosest for large. Hece, the optimal capacity-delay tradeoff is upper bouded by λ 3 O( D log3 ). V. AN ACHIEVABLE LOWER BOUND ON THE CAPACITY-DELAY TRADEOFF The capacity-delay tradeoff i Propositio 5 is better tha those reported i [3] ad [4]. Assumig that the delay boud is Θ( d ), 0 d <, the achievable perode capacity is O( ( d) ) by the scheme i [3], ad O( ( d)/ ) by the scheme i [4]. Our upper boud, however, implies a per-ode capacity of O( ( d)/3 ) (we have igored all log factors). Sice d <, there is clearly room to substatially improve existig schemes (see Fig. ). I this sectio, we will show how the study of the upper boud also helps us i developig a ew scheme that ca achieve a capacity-delay tradeoff that is close to the upper boud. The, i the ext sectio, we will idetify the limitig factors of existig schemes i [3], [4] that prevet them from achievig the optimal capacity-delay tradeoff. A. Choosig the Optimal Values of the Key Parameters Assume that the mea delay is bouded by d, d <. By Propositio 5, we have, λ Θ( 3 D log3 ) = Θ( d 3 log ). TABLE I THE ORDER OF THE OPTIMAL VALUES OF THE PARAMETERS WHEN THE MEAN DELAY IS BOUNDED BY d. R b : # of Duplicates Θ( ( d)/3 ) l b : Distace to Destiatio h b : # of Hops q Sb h : Trasmissio Rage of Each Hop Θ( Θ( (+d)/6 log ) Θ( ( d)/3 ) log log ) I order to achieve the maximum capacity o the right had side, all iequalities (0)-(8) should hold with equality. By studyig the coditios uder which these iequalities are tight, we will be able to idetify the optimal choices of various key parameters of the schedulig policy. For example, by checkig the coditios whe (0)-(4) are tight, we ca ifer that the parameters (such as S h b, E[l b], E[D b ]) of each bit b should be about the same ad should cocetrate o their respective average values. This implies that the schedulig policy should use the same parameters for all bits. Further, ote that equality holds i (8) if ad oly if 4c log ( ) 3 = E[l b ]) D( π W T ( E[l b ]). From here we ca solve for l b, h b, Sb h ad R b. The results are summarized i Table I. The details of the derivatio is available i [6]. Several remarks are i order. Sice it is sufficiet to cotrol all parameters aroud these optimal values, simple cell-based schemes such as the oe i Example B of Sectio III suffice. Secodly, the optimal values for R b ad l b ca provide guidelies o how to choose the cell partitioig. Thirdly, the optimal value for Sb h is roughly the average distace betwee eighborig odes whe odes are uiformly distributed i a uit square. Hece, it is desirable to use multi-hop trasmissio over eighborig odes to forward the iformatio from the last mobile relay to the destiatio. These guidelies have sketched a blueprit of the optimal schedulig scheme for us. We ext preset schemes that ca achieve capacity-delay tradeoffs that are close to the upper boud up to a logarithmic factor. B. Achievable Capacity with Θ() Delay We first focus o the case whe the mea delay is bouded by a costat, i.e., the expoet d = 0. By Propositio 5, the per-ode throughput is bouded

9 by O( /3 log ). We ow preset a scheme that ca achieve Θ( /3 / log ) capacity with Θ() delay for large. This is a ecouragig result for mobile etworks because we kow that the per-ode capacity of static etworks is O(/ log ) []. Hece, mobility icreases the capacity eve with costat delay. We will eed the followig Lemma before statig the mai schedulig scheme. We will repeatedly use the followig type of cell-partitioig. Let m be a positive iteger. Divide the uit square ito m m cells (i m rows ad m colums). Each cell is a square of area /m. As i [4], we call two cells eighbors if they share a commo boudary, ad we call two odes eighbors if they lie i the same or eighborig cells. We say that a group of cells ca be active at the same time whe oe ode i each cell ca successfully trasmit to or receive from a eighborig ode, subject to the iterferece from other cells that are active at the same time. Let x be the largest iteger smaller tha or equal to x. The proof of the followig Lemma is available i [6]. Lemma 6: There exists a schedulig policy such that each cell ca be active for at least /c 4 amout of time, where c 4 is a costat idepedet of m. The capacity achievig scheme is as follows. Capacity Achievig Scheme: ) At each odd time slot, we schedule trasmissios from the sources to the relays. We divide the uit square ( ) ito g () = /3 8 log cells. We refer to each cell i the odd time slot as a sedig cell. By Lemma 6, each cell ca be active for c 4 amout of time. Whe a cell is scheduled to be active, each ode i the cell broadcasts a ew message to all other odes i the same cell for 3c 4 /3 log amout of time. These other odes the serve as mobile relays for the message. The odes withi the same sedig cell coordiate themselves to broadcast sequetially. If ay sedig cell has more tha 3 /3 log odes, we refer to it as a Type-I error [4]. ) At each eve time slot, we schedule trasmissios from the mobile relays to the destiatio odes. Note that the positios of the mobile relays have chaged ad are ow idepedet of their positios i the previous time slot. We divide the uit square ito g () = ( /3) cells. We refer to each cell i the eve time slot as the receivig cell. For ay receivig cell i =,..., g () ad ay sedig cell j =,..., g (), pick a ode Y ij that is i the receivig cell i i the curret time slot ad that was i the sedig cell j i the previous time slot. We refer to this ode Y ij as the desigated mobile relay i receivig cell i ad for sedig cell j. If there is o such ode Y ij for ay i or j, we refer to it as a Type-II error. There may be multiple odes that ca serve as the desigated mobile relay for some i, j. I this case we oly pick oe. Note that if o Type-II errors occur, each destiatio ode ca ow fid a desigated mobile relay that holds the message iteded for the destiatio ode ad that resides i the same receivig cell. We the schedule multihop trasmissios i the followig fashio to forward each message from the desigated mobile relay to its destiatio i the same receivig cell. We further divide each receivig cell i ito g 3 () = ( /3 4 log ) miicells (i g 3 () rows ad g 3 () colums). Each miicell is a square of area /(g ()g 3 ()). By Lemma 6, there exists a schedulig scheme where each mii-cell ca be active for c 4 amout of time. Whe each mii-cell is active, it forwards a message (or a part of a message) to oe other ode i the eighborig mii-cell. If the destiatio of the message is i the eighborig cell, the message is forwarded directly to the destiatio ode. The messages from each desigated mobile relay are first forwarded towards eighborig cells alog the X-axis, the to their destiatio odes alog the Y-axis. I this fashio, a successful schedule will allow each destiatio W ode to receive a message of legth 3c 4 /3 log from its respective desigated mobile relay residig i the same receivig cell. For details o costructig such a schedule, see [6]. If o such schedule exists, we refer to it as a Type-III error. At the ed of each eve time slot, if there are ay packets that caot be delivered to the destiatio odes due to Type-II or Type-III errors, they are dropped. We ca show that, as, the probabilities of errors of all types will go to zero. The followig propositio thus holds. For space cosideratio, we do ot provide the proof here. It is available i [6]. Propositio 7: With probability approachig oe, as, the above scheme allows each source to W 3c 4 /3 log sed a message of legth to its respective destiatio ode withi two time slots. Remark: Our scheme uses differet cell-partitioig i the odd time slots tha that i the eve time slots. Note that i previous works [3], [4], the cell structure remais the same over all time slots. Our judicious choice of the cell-structures is the key to our tighter lower boud for the capacity. I particular, the size of the sedig cell is chose such that the average umber of odes i each cell, /g () = Θ( /3 log ), is close to the optimal value of R b i Sectio V-A (with d = 0). The size of the receivig cell is chose such

10 that its area, /g () = Θ( /3 ), is close to the optimal value of lb. Fially, the size of the mii-cell is chose such that each hop to the eighborig cell is of legth / g ()g 3 () = Θ( log /), which is close to the optimal value of Sb h. C. The Effect of Queueig Whe we defied the delay D b of each bit b i Sectio III, it icludes the possible queueig delay at the source ode ad at the relay odes. The upper boud o the capacity-delay tradeoff (Propositio 5) thus holds regardless of the queueig disciplie used i the system, ad D also icludes the queueig delay. We ow show how to aalyze the queueig delay of the capacityachievig scheme i Sectio V-B. This scheme attempts W to deliver oe message of legth 3c 4 /3 log for each source-destiatio pair every two time slots. Let p e be the probability that a message is successfully delivered to the destiatio at the ed of the eve time slot. (Note that p e is the same for all source-destiatio pairs due to symmetry, ad by Propositio 7, p e as.) Assume that if such delivery is usuccessful, messages that have ot bee delivered to the destiatios at the ed of each eve time slot are discarded ad have to be retrasmitted at the source odes. Further, assume W that packets of legth 3c 4 /3 log arrive at each source accordig to certai stochastic process. The packets may get equeued at the source odes. If we observe the system at the ed of each eve time slot, the umber of packets queued for each source-destiatio pair will evolve as that of a discrete-time queue with geometric service time distributios [7], ad the queues for each source-destiatio pair ca be studied idepedetly. If we kow the packet arrival process, we ca the compute the queueig delay. For example, if the arrival process is Beroulli, i.e., oe ew packet for each sourcedestiatio pair arrives at the source every two time slots with probability Λ, the usig stadard results for discrete time M/M/ queues [7, p8], we ca compute the queueig delay as, As, p e. Hece, D = Λ p e Λ. D, as. O the other had, if the arrival process is Poisso with rate Λ, the the umber of packets arrivig at a sourcedestiatio pair every two time slots is a Poisso radom variable with mea Λ. Hece, usig results for discrete time M a /M/ queues [7, p89], we ca compute the queueig delay as D = Λ p e Λ. Assume Λ ɛ, where 0 < ɛ <. As, p e. Hece, D Λ, as. ɛ Note that i both cases, the queueig delay is at most a costat multiple of (time slots) provided that ɛ (i.e., the differece betwee the arrival rate ad the capacity) is positive ad bouded away from zero as. Hece, the capacity-achievig scheme i Sectio V-B ca sustai Θ( /3 / log ) throughput (i bits per time slot) with O() queueig delay. D. The Capacity Achievig Scheme for Arbitrary Delay Boud The above scheme ca be geeralized to arbitrary delay bouds. Let the mea delay be bouded by D = Θ( d ), 0 d <. We ca group every d + time slots ito a super-frame. I each odd super-frame, we schedule trasmissios from the sources to the relays. We divide the uit square ito Θ( (+d)/3 / log ) sedig cells of equal area. Withi each sedig cell, each source broadcasts a ew message to all other odes withi the same cell for a duratio of Θ( ) every time ( d)/3 log slot. I each eve super-frame, we schedule trasmissios from the relays to the destiatio odes. We divide the uit square ito Θ( (+d)/3 ) receivig cells of equal area. I every time slot, some mobile relays will have messages iteded for some other destiatio odes i the same receivig cell. We the schedule multi-hop trasmissios to deliver the messages from the mobile relays to the destiatio odes i the same receivig cell. Usig similar techiques as the oe i [4] ad the oe i Sectio V-B, we ca show that, with probability approachig oe as, each source ca sed d + messages of legth Θ( ( d)/3 / log ) to its destiatio withi ( d + ) time slots. The queueig delay ca also be studied i a similar fashio as i Sectio V-C. The details are omitted because of space costraits. VI. THE LIMITING FACTORS IN EXISTING SCHEMES I Sectio V, we have show that choosig the optimal values of the schedulig parameters is the key to achievig the optimal capacity-delay tradeoff. I this

11 sectio, we will show that deviatig from these optimal values will lead to suboptimal capacity-delay tradeoffs. I particular, we will idetify the limitig factors i the existig schemes i [3] ad [4] by comparig the optimal values of schedulig parameters i Sectio V-A with those used by the existig schemes. Our model i Sectio IV ca be exteded to study the upper bouds o the capacity-delay tradeoff whe oe imposes additioal restrictive assumptios that correspod to these limitig factors. We will see that these ew upper bouds are iferior to the capacity-delay tradeoff reported i Sectios IV ad V. The existig schemes of [3] ad [4] i fact achieve capacity-delay tradeoffs that are close to the respective upper bouds. These results will give us ew isight o which schemes to use uder differet coditios. A. The Limitig Factor i the Scheme of [3] The scheme by Neely ad Modiao [3] divides the uit square ito cells each of area /. A mobile relay will forward messages to the destiatio oly whe they both reside i the same cell. Hece, the distace from the last mobile relay to the destiatio, l b, is o average o the order of O(/ ), regardless of the delay costraits. However, we have show i Sectio V-A that the optimal choice for l b should be o the order of Θ( (+d)/6 log / ), whe the mea delay is bouded by Θ( d ). The ext Propositio shows that the restrictive choice of l b is ideed the limitig factor of the scheme i [3]. The proof is available i [6]. Propositio 8: Let D be the mea delay averaged over all bits ad all source-destiatio pairs, ad let λ be the throughput of each source-destiatio pair. If D = O( d ), 0 d < ad E[l b ] = O(/ ), the for ay causal schedulig policy, λ O( D log ). Remark: The scheme of [3] achieves the above upper boud up to a logarithmic factor. B. The Limitig Factor i the Scheme of [4] I the scheme by Toumpis ad Goldsmith [4], a mobile relay will always use sigle-hop trasmissio to forward the messages directly to the destiatio. That is, the umber of hops from the last mobile relay to the destiatio ode, h b, is always. However, we have show i Sectio V-A that the optimal value of h b is Θ( ( d)/3 / log ) whe the mea delay is bouded by Θ( d ). The ext Propositio shows that the restrictio o h b is ideed the limitig factor of the scheme i [4]. The proof is available i [6]. Propositio 9: Let D be the mea delay averaged over all bits ad all source-destiatio pairs, ad let λ be the throughput of each source-destiatio pair. If D = O( d ), 0 d < ad h b = O(), the for ay causal schedulig policy, λ O( D log3 ). Remark: The scheme of [4] achieves the above upper boud up to a logarithmic factor. Propositios 5, 8 ad 9 preset three differet upper bouds o the capacity-delay tradeoff of mobile ad hoc etworks uder differet assumptios. Assume that the mea delay is bouded by d, 0 d <. Whe the capacity is the mai cocer, Propositio 5 shows that the per-ode throughput is at most O( ( d)/3 log ). The capacity-achievig scheme reported i Sectio V ca achieve close to this upper boud up to a logarithmic factor. However, this capacity-achievig scheme requires sophisticated coordiatio amog the mobile odes. Hece, it may ot be suitable whe simplicity is the mai cocer. O the other had, the scheme of [3] oly requires coordiatio amog odes that are withi a cell of area /. Note that the average umber of odes i such a cell is Θ(). Propositio 8 the shows that, whe coordiatio amog a large umber of odes is prohibited, the scheme of [3] is close to optimal. Similarly, the scheme of [4] oly requires sigle-hop trasmissios from the mobile relays to the destiatios. Propositio 9 shows that, whe multi-hop trasmissios are udesirable, the scheme of [4] is close to optimal. Therefore, the results reported i this paper preset a relatively complete picture of the achievable capacitydelay tradeoffs uder differet coditios. A iterestig ope problem for future work is to ivestigate whether these isights apply to the capacitydelay tradeoff uder mobility models other tha the i.i.d. model. For example, [8] ad [9] have studied the capacity-delay tradeoff uder the Browia Motio mobility model. I these works, the authors also have implicit restrictios o the schedulig policy. I particular, the scheme i [8] uses at most oe mobile relay at ay time (i.e., R b = ), ad the scheme i [9] schedule a trasmissio from the mobile relay to the destiatio oly whe they are at a distace of O(/ ) away (i.e., l b = O(/ )). I this paper, we have show uder the i.i.d. mobility model that, i order to achieve the optimal capacity-delay tradeoff, R b, l b ad h b should all vary with the delay expoet d. Puttig restrictios o ay

12 oe of these variables will lead to suboptimal capacity for a give delay costrait. For our future work, we pla to study whether these kid of restrictios will also limit the capacity-delay tradeoff obtaied i existig works uder other mobility models. VII. CONCLUSION AND FUTURE WORK I this paper, we have studied the fudametal capacity-delay tradeoff i mobile ad hoc etworks uder the i.i.d. mobility model. Our cotributios are threefold. We have established the upper boud o the optimal capacity-delay tradeoff over all causal schedulig policies. The upper boud ot oly provides the fudametal limits of capacity ad delay, but also helps to idetify the optimal values of the key schedulig parameters i order to achieve the optimal capacitydelay tradeoff. Our secod cotributio is to develop a ew schedulig scheme that ca achieve a capacitydelay tradeoff that differs from the upper boud oly by a logarithmic factor, which also implies that our upper boud is fairly tight. The capacity achievable by our ew scheme is larger tha that of the existig schemes i [3] ad [4]. I particular, whe the delay is bouded by a costat, our scheme achieves a per-ode capacity of Θ( /3 / log ). This idicates that, uder the i.i.d. mobility model, mobility icreases the capacity eve with costat delays. Our third cotributio is to use the isight draw from the upper boud to idetify the limitig factors i the existig schemes. These results preset a relatively complete picture of the achievable capacity-delay tradeoffs uder differet cosideratios. I this paper, we have assumed a i.i.d. mobility model. For future work, we pla to study the optimal capacity-delay tradeoff for mobile ad hoc etworks uder other mobility models. Amog the properties that we proved i Sectio III, we expect that the Tradeoffs II to IV will be relatively ivariat to the choice of mobility models, while Tradeoff I is likely to deped o a specific model. Hece, future work will cocetrate o how to tailor Tradeoff I for other mobility models. Some immediate extesios to the i.i.d. mobility model are possible. For example, at each time slot, each ode may idepedetly choose to stay i its old positio with probability p, ad to move to a ew radom positio with probability p. This model may approximate scearios where odes move at a fast speed ad the stay for a relatively log period of time. Tradeoff I will hold for this extesio of the i.i.d. mobility model, ad hece our mai results will hold as well. Other mobility models that we pla to ivestigate are, the Browia motio mobility model [8], [9], the radom waypoit model [9], [0], ad the liear mobility model [], etc. Other aspects to cosider are how the upper boud will be impacted by the use of diversity codig [], effect of fadig [4], ad the use of iformatio-theoretic approaches [3], [4]. REFERENCES [] P. Gupta ad P. R. Kumar, The Capacity of Wireless Networks, IEEE Trasactios o Iformatio Theory, vol. 46, o., pp , March 000. [] M. Grossglauser ad D. Tse, Mobility Icreases the Capacity of Ad Hoc Wireless Networks, IEEE/ACM Trasactios o Networkig, vol. 0, o. 4, August 00. [3] M. J. Neely ad E. Modiao, Capacity ad Delay Tradeoffs for Ad-Hoc Mobile Networks, submitted to IEEE Trasactios o Iformatio Theory, available at mjeely/, 003. [4] S. Toumpis ad A. J. Goldsmith, Large Wireless Networks uder Fadig, Mobility, ad Delay Costraits, i Proceedigs of IEEE INFOCOM, Hog Kog, Chia, March 004. [5] R. Durrett, Probability : Theory ad Examples, d ed. Belmot, CA: Duxbury Press, 996. [6] X. Li ad N. B. Shroff, The Fudametal Capacity-Delay Tradeoff i Large Mobile Ad Hoc Networks, Techical Report, Purdue Uiversity, lix/papers.html, 004. [7] M. E. Woodward, Commuicatio ad Computer Networks: Modellig with Discrete-Time Queues. Los Alamitos, CA: IEEE Computer Society Press, 994. [8] A. E. Gamal, J. Mamme, B. Prabhakar, ad D. Shah, Throughput-Delay Trade-off i Wireless Networks, i Proceedigs of IEEE INFOCOM, Hog Kog, Chia, March 004. [9] G. Sharma ad R. R. Mazumdar, Delay ad Capacity Tradeoffs for Wireless Ad Hoc Networks with Radom Mobility, preprit available at mazum/, October 003. [0] N. Basal ad Z. Liu, Capacity, Delay ad Mobility i Wireless Ad-Hoc Networks, i Proceedigs of IEEE INFOCOM, Sa Fracisco, CA, April 003. [] S. Diggavi, M. Grossglauser, ad D. Tse, Eve Oe- Dimesioal Mobility Icreases Ad Hoc Wireless Capacity, i ISIT 0, Lausae, Switzerlad, Jue 00. [] E. Perevalov ad R. Blum, Delay Limited Capacity of Ad hoc Networks: Asymptotically Optimal Trasmissio ad Relayig Strategy, i Proceedigs of IEEE INFOCOM, Sa Fracisco, CA, April 003. [3] M. Gastpar ad M. Vetterli, O the Capacity of Wireless Networks: The Relay Case, i Proceedigs of IEEE INFOCOM, New York, Jue 00. [4] P. Gupta ad P. R. Kumar, Towards a Iformatio Theory of Large Networks: A Achievable Rate Regio, IEEE Trasactios o Iformatio Theory, vol. 49, o. 8, pp , August 003.