Chapter ON THE FUNDAMENTAL RELATIONSHIP BETWEEN THE ACHIEVABLE CAPACITY AND DELAY IN MOBILE WIRELESS 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 I this work, we establish the fudametal relatioship betwee the achievable capacity ad delay i mobile wireless etworks. Uder a i.i.d. mobility model, we first obtai the followig upper boud o the achievable capacity give a delay costrait. For a mobile wireless etwork with odes, if the per-bit-averaged mea delay is bouded by D, the the upper boud o the per-ode capacity is o the order of q 3 D log. By studyig the coditios uder which the upper boud is tight, we are able to idetify the optimal values of several key schedulig parameters. We the develop a schedulig scheme that ca almost achieve the upper boud (up to a logarithmic factor). This suggests that the upper boud is tight. Our schedulig scheme also achieves a provably larger per-ode capacity tha schemes reported i previous works. I particular, whe the delay is bouded by a costat, our schedulig scheme achieves a per-ode capacity that is iversely proportioal to the cube root of (up to a logarithmic factor). This implies that, for the i.i.d. mobility model, mobility improves the achievable capacity of static wireless etworks, eve with costat delays! Fially, the isight draw from the upper boud allows us to idetify limitig factors i existig schedulig schemes. These results preset a relatively com- This work has bee partially supported by the NSF grats ANI-00778, EIA-030599 ad the Idiaa st Cetury Fud for Wireless Networks.
plete picture of the achievable capacity-delay tradeoffs uder differet settigs. Keywords: Mobile wireless etworks, mobile ad hoc etworks, capacity-delay tradeoff, large system asymptotics. Itroductio 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 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 []. The capacity of wireless etworks ca be improved whe mobility is take ito accout. Whe the odes are mobile, Grossglauser ad Tse show that per-ode 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 wireless etworks. I the work by Neely ad Modiao [3], it was show that the maximum achievable per-ode capacity of a mobile wireless 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-to-ed delay be bouded by D. For D betwee Θ() ad Θ(), [3] shows that the maximum per-ode capacity λ is upper bouded We use the followig otatio throughout: f() f() = o(g()) lim g() = 0, f() = O(g()) lim sup f() g() <, f() = ω(g()) g() = o(f()), f() = Θ(g()) f() = O(g()) ad g() = O(f()).
The Fudametal Relatioship Betwee Capacity ad Delay 3 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 wireless etwork ca sustai at most O(/) 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.6). 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 wireless etworks. A ope questio that still remais is: what is the optimal capacitydelay tradeoff i mobile wireless 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 wireless etworks. We will show that, if the per-bit-averaged mea delay is bouded by D, the the per-ode
4 Capacity O() O( 3 ) O( ) O( ) O() O( d ) O() Delay Figure.. The achievable capacity-delay tradeoffs of existig schemes compared with the upper boud (igorig the logarithmic terms). 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]. The top lie correspods to our upper boud (achievable by the scheme outlied i Sectio.5 up to a logarithmic factor), the middle (dashed) lie is achieved by the scheme i [4], ad the bottom (dash-dotted) lie is achieved by the scheme i [3]. There is obviously a gap betwee the upper boud ad what ca be achieved by existig schemes. 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 perode 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 wireless 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., we outlie the etwork ad mobility model. I Sectio.3, we prove several key properties that capture various tradeoffs iheret i mobile wireless
The Fudametal Relatioship Betwee Capacity ad Delay 5 etworks. We establish the upper boud o the optimal capacity-delay tradeoff i Sectio.4 ad preset a scheme i Sectio.5 that achieves a capacity-delay tradeoff close to the upper boud. I Sectio.6, we discuss the existig schemes described i [3] ad [4]. The we coclude.. Network ad Mobility Model We cosider a mobile wireless etwork with odes movig withi a uit square. We assume that time is divided ito slots of uit legth. We assume the 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 source-destiatio 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-to-iterferece Note that chagig the shape of the area from a square to a circle or other topologies will ot affect our mai results.
6 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..3 Properties of the Schedulig Policies I this sectio, we will prove several key results that capture the various tradeoffs iheret i mobile wireless 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
The Fudametal Relatioship Betwee Capacity ad Delay 7 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. 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
8 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 this broad class of schedulig policies. 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 wireless etworks..3. Notatios Let (Ω, F, P ) be the probability space o which the radom mobility of the mobile odes is defied. Let X(i, t) be the radom variable that deotes the positio of ode i at time slot t. Let b deote a bit that eeds to be commuicated from a source ode S(b) to destiatio ode D(b). Let t 0 (b) be the time slot whe bit b first eters the system. Let I b (i, t) be a idicator fuctio, where I b (i, t) = if ode i has a copy of bit b at the begiig of time slot t, I b (i, t) = 0 otherwise. By defiitio, I b (S(b), t 0 (b)) =, ad I b (i, t) = 0 for all i ad t < t 0 (b). Let F t be the σ-algebra geerated by the radom variables X(i, s) ad I b (i, s) for all s t. Hece {F t, t = 0,,...} is a filtratio [5, p3] ad F t captures all iformatio about the history up to time slot t. Fix ay schedulig policy ad fix a bit b that eters the system at time slot t 0 (b). For ay time slot t t 0 (b), let C b (t) = if the scheduler
The Fudametal Relatioship Betwee Capacity ad Delay 9 decides that a successful capture occurs at this time slot. C b (t) = 0, otherwise. If C b (t) =, the scheduler the picks oe mobile relay that has a copy of the bit b at the begiig of the time slot to forward the bit towards the destiatio withi the same time slot t, usig possibly multihop trasmissios. Let l b (t) be the distace from the chose mobile relay to the destiatio of the bit b. Let l b (t) = if C b (t) = 0. Fially, let r b (t + ) deote the umber of mobile relays holdig the bit b at the ed of the time slot t, i.e., r b (t + ) is the cardiality of the set {i : I b (i, t + ) = }. Sice the radom variables C b (t), l b (t) ad r b (t + ) are all outcomes of the schedulig policy, the causality assumptio implies that they are all F t -measurable 3. Let s b mi{t : t t 0 (b) ad C b (t) = } be the first time whe a successful capture for bit b occurs. Thus s b is a stoppig time [5, p34] with respect to the filtratio {F t, t = 0,,...}. Let R b r b (s b ) deote the umber of mobile relays holdig the bit b at the time of capture. Let D b s b t 0 (b) deote the umber of time slots from the time bit b eters the system to the time of capture. Let l b l b (s b ) deote the distace from the chose last mobile relay ode to the destiatio. The quatities R b, D b, ad l b are essetial for the tradeoffs that follow. Note that D b icludes possible queueig delay at the source ode or at the relay odes..3. 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 ] where c is a positive costat. (E[l b ] + ) E[R b ] for all bits b, (.4) The proof is available i Appedix.A. This ew result is oe of the corerstoes for derivig the optimal capacity-delay tradeoff i mobile wireless 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 3 Here we have excluded probablistic schedulig policies. Otherwise, F t should be augmeted with a σ-algebra that is idepedet of ode mobility i future time slots.
0 Example A at the begiig of Sectio.3. Whe R b ad the area of the capture eighborhood A b are costats, the ( A b ) R b 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 ) R. b 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 By Hőlder s Iequality [5, p5], Hece, E[D b R b, A b ] A b R b. E [ ] E[R b ]E[ ]. Ab A b R b 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. For details, please refer to Appedix.A..3.3 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
The Fudametal Relatioship Betwee Capacity ad Delay 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 Sb h l b. (.6) h=.3.4 Tradeoff III : Radio Resources 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 λt. 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, λt b= 4 E[R b ] λt + E[ h b π b= h= 4 (Sh b ) ] c 3 W T log. (.7) The assumptio that D b c for large is ot as restrictive as it appears. It has bee show i [3] that the maximal achievable perode capacity is Θ() ad this 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 Appedix.B. Cosider odes i, j that directly trasmit to odes k ad l, respectively, at the same time. The, accordig to the iterferece costrait: Hece, 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. Similarly, Therefore, X i X j X j X l. X i X j ( X i X k + X j X l ). That is, disks of radius times the trasmissio rage 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 ] 4 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 carried out over multiple time slots, because the average umber of ew mobile relays each broadcast ca cover is at 4 A similar observatio is used i [] except that they take a receiver poit of view.
The Fudametal Relatioship Betwee Capacity ad Delay 3 most proportioal to the area cosumed by the broadcast. Therefore, ispired by [], the amout of radio resources cosumed must satisfy λt b= 4 E[R b ] λt + E[ h b π b= h= 4 (Sh b ) ] c 3W T, (.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 oopportuistic scheme to duplicate the bit to the same umber of mobile relays. Noetheless, Propositio.3 shows that o schedulig policies ca improve the tradeoff by more tha a log factor. For details, please refer to Appedix.B..3.5 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 λt b= h= W T. (.9).4 The Upper Boud o the Capacity-Delay Tradeoff Our first mai result is to derive, from the above four tradeoffs, the upper boud o the optimal capacity-delay tradeoff of mobile wireless etworks uder the i.i.d. mobility model. Sice the maximal 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 sourcedestiatio pair. If D = O( d ), 0 d <, the followig upper boud
4 holds for ay causal schedulig policy, λ 3 O( D log3 ). Proof: Usig the Cauchy-Schwartz iequality, we have ( λt h b ) Sb h b= h= ) ( h b λt ) h b (Sb h ) ( λt b= h= W T h b b= h= λt (Sb h ), (.0) b= 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, (Sb h ) ] λt E[ h b b= h= ( λt ) h b W T E[ Sb h ] W T W T b= h= ( ) λt h b E[ Sb h ] (.) b= h= ( λt E[l b ]), (.) b= where i the last two steps we have used Jese s Iequality ad the Tradeoff II (.6), respectively. Iequality (.) is tight whe λt h b is almost surely a costat, ad (.) is tight whe (.6) is tight. From Tradeoff I (.4), we have λt E[R b ] b= λt b= Sb h b= h= c log (E[l b ] + ) E[D b ]. (.3) Let D = λt b= λt E[D b ] b= = λt b= E[D b ]. λt
The Fudametal Relatioship Betwee Capacity ad Delay 5 Usig Jese s Iequality ad Hőlder s Iequality, we have, λt P (E[l b ]+ b= ) λt P b= λt b= λt b= (E[l b ]+ ) E[D b ] λt b= (E[l b ]+ λt b= λt b= λt ) 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 b= λt E[R b ] b= c log ( λt ) 3 b= ( λt ). (.5) D (E[l b ] + ) b= Substitutig (.) ad (.5) ito Tradeoff III (.7), we have 4c 3 W T log λt b= E[R b ] c log λt + πe[ h b (Sb h ) ] b= h= (λt ) 3 ( λt D (E[l b ] + ) b= ) ( + π λt E[l b ]) λt. W T b= There are two cases that we eed to cosider. Case : If λt E[l b ] λt, the b= 4c 3 W T log = c log 4c log (λt ) 3 D ( ) λt λt λt 4 D λt.
6 Whe D = O( d ), d <, the first term domiates whe is large. Hece, for large eough, Case : If λt b= Therefore, either 4c 3 W T log 4c 3 W T log E[l b ] λt, the λt 4 8c log D λ 3c c 3 W c log ( D ( λt D log 4. (.6) (λt ) 3 E[l b ] λt b= ) + π E[l b ]) λt W T b= (.7) π (λt ) 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 c log λ 3 T 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 ). Q.E.D.
The Fudametal Relatioship Betwee Capacity ad Delay 7.5 A Achievable Lower Boud o 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 per-ode 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 to develop a ew scheme that ca achieve a capacity-delay tradeoff that is close to the upper boud. Precisely, we met several iequalities (.0)-(.8) durig the derivatio of the upper boud. 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. I the ed, the kowledge of the optimal choices of the parameters will help us develop a ew scheme that is superior to existig oes..5. 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 ). I order to achieve the maximum capacity o the right had side, all iequalities (.0)-(.8) should hold with equality. By checkig the coditios whe (.0)-(.4) are tight, we ca ifer that the parameters (such as Sb h, 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. From ow o, we will assume that all key parameters (such as R b, l b, etc.) are ideed the same for all bits. The iequality (.8) is essetial for derivig the optimal values of these parameters. Note that equality holds i (.8) if ad oly if 4c log (λt ) 3 = E[l b ]) D( λt b= λt π W T ( E[l b ]). b=
8 Substitutig λt b= E[l b ] = λt l b, we ca solve for l b, λt 4c log Dl b = l 4 b = π W T (λt ) l b 8πc W Dλ log. Substitutig λ = Θ( (d )/3 log ) ad D = d, we obtai the optimal value of l b, l b = Θ( +d 6 log ) A reasoable choice for the area of capture eighborhood, A b, is the, A b = l b = Θ( +d 3 / log ). By settig (.9) of Tradeoff IV to equality, we have λt h b = W T h b = W λ = Θ( d 3 / log ). By settig (.6) of Tradeoff II to equality, we have Sb h = l b log = Θ( h b ). Fially, by settig (.4) of Tradeoff I to equality, we have R b = Θ( c log l b d ) = Θ( 3 ). D The optimal values of these parameters are summarized i Table.. 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.3 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.
The Fudametal Relatioship Betwee Capacity ad Delay 9 Table.. The order of the optimal values of the parameters whe the mea delay is bouded by d. R b : # of Duplicates Θ( ( d)/3 ) l b : Distace to Destiatio Θ( (+d)/6 / log / ) h b : # of Hops Θ( ( d)/3 / log ) q Sb h : Trasmissio Rage of Each Hop Θ( log ).5. 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 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, see Fig..). 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 Appedix.C. 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 cells. 8 log Each cell is a square of area /g (). We refer to each cell i the odd time slot as a sedig cell. By Lemma.6, each cell ca be active for amout of time. Whe a cell is scheduled to be active, each ode i c 4
0 / m ( + 6) / m ( + 3) / m / m Figure.. Cells that are + 6 /m apart (i.e., the shaded cells i the figure) ca be active together. the cell broadcasts a ew message to all other odes i the same cell for amout of time (Fig..3). These other odes the serve as 3c 4 /3 log 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]. Uless a Type-I error occurs, each source ca broadcast a message of legth to all other odes i the same sedig cell. W 3c 4 /3 log ) 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. Each cell is a square of area /g (). 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. Uless a Type-II error occurs, each receivig cell will cotai oe desigated mobile relay from every sedig cell. Therefore, each destiatio ode ca ow fid a desigated mobile relay that holds the message iteded for the destiatio ode ad that resides
The Fudametal Relatioship Betwee Capacity ad Delay Source g_() colums Relay move g_() rows Figure.3. Trasmissio schedule i the odd time slot. g () = /3. 8 log i the same receivig cell (see Fig..4). We the schedule multi-hop 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 mii-cells 4 log (i g 3 () rows ad g 3 () colums, see Fig..5). Each mii-cell 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 (see Fig..5). I this fashio, a successful schedule will allow each destiatio ode to receive a message W of legth from its respective desigated mobile relay residig 3c 4 /3 log i the same receivig cell. For details o costructig such a schedule, see Appedix.D. 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. The proof is available i Appedix.D.
g_() colums g_() rows moved Desigated Mobile Relay Static Relay Destiatio Figure.4. Trasmissio schedule i the eve time slot. g () = /3. A Receivig Cell g_3() colums Desigated Mobile Relay Static Relays g_3() rows Destiatio Figure.5. Multi-hop trasmissios withi a receivig cell.
The Fudametal Relatioship Betwee Capacity ad Delay 3 Propositio.7 With probability approachig oe, as, the W above scheme allows each source to sed a message of legth 3c 4 /3 log 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.5. (with d = 0). The size of the receivig cell is chose such 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..5.3 The Effect of Queueig Whe we defied the delay D b of each bit b i Sectio.3, 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 capacity-achievig scheme i Sectio.5.. This scheme W attempts to deliver oe message of legth for each sourcedestiatio pair every two time slots. Let p e be the probability that a 3c 4 /3 log 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 that W packets of legth arrive at each source accordig to certai 3c 4 /3 log 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 [6], 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 source-destiatio pair arrives at the source every two time slots with probability Λ, the usig stadard results for discrete
4 time M/M/ queues [6, 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 source-destiatio pair every two time slots is a Poisso radom variable with mea Λ. Hece, usig results for discrete time M a /M/ queues [6, 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.5. ca sustai Θ( /3 / log ) throughput (i bits per time slot) with O() queueig delay..5.4 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
The Fudametal Relatioship Betwee Capacity ad Delay 5 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 Appedix.D, 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.5.3. The details are omitted because of space costraits..6 The Limitig Factors i Existig Schemes I Sectio.5, we have show that choosig the optimal values of the schedulig parameters is the key to achieve the optimal capacity-delay tradeoff. I this 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.5. with those used by the existig schemes. Our model i Sectio.4 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.4 ad.5. 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 isights o which schemes to use uder differet coditios..6. The Limitig Factor i the Scheme of Neely ad Modiao 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.5. 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 Appedix.E 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 sourcedestiatio pair. If D = O( d ), 0 d < ad E[l b ] = O(/ ), the
6 for ay causal schedulig policy, λ O( D log ). Remark: The scheme of [3] achieves the above upper boud up to a logarithmic factor..6. The Limitig Factor i the Scheme of Toumpis ad Goldsmith 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.5. 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 Appedix.F. 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 sourcedestiatio 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 wireless 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 capacityachievig scheme reported i Sectio.5 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
The Fudametal Relatioship Betwee Capacity ad Delay 7 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 capacity-delay tradeoffs uder differet coditios. A iterestig ope problem for future work is to ivestigate whether these isights apply to the capacity-delay tradeoff uder mobility models other tha the i.i.d. model. For example, [7] ad [8] 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 [7] uses at most oe mobile relay at ay time (i.e., R b = ), ad the scheme i [8] schedules a trasmissio from the mobile relay to the destiatio oly whe they are at a distace of O(/ ) away (i.e., l b = O(/ )). As we have show i this paper, uder the i.i.d. mobility model, the optimal capacity-delay tradeoff ca oly be achieved whe R b, l b ad h b all vary as fuctios of the delay expoet d. Puttig restrictios o ay 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..7 Coclusio ad Future Work I this paper, we have studied the fudametal capacity-delay tradeoff i mobile wireless etworks uder the i.i.d. mobility model. Our cotributios are three-fold. 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 capacity-delay tradeoff. Our secod cotributio is to develop a ew schedulig scheme that ca achieve a capacity-delay 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 demostrates 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.
8 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 wireless etworks uder other mobility models. Amog the properties that we proved i Sectio.3, 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 [7, 8], the radom waypoit model [8, 9], ad the liear mobility model [0], 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]. Appedix: (.A) Proof of Propositio. We will eed the followig lemma o the miimum distace from the mobile relays to the destiatio at ay time slot. Fix a bit b that eters ito the system at time slot t 0(b). At each time slot t t 0(b), recall that r b (t) is the umber of mobile relays holdig the bit b at the begiig of the time slot. Amog these r b (t) mobile relays, there is oe mobile relay whose distace to the destiatio of bit b is the smallest. Let L b (t) deote this miimum distace, ad let L b (t) = max{, L b (t)}. It is easy to verify that lb (t) L b (t) L b (t). Lemma A. Uder the i.i.d. mobility model, if 3, the» E L b (t)r b(t) Ft 8π log for all t t 0(b). Proof: Let I A be the idicator fuctio o the set A. By the defiitio of L b (t), we have,» h i E L Ft = E 4 I b (t) { Lb (t) } Ft +E " L b (t) I { L b (t)> } Ft #.