The Throughput ad Delay Trade-off of Wireless Ad-hoc Networks. Itroductio I this report, we summarize the papers by Gupta ad Kumar [GK2000], Grossglauser ad Tse [GT2002], Gamal, Mame, Prabhakar, ad hah [GMP2004], ad Neely ad Modiao [NM2004] respectively. These papers cosider the problem of the throughput ad delay of the wireless ad-hoc etworks: Gupta ad Kumar studies the throughput of wireless etworks where the odes are radomly located but immobile. The mai result shows that as the umber of odes icreases, the throughput per source-to-destiatio (-D) pair decreases approximately like /. Grossglauser ad Tse address the throughput issue of mobile etworks. They show that by usig the mobility ad a sigle relay, a -D pair o average ca have a costat throughput. [GMP2004] ad [NM2004] take differet approaches to study the delay ad the throughput-delay tradeoff. I [GMP2004], the trade-off is achieved by varyig the trasmissio rage, while i [NM2004] it is obtaied by usig the cocept of redudacy. 2. The Capacity ad Delay Trade-off of a Fixed Wireless Network: A. Fixed Network, uccessful Trasmissio, ad Throughput I [GK2000] odes are fixed over time. Their positios are i.i.d. ad uiformly distributed i the 2 surface of a three-dimesioal sphere (or a disk i the plae) of uit area. The destiatio for each source is a radomly chose ode i the etwork ad the destiatios are chose idepedetly. The packet from a source ca reach its destiatio either i oe hop or i multiple hops. That is, each ode acts simultaeously as a source, a destiatio for some other ode, as well as relays for others packets. To model what costitutes a successful receptio of a trasmissio over a sigle hop, [GK2000] proposes two models - the Protocol Model ad the Physical Model. The Protocol Model: suppose X i trasmit to ode X j. The this trasmissio is successful received by ode X j if Xk X j ( + ) Xi X j () for every other ode k simultaeously trasmittig over the same frequecy. The Physical Model: The trasmissio from X i to X j is successful if Pi α Xi X j β, (2) Pk N + α k i, k Ψ X X k where Ψ is the subset of the odes that simultaeously trasmit o the same frequecy, P k is the power level chose by ode X k, ad the sigal power decays with distace r as /r α j
I [GK2000] the otio of throughput is defied as the time average of the umber of bits per secod that ca be trasmitted by every ode to its destiatio. Thus a feasible throughput of λ( ) bits per secod for each ode meas that there is a spatial ad temporal scheme, such that by operatig the etwork i multi-hop fashio ad bufferig at itermediate odes, every ode ca sed λ( ) bits per secod o average to its chose destiatio ode. That is, for time iterval [( i ) t, it], where t <, every ode ca sed tλ( ) bits to its correspodig destiatio ode. B. Mai Result: Result (Mai Result 4 i [GK2004]): there exist costats c ad c such that, cr cr ' lim Pr λ( ) = is feasible = ad lim Pr λ( ) = is feasible = 0. log This mai result demostrates that as the umber of odes per uit area icreases, the throughput per -D pair decreases approximately like /. This is the best performace achievable eve allowig for optimal schedulig, routig, ad relayig of packets i the etworks ad is a somehow pessimistic result o the scalability of such etworks. The ituitio behid this pheomeo is that a trasmissio may travel either through a sigle direct trasmissio or through multiple hops via relay odes. As show i the protocol model, the successful receptio of the trasmissio of a give -D pair prohibits simultaeous trasmissio with i the disk of radius proportioal to the trasmissio distace of the pair: a successful trasmissio o the rage r icurs a cost proportioal to r 2 by excludig other trasmissios i the viciity of the seder. I order to maximize the trasport throughput of a etwork, i.e., the total umber of meters traveled by all bits per time uit, it is therefore beeficial to schedule a large umber of short trasmissios. The best we ca do is to restrict trasmissios to eighbors, which are at a typical distace of /. (ice the expected distace for each sessio is Θ (), the umber of relays a packet has to go through scales as ). The trasport capacity is the at most bitsi m/s. As there are sessios, it follows that the throughput per sessio ca at best be O( / ). D a ( ) Figure The uit torus is divided ito cells of size a() for cheme. 2
C. The Capacity ad Delay Trade-off I [GMP2004] the authors recover the results i [GK2000] ad aalyze the throughput ad delay trade-off usig the followig scheme: cheme (cheme i [GMP2004]): Divide the uit torus usig a square grid ito square cells, each of area a(). A TDMA scheme is used, i which each cell becomes active, i.e., the odes i the give cell ca trasmit successfully to odes i the cell or i eighborig cells, at regularly scheduled cell time slots. A source trasmits data to its destiatio D by hoppig alog the adjacet cells o its -D lie. Whe a cell becomes active, it trasmits a sigle packet for each of the -D lies passig through it. Result 2 (Theorem i [GMP2004]): for cheme, with a ( ) 2log /, i.e., the achievable throughput-delay trade- a ( ) λ( ) =Θ ad D ( ) a( ) =Θ D( ) off is λ( ) =Θ. R /log Q / / log Figure 2 The throughput-delay trade-off curve. We ca iterpret the above results as follows: the highest throughput per ode achievable i a fixed ode is Θ (/ log ), cosistet with the result by [GK2004]. At this throughput the average delay is D =Θ ( /log ) (correspodig to the poit Q i the throughput-delay trade-off curve plotted i Fig.2). The delay of a fixed etwork is proportioal to the umber of hops that a packet has to travel. By icreasig the trasmissio radius the average umber of hops (thus the delay) ca be reduced. ice the iterferece is higher ow, the throughput would decrease. If we oly allow oe-hop trasmissio (a()=), the throughput will go as low as / (correspodig to poit i Fig.2). Next we outlie the proof of Result 2: 3
Outlie of the proof: () Throughput: To aalyze the scalig of the throughput with the setup of cheme, we use the followig lamas: Lemma (Lemma i [GMP2004]): if a ( ) 2log /, the all cells have at least oe ode with high probability (whp). Lemma 2 (Lemma 2 i [GMP2004]): the umber of cells that iterfere with ay give cell is bouded by a costat c, idepedet of. Each cell ca be active for a guarateed fractio of time, i.e., it ca have a costat throughput. Lemma 3 (Lemma 3 i [GMP2004]): the umber of -D lies passig through ay cell is O( a( )). If each cell divides its cell time-slot i to O( a( ) ) packet time slots, each -D pair hoppig through it ca use oe packet time slot. Equivaletly, each -D pair ca successfully trasmit to Θ( / a( ) ) fractio of time. That is, the achievable throughput per -D pair is λ ( ) =Θ ( / a( ) ). (2) Delay: Packet delay is the amout of time spet i each hop. ice each hop covers a distace of Θ ( a ( )), the umber of hops per packet for -D pair i is Θ ( di / a( ) ), where d i is the legth of the -D lie i. Thus the umber of hops take by a packet averaged over all -D pairs is Θ d / a( ) i= i. For large, the average distace betwee a -D pair is d =Θ() Θ / a ( )., thus the average umber of hops is ( ) i= i 3. The Capacity ad Delay Trade-off of a Mobile Wireless Network-igle Relay Case: A. The Mobile Wireless Network The capacity of a mobile wireless etwork was first studied i [GT2002]. The etwork i [GT2002] cosists of odes all lyig i the disk of uit area. The locatio of the ith user at time t is give by X i (t), which is modeled as a statioary ad ergodic process with statioary distributio; moreover, the trajectories of differet users are idepedet ad i. i. d. Each ode is the source for oe sessio ad the destiatio of aother sessio, ad the -D associatio does ot chage with time. The cetral ituitio of [GT2002] is that ay two odes ca be expected to be close to each other from time to time so that we may improve the capacity of the etwork the delay tolerace ca be usefully exploited i a mobile wireless etwork. 4
B. The Role of Relayig I [GT2002] the authors first show that without relayig, there is o way to achieve a Θ() throughput per -D pair. (The results are preseted i Lemma III-2 ad Theorem of III-3 of [GT2002]. For the brevity of this summary, they are ot repeated here.) The explaatio is that the umber of simultaeous log-rage commuicatio is limited by iterferece. If trasmissios over log distace are allowed, the there are may -D pairs that are withi the rage - iterferece limits the umber of cocurret trasmissios overlog distaces; the throughput is iterferece limited. O the other had, if we costrai commuicatio to eighborig odes, the there is oly a small fractio of -D pairs that are sufficietly close for trasmittig a packet. Hece, the throughput is distace limited. To icrease the throughput, oe eed to fid a way to limit the trasmissio locally, while guarateeig that there would be eough seder-receiver pairs that have packets to sed. Thus [GT2002] proposes to spread the traffic stream betwee the source ad the destiatio to a large umber of itermediate relay odes. The goal is that i the steadystate, the packet of every source ode will be distributed across all the odes i the etwork, esurig that every ode will have packets buffered destied to every other ode. This esures that a scheduled seder-receiver pair always has a packet to sed, i cotrast to the case of direct trasmissios. A questio that aturally follows is that how may times a packet eeds to be relayed. I fact, as the ode locatio processes are idepedet, statioary, ad ergodic, it is sufficiet to relay oly oce. This is because the probability for a arbitrary ode to be scheduled to receive a packet from a source ode is equal for all odes ad idepedet of. Each packet the makes two hops, oe from the source to its radomly chose relay ode ad oe from the relay ode to the destiatio as show i Fig. 3. Relay Nodes ource Destiatio D - routes Figure 3 The two-phase schedulig scheme viewed as a queuig system. C Throughput [GT2002] proposes a schedulig algorithm that cosists of two phases: the schedulig of packet trasmissios from sources to relays ad the schedulig of trasmissios form relays to fial destiatios. Note that i both phases a trasmissio from a source directly to a destiatio is possible. This two-phased algorithm leads to the followig result. 5
Result 3 (Theorem III-5 i [GT2002]): the two-phased algorithm achieves a throughput per -D pair of Θ (), i.e., there exists a costat c > 0 such that lim Pr λ( ) = cr is feasible = { } Outlie of the proof: The proof of the Result 3 depeds o a importat result that states that the probability that give two odes are selected as a -D pair is Θ (/ ) (This result is formally stated as Theorem III-4 i [GT2002], the outlie of the proof is ot icluded here for brevity of this summary). For a give -D pair, there is oe direct route ad -2 two-hop routes. The throughput of the direct route is Θ (/ ). Each of the two-hop route ca be treated as a sigle server queue, each with arrival ad service rate of Θ (/ ). The total throughput is Θ (/ ) by summig all the throughputs of - routes. D. The Capacity ad Delay Trade-off [GMP2004] also assumes that odes formig -D pairs i a torus of uit area ad assumed slotted trasmissio time. Each ode moves idepedetly ad uiformly o the uit torus. The authors recover the results by [GT2002] ad aalyze the throughput ad delay trade-off usig the followig scheme: cheme 2 (cheme 2 of [GMP2004]) Divide the uit torus ito square cells, each of area /. Each cell becomes active oce i every +c cell time slots. Trasmissio is limited with i a active cell. I a active cell, if two or more odes are preset pick oe at radom. Each cell time-slot is divided ito two subslots A ad B. A. Trasmit to destiatio if it is i the same cell. Otherwise, it trasmits to a radomly chose ode i the same cell, which acts as a relay, B. The radomly chose ode picks aother ode at radom form the same cell ad trasmits to it a packet that is destied to it. Result 4 (Theorem 3 ad 4 of [GMP2004]): the throughput ad delay usig cheme 2 is give by λ ( ) =Θ() ad D( ) =Θ ( ). This poit is show as R i Fig.2. Outlie of the proof: () Throughput Each packet is trasmitted directly to its destiatio or relayed at most oce ad hece the et traffic is at most twice the origial traffic. ice: (a) a relay ode is chose radomly; (b) the odes have idepedetly ad uiformly distributed motio, each source s traffic gets spread uiformly amog all other odes. As a result, i steady state, each ode has packets for every other ode for a costat fractio of time c 2. 2e fractio of the cells cotai at least 2 odes. 0.26 c2 /( + c) fractio of cells ca execute the scheme successfully. ice each cell has a throughput of Θ (), the et 6
Θ is divided equally amog pairs, we have a throughput T=Θ ( ) (). (2) Delay ice the torus has square cells, the area of each cell is /, the side of each cell is /. The movemet of the odes is modeled as a radom walk o a 2-D torus of throughput i ay time-slot is Θ( ) whp. ( ) size t ( ) =Θ / v ( ). ice most of the -D pairs have to go through two hops, the delay for this scheme has two compoets: hop-delay ad mobile delay. ice ode velocity is much lower tha the speed of electromagetic propagatio, the delay is domiated by the mobile delay. [GMP2004] first models the queues formed at a relay ode for each -D pair as GI/GI/-FCF, the characterizes the iter-arrival ad iter-departure times of the queue to obtai the average mobile delay. By boudig the first ad the secod momets of the iter-arrival ad departure process ad usig the followig average delay boud (Lemma 8 of [GMP2004]), oe ca upper boud the umber of radom walks required as 2 2 σa + σ E[# of rad. wlk] max, s µ =Θ( ) 2µε. (3) The delay ca be derived as D ( ) =Θ( ) Θ =Θ v( ) v ( ). (4) D( ) =Θ whe v( ) =Θ (/ )., where each move occurs every t() time slots, where ( ) We have ( ) I [NM2004], the authors propose a Cell Partitioed Relay Algorithm ad obtai the same capacity-delay trade-off relatioship for the mobile ad sigle relay case. The problem setup assumes that the etwork is partitioed ito C o-overlappig cells of equal size. There are mobile users idepedetly roamig from cell to cell over the etwork: for a give time slot users remai i their curret cells for a timeslot ad potetially move to a ew cell at the ed of the slot. As i cheme 2, the trasmissio is also limited i the cell. The mai differece of the setup i [NM2004] is that the authors assume ifiite mobility - the etwork topology dramatically chages over timeslot, so that etwork behavior caot be predicted. The other differece of the setup is that queuig delay at the source ode is cosidered i [NM2004], while this is ot take ito accout i [GMP2004]. Result 5a (Theorem of [NM2004]): let p 0 deote the probability that there are at least 2 odes i a cell, q deote the probability that there is a -D pair i a cell, ad d deote the ode desity (the umber of odes per cell). The capacity of the etwork is bouded d d Cp0 + Cq + 2ε Cp0 + Cq + 2ε e de by λ( ) ad lim = 2 2 2d 7
Commets: ote that d d e de lim = 0 (5) 2d d, d 0 That is, if d is too large, there will be may users i each cell, most of which will be idle as a sigle trasmitter ad receiver are selected. If d is too small, the probability of two users beig i a give cell vaishes. I both cases, the capacity dimiishes. [NM2004] calculates the optimal ode desity ad throughput: d * =.7933 ad µ * =0.492. R p0 q 2 d ( 2) λ λ() i R -2 p0 q 2 d ( 2) D Figure 4 A two-stage queue, the first stage are Beroulli with rate λ ( ). ervice at the secod stage (relay) queues is Beroulli with rate (p-q)/(2d(n-2)). Result 5b (Theorem 3 of [NM2004]) Cosider a cell partitioed etwork (with users ad C cells) uder the 2-hop relay algorithm, ad assume that users chage cells i. i. d. ad uiformly over each cell every timeslot. If the exogeous iput stream to user i is a Beroulli stream of rate λ i (where λi < λ( ), the the total etwork delay T i for user i traffic satisfies: N λi ET { i} = λ( ) λi Commets: () The radomized ature of the cell partitioed relay scheme admits a ice decouplig betwee the sessios, where idividual users see the etwork as a twostage queues, the first stage are Beroulli with rate λ ( ). ervice at the secod stage (relay) queues is Beroulli with rate (p-q)/(2d(n-2)). This is differet from the sigle-stage queue treatmet i [GMP2004], i which the queuig delay at the source ode is ot cosidered. (2) Because of the ifiite mobility assumptio, the delay is ot modeled explicitly as a fuctio of ode velocity. For the detail of the proof for Result 5a ad Result 5b, please see [NM2004], the proof is ot outlied i this summary. 8
4. The Capacity ad Delay Trade-off of a Mobile Wireless Network Trasmissio Rage ad Redudacy: A. Varyig Trasmissio Rage [GMP2004] proposes the followig schemes to achieve the capacity ad delay tradeoff through varyig trasmissio rage ad multi-hoppig. cheme 3a (cheme 3a of [GMP2004]) Divide the uit torus usig a square grid ito square cells, each of area a(). A TDMA scheme is used, i which each cell becomes active, i.e., its odes ca trasmit successfully to odes i the cell or i eighborig cells, at regularly scheduled cell time slots. A source trasmits data to its destiatio D either by direct trasmissio of by relayig to the odes i the adjacet cells alog its -D lie. Commets: this scheme is basically the same as the cheme, except that ow the odes are movig aroud, so the -D ad R-D lies are chagig, as illustrated by Fig. 4 D D R' D R R R' Figure 5 cheme3a is basically the same as cheme Result 6 (Theorem 5 of [GMP2004]) If v ( ) = o( log / ) is satisfied, achieves the followig trade-off: D( ) T ( ) =Θ, for T ( ) = O log Commets: this trade-off correspods to the poit Q of Fig.2. Outlie of the proof: The proof follows the same ratioale as the proof for Result. () The coditio that v ( ) = o( log / ) is ecessary for every packet to be evetually delivered. Note that lim v ( ) = 0, the mobile etwork behaves more like a fixed etwork as the umber of odes icreases. The proof follows the same ratioale as the proof for Result. 9
(2) Delay: the average umber of times a packet has to be relayed i order to reach its Θ / a ( ), which is same as i cheme for fixed destiatio is of order ( ) etworks. Hece the delay scales as D( ) ( / a( ) ) =Θ. (3) Throughput: the proof for throughput is the same as that for cheme, the umber of -D paths passig through ay cell at ay give time-slot is / a ( ) Θ / a( ). Θ ( ), thus the throughput per -D pair is at least ( ) I [GMP2004], the followig scheme is also proposed to achieve the poits o the trade-off segmet QR. cheme 3b (cheme 3b of [GMP2004]) Divide the uit torus usig a square grid ito square cells, each of area a(). Divide each cell ito sub-cells, each of area b(). A cellular TDMA scheme is used, with each cell slot cotais Θ( a( ) ) packet slots. Each active packet time-slot is divided ito two sub-slots: A. Trasmit to destiatio if it is i the same cell. Otherwise, it trasmits to a radomly chose ode i the same cell, which acts as a relay. The packet is set usig hops alog sub-cells as i cheme 3a. B. The radomly chose ode picks aother ode at radom form the same cell ad trasmits to it a packet that is destied to it. The packet is set usig hops alog sub-cells as i cheme 3a. D R' R' a ( ) R R b ( ) Figure 6 The uit torus is divided ito cells of size a() ad sub-cells of b()for cheme3b. This scheme, as illustrated i Fig.6, cosists of iter-cell travel ad itra-cell trasmissio. The iter-cell travel, which eable the packets evetually to be delivered, is mostly similar to that of cheme2, with the differece that the cell size a() is adjustable, istead of / i cheme 2. The itra-cell trasmissio is mostly similar to that of cheme 3a, with the differece that the hop size is b(), istead of a() i cheme 3a. 0
Result 7 (Theorem 6 of [GMP2004]) If v ( ) = o( log / ) is satisfied, cheme 3b achieves the followig trade-off: T ( ) =Θ ad D ( ) = O a( )log av ( ) ( ) where a ( ) = O() ad a ( ) =Ω ( log / ) Outlie of the proof: The ituitio behid the cheme 3b ad the proof is that itra-cell trasmissio would provide us the desired throughput ad iter-cell travel provides desired delay. The poits o the segmet QR is achieved by varyig the size of the cell a ( ). () I steady state, each ode has packets for every other ode for a costat fractio of time ad the traffic betwee each source destiatio pair is spread uiformly across all other ode, as for cheme 2. (2) Throughput: sice the trasmissio is limited withi the cell (size of a()), the throughput ca be derived i the similar fashio as i that for cheme, with the followig differece, is replaced by a() ad a() is replace by b(). (3) Delay: the mobile delay domiates the total delay. The mobile delay ca be aalyzed i the same fashio as for cheme 2, albeit with the followig differeces: The radom walk is of discrete size of / a ( ) / a ( ), istead of / /. a ( ) A ode travels to its eighborig slots every t ( ) =Θ time-slots, istead of v ( ) t ( ) =Θ. v ( ) B. Redudacy ad igle Relay I [NM2004] the authors adopt a differet approach to achieve the throughput-delay trade-off sedig duplicates of the same packet to differet users. The throughput will decrease but the delay ca be improved. I-Cell Feedback cheme with redudacy: i every cell with at least two users, a radom seder ad a radom receiver are selected, with uiform probability over all users i the cell. With probability ½, the seder is scheduled to operate i either source-torelay mode, or relay-to-destiatio mode, described as follows: ) ource-to-relay Mode: The seder trasmits packet N, ad does so upo every trasmissio opportuity util replicas have bee delivered to distict users, or util the seder trasmits N directly to the destiatio. After such a time, the sed umber is icremeted to N+. If the seder does ot have a ew packet to sed, remai idle. 2) Relay-to-Destiatio Mode: Whe a user is scheduled to trasmit a relay packet to its destiatio, the followig hadshake is performed: The receiver delivers its curret RN umber for the packet it desires.
The trasmitter deletes all packets i its buffer destied for this receiver that have N umbers lower tha RN. The trasmitter seds packet RN to the receiver. If the trasmitter does ot have the requested packet RN, it remais idle for that slot. Result 8 (Theorem 6 i [NM 2004]) The I-Cell Feedback cheme achieves the O( ) delay boud ad the throughput of O(/ ). Commets: () The hadshakes as described i the above scheme are ecessary so that the old versios of packets already delivered are removed from the etwork to avoid uecessary cogestio. (2) The delay ca t be better tha O( ), if oly sigle relay is allowed. For the detail of the proof for Result 8, please see [NM2004], the proof is ot outlied i this summary. 5. ummary The schemes used i the report are summarized i the followig table. Ad Fig. 8 plots the mai results. cheme [GK] & [GMP] cheme 2 [GT], [GMP] & [NM] cheme 3a [GMP] cheme 3b [GMP] Node Mobility Fixed Mobile Mobile Mobile Cell size a() / a() a() ub-cell size N.A. N.A. N.A. b() Multiple Access cheme Trasmissio Rage Deliverig Data to Destiatio Multiple hops TDMA +c slots To adjacet cells -D lie > 2 hops possible TDMA +c slots with sub-slots Withi the cells -D or -R ad R-D 2 hops maximum TDMA +c slots To adjacet cells -D lie > 2 hops possible TDMA +c slots with sub-slots With i the cells; to adjacet subcells Iter Cells usig 2, itra cells usig 3(a) > 2 hops possible 2
R [GT], [GMP] & [NM] R' [GT], [GMP] & [NM] /log Q [GK] & [GMP] Icrease a() Q' [NM] Redudacy [GK] Icrease a() / / log / Figure 8 The throughput-delay trade-off curve. Left: the results from [GMP2004]. Right: the results from [NM2004]. Referece: [GK2000] "The capacity of wireless etworks", P. Gupta ad P.R. Kumar, IEEE Trasactios o Iformatio Theory, March 2000, 46(2): 388-404. [GMP2004] "Throughput-delay trade-off i wireless etworks", A. El Gamal, J. Mamme, B. Prabhakar, ad D. hah, Proceedigs of IEEE Ifocom 2004. [GT2002] "Mobility icreases the capacity of ad-hoc wireless etworks", M. Grossglauser ad D.N.C. Tse, IEEE/ACM Trasactios o Networkig, Aug 2002, 0(4): 477-486. [NM2004] "Capacity ad delay tradeoffs for ad-hoc mobile etworks", M.J. Neely ad E. Modiao, submitted to IEEE Trasactios o Iformatio Theory, Nov. 2003. 3