Throughput-Delay Scalig i Wireless Networks with Costat-Size Packets Abbas El Gamal, James Mamme, Balaji Prabhakar, Devavrat Shah Departmets of EE ad CS Staford Uiversity, CA 94305 Email: {abbas, jmamme, balaji, devavrat}@staford.edu Abstract I previous work 2004), we characterized the optimal throughput-delay trade-off i static wireless etworks as D) = ΘT )), where D) ad T ) are the average packet delay ad throughput i a etwork of odes, respectively. While this trade-off captured the essetial etwork dyamics, packets eeded to scale dow with the etwork size. I this fluid model, o buffers were required. Due to this packet scalig, D) did ot correspod to the average delay per bit. That led to the questio whether the trade-off remais the same whe the packet size is kept costat, which ecessitates buffers ad packet schedulig i the etwork. I this paper, we aswer this questio i the affirmative by showig that the optimal throughput-delay trade-off is still D) = ΘT )), where ow D) is the average delay per bit. Packets of costat size ecessitate the use of buffers i the etwork, which i tur requires schedulig packet trasmissios i a discrete-time queueig etwork ad aalyzig the correspodig delay. Our method cosists of derivig packet schedules i the discrete-time etwork by lookig at a correspodig cotiuous-time etwork ad the aalyzig the delay iduced i the actual discrete etwork usig results from queueig theory for cotiuous-time etworks. packet size remais costat, the throughput-delay trade-off ca be o better tha that i the fluid model. However, a priori, it is ot clear whether the same throughput-delay tradeoff as i the fluid case ca be achieved, sice ow, routig packets through the etwork also ivolves the additioal task of schedulig i the etwork. I [7], it was show that i a mobile etwork model with i.i.d. mobility each ode is distributed uiformly at radom i each time-slot idepedet of others ad the past), a two-hop scheme like the oe i [2] achieves the optimal trade-off usig packets of costat size. However, this method does ot exted to static etworks or mobile etworks with o-i.i.d. mobility. I this paper, we exted our previous work to the case of wireless etworks with buffers ad costat-size packets ad show that the optimal trade-off is still D) = ΘT )) as show i Figure ) where ow D) is the average delay per bit. D) I. INTRODUCTION PSfrag replacemets Gupta ad Kumar, i their semial work [3], itroduced / log a radom etwork model for studyig throughput scalig i a static wireless etwork, i.e., whe the odes do ot move. They showed that the throughput per source-destiatio pair scales as Θ / log ). They implicitly used a fluid model ad later work by Kulkari ad Viswaath [5] cosolidated the result with a explicit costat packet size model. I previous work [], we studied the throughput-delay tradeoff i wireless etworks. The optimal throughput-delay tradeoff was established to be D) = ΘT )), which is graphically preseted i Figure. I this work, packet size eeded to scale dow with the umber of odes i the etwork. This led to a fluid model for trasmittig packets ad allowed us to obtai the essetial trade-off by skirtig the issue of bufferig ad the resultat queueig delay at the odes. The delay that was cosidered i [] was the average packet delay ad sice the packet size was allowed to scale dow with, it did ot correspod to the average delay per bit. This paper ivestigates the throughput-delay trade-off whe the packet size remais costat, i.e., does ot scale dow with. This is a importat questio, sice i real etworks, packet size does ot chage whe more odes are added to the etwork. Note that with the additioal costrait that the / / log T ) Fig.. Throughput-delay scalig trade-off i the static radom etwork model. The scales of the axes are i terms of orders i. The mai cotributio of this paper is to determie the exact order of delay by couplig the evolutio of a discrete-time queueig etwork with that of a cotiuous-time queueig etwork. This provides both a packet schedulig policy see item 6 of Policy Σ i Sectio II) ad a method for aalyzig the delay. Packets i a wireless etwork have fixed routes depedig o the source-destiatio pair to which they belog. The etire wireless etwork the correspods to a discretetime, ope queueig etwork with geeral customer routes, i the termiology of queueig theory e.g. see [4], [8]). I the case of cotiuous-time queueig etworks, these are
also kow as Kelly or BCMP etworks ad the equilibrium distributio is kow to have a product form. Based o packet arrival times i a cotiuous-time queueig etwork with a Preemptive LIFO queue maagemet at each server, we derive a schedulig policy for the wireless etwork. Fially, usig product form equilibrium results for cotiuous-time etworks, we determie the exact order of queueig delay i the discrete-time wireless etwork. A. Model ad Defiitios We first preset the static radom etwork model, the model for successful wireless trasmissio ad the the defiitios of the performace metrics throughput ad delay. Defiitio Static radom etwork model): The static radom etwork cosists of a uit torus i which odes are distributed uiformly at radom. These odes are split ito distict source-destiatio S-D) pairs at radom. Time is slotted for packetized trasmissio. For simplicity, we assume that the time-slots are of uit legth. Defiitio 2 Model for successful trasmissio): Uder our Relaxed Protocol model, a trasmissio from ode i to ode j i a time-slot is successful if for ay other ode k that is trasmittig simultaeously, dk, j) + )di, j) for > 0 where di, j) is the distace betwee odes i ad j. Durig a successful trasmissio, odes sed data at a costat rate of W bits per secod. With time-slots of uit legth, this meas that the size of packets trasmitted i each slot is W bits. Defiitio 3 Scheme): A scheme Π, for a radom etwork, is a sequece of commuicatio policies, Π ), where policy Π determies how commuicatio occurs i a etwork of odes. Defiitio 4 Throughput of a scheme): Let B Π i, t) be the umber of bits of S-D pair i, i, trasferred i t time-slots uder policy Π. Note that this could be a radom quatity for a give realizatio of the etwork. Scheme Π is said to have throughput T Π ) if there exists a sequece of sets A Π ) such that { } A Π ) = ω : mi lim if i t t B Π i, t) T Π ) ad P A Π )) as. We use the term whp with high probability) to deote this. Defiitio 5 Delay of a scheme): The delay of a bit is the time it takes for the bit to reach its destiatio after it leaves the source. Let DΠ i j) deote the delay of bit j of S-D pair i uder policy Π, the the sample mea of delay for S-D pair i is D i Π = lim sup k k k DΠ i j). j= The average delay over all S-D pairs for a particular realizatio of the radom etwork is the D Π = 2 D i Π. The delay for a scheme Π is the expectatio of the average delay over all S-D pairs, i.e., D Π ) = E[ D Π ] = 2 E[ D Π i ]. Defiitio 6 Throughput-delay optimality): A pair T ), D)) is said to be Throughput-Delay T-D) optimal if there exists a scheme Π with T Π ) = ΘT )) ad D Π ) = ΘD)) ad scheme Π with T Π ) = ΩT )), DΠ )) = ΩD)). Regardig delay, we would like to ote that i the defiitio we used bit delay whereas i the schemes we preset later we refer to packet delay sice commuicatio is packetized. But sice the packet size is costat, these quatities are of the same order. Our mai result is as follows. Theorem : The optimal throughput-delay trade-off i the static radom etwork model is give by T ) = Θ D)/), for T ) = O / log ). The above result says that uder a delay scalig costrait of D) the optimal throughput scalig is ΘD)/). Ad this holds for T ) = O / log ), that is, the etire rage of achievable throughputs i the static radom etwork model. The rest of this paper is orgaized as follows. I Sectio II, we itroduce Scheme Π ad show that it achieves the throughput-delay trade-off stated i Theorem. Fially we preset a coverse that shows that o scheme ca provide a better throughput-delay trade-off tha Scheme Π, thus establishig Theorem. II. THROUGHPUT-DELAY TRADE-OFF IN STATIC NETWORKS Our trade-off scheme is a multi-hop, time-divisiomultiplexed TDM), cellular scheme with square cells of area a) so that the uit torus cosists of /a) cells as show i Figure 2. I the followig aalysis, we igore the edge effects due to /a) ot beig a perfect square. Before presetig the trade-off scheme, we preset three lemmas about the geometry of the odes o the torus divided ito square cells of area a). See [] for the proofs. Lemma : If a) 2 log /, the each cell has at least oe ode whp. We say that cell B iterferes with aother cell A if a trasmissio by a ode i cell B ca affect the success of a simultaeous trasmissio by a ode i cell A. Lemma 2: Uder the Relaxed Protocol model, the umber of cells that iterfere with ay give cell is bouded above by a costat c, idepedet of.
ag replacemets S S Cell of size a) S-D lies Fig. 2. The uit torus is divided ito cells of size a) for Scheme Π. The S-D lies passig through the shaded cell i the ceter are show. We say that a cell is active i a time-slot if ay of it odes trasmits i that time-slot. A cosequece of Lemma 2 is that, there exists a iterferece-free schedule where each cell becomes active regularly, oce i +c time-slots ad o cell iterferes with ay other simultaeously trasmittig cell. Let the straight lie coectig a source S to its destiatio D be called a S-D lie. Lemma 3: The umber of S-D lies passig through each cell is O ) a), whp. The above lemma shows that the umber of S-D lies passig through each cell is c 2 a) whp, for a appropriate choice of the costat c 2. Now we are ready to describe Scheme Π, which is parameterized by the cell area a) where a) = Ωlog /) ad a). Recall that by defiitio, Scheme Π is a sequece of commuicatio policies Π ). For ay particular realizatio of the radom etwork with odes, policy Π differs based o the followig two coditios. Coditio : No cell is empty. Coditio 2: The umber of S-D lies through each cell is at most c 2 a). If both the above coditios above are satisfied the Π is the policy Σ, described below. Otherwise, Π is a time-divisio policy where each of the sources trasmits directly to its destiatio i a roud-robi fashio. Policy Σ : ) Divide the uit torus usig a square grid ito square cells, each of area a) see Figure 2). 2) Each ode geerates packets accordig to a Poisso D D process of rate T ) = Θ / ) a). The radom etwork is a discrete-time system whereas the packet geeratio is a cotiuous-time process. So if a packet is geerated at time t, it is available for trasmissio from time-slot t owards. 3) Each cell becomes active at a regular iterval of + c time-slots see Lemma 2). Several cells which are sufficietly far apart become active simultaeously. Thus the scheme uses TDM betwee earby cells. 4) A source S seds packets to its destiatio D by relayig or hoppig alog the adjacet cells lyig o its S- D lie as show i Figure 2. Thus, i this scheme, direct trasmissio of packets is oly betwee odes i adjacet cells. 5) Oe of the odes i a cell acts as a relay by maitaiig a buffer for the packets of all the S-D lies passig through that cell. I each time-slot oly oe packet ca be trasmitted. However, a relay ode may receive up to four packets from its adjacet cells before it gets a chace to relay them. Moreover multiple packets may be geerated withi the cell which will be available for trasmissio i the ext time-slot. Hece a virtual queue is formed i each cell which cosists of packets geerated withi the cell as well as the packets to be relayed through the cell. 6) Whe the cell becomes active, oe packet from this queue if ot empty) is trasmitted to a adjacet cell accordig to a Last-I-First-Out LIFO) type of queue service policy. However, the arrival times cosidered by this policy are ot the actual arrival times of the packets, but the arrival times that would occur i a cotiuous-time etwork with the same arrivals ad a PL Preemptive LIFO) queue maagemet at each server. This is elaborated later i this sectio durig the aalysis of delay. The trade-off achieved by Scheme Π is give by the followig theorem. Theorem 2: Scheme Π with a) = Ωlog /) has T ) = Θ / a) ) ad D) = Θ / a) ), i.e., it achieves the trade-off T ) = Θ D)/). Throughput aalysis: If the time-divisio policy with direct trasmissio is used, the the throughput is 2W/ ad delay of. But sice it happes with a vaishigly low probability, as show by Lemmas ad 3, the throughput ad delay for Scheme Π are determied by that of policy Σ. Whe policy Σ is used, sice Coditio is satisfied, each cell has at least oe ode. This guaratees that each source ca sed data to its destiatio by hops alog adjacet cells o its S-D lie. From Lemma 2, it follows that each cell gets to trasmit a packet every + c time-slots, or equivaletly, the cell throughput is Θ). The total traffic through each cell is that due to all the S-D lies passig through the cell, which is O ) a) sice Coditio 2 is also satisfied. This suggests
that T ) = Θ ) a) is achievable, if the average delay is fiite. Fig. 3. The torus o the left with has 6 cells ad each cell cotais at least oe ode. The circled ode i each cell acts as a relay. The correspodig queueig etwork of 6 servers, with each server correspodig to a cell i the wireless etwork, is show o the right. Delay Aalysis: Next we aalyze the packet delay i the wireless etwork whe policy Σ is used. Dividig the uit torus ito square cells of area a) results i /a) cells. Oe of the odes i each cell maitais a buffer ad acts as a relay for all the S-D lies passig through that cell. These relay odes are the circled odes i Figure 3. The buffer i each cell correspods to a queue ad the cell itself correspods to a server that ca trasmit oe packet from this queue oce i + c time-slots. This is because each cell becomes active oce i + c time-slots as described earlier. Sice policy Σ restricts direct trasmissios to be betwee adjacet cells, each cell ca receive from or trasmit to ay four of its adjacet cells. This determies the coectivity betwee the servers so that the etire wireless etwork correspods to a discretetime queueig etwork of /a) servers, where each server is coected to four others as show i Figure 3. Note that the TDM betwee cells is such that i the c slots before each cell becomes active agai each of its eighbors becomes active exactly oce. Hece we ca igore the effect of cells becomig active at regular itervals ad istead cosider a discrete-time etwork of queues N D where D sigifies the discrete time ature of this etwork. The actual delay i the wireless etwork would the be +c times the delay i N D. Queueig etwork N D : The discrete-time queueig etwork N D cosists of /a) servers, each of which ca service oe packet from its queue i a time-slot if it is ot empty. Moreover, each server is coected to four others as explaied above. I the wireless etwork, packets travel from their sources to their destiatios by hops alog adjacet cells o their S-D lies. Thus the route of a packet depeds o the S-D pair to which it belogs. This meas that i N D there are customer routes correspodig to the S- D pairs. Recall that packets arrive i the wireless etwork at the sources accordig to idepedet Poisso processes of rate T ). These correspod to exogeous arrivals at the queues i N D. The remaiig arrivals at the queues are due to the departures from other queues. I the termiology of queueig theory, N D is a discrete-time, ope etwork of queues with geeral customer routes see Chapter 6.6 of [8]). Delay aalysis for such discrete-time etworks with geeral customer routes is ot kow, which prevets us from usig a simple First-I-First-Out FIFO) order of service i N D. We leverage results kow about cotiuous-time etworks to obtai the queue maagemet policy for N D i such a way that the average delay ca be computed. Queueig etwork N C : Cosider a cotiuous-time ope etwork of /a) servers havig the same coectivity structure as N D ad the same customer routes see Figure 3). Let this etwork be called N C. Further, let the exogeous arrivals i both the etworks N C ad N D are the same. Ad let the service requiremet of each packet at each server be determiistically equal to uit time. From the descriptio util ow, it is clear that N C is the cotiuous-time aalog of the discrete-time etwork N D. A Preemptive LIFO PL) queue maagemet is used at each server i N C see Chapter 6.8 of [8] for more details). The queue size distributio for the cotiuous time etwork N C with PL queue maagemet at each server has a product form i equilibrium as show i [4] see Theorems 3.7 ad 3.8 of Chapter 3) provided that the followig two coditios are satisfied. First, the service time distributio should be either phase-type or the limit of a sequece of phase-type distributios. I our case the service time is costat ad equal to. The secod coditio is that, the total traffic at each server is less tha its capacity, which is oe i our case. Cosider the sum of expoetial radom variables each with mea /. This sum has a phase-type distributio ad i the limit as teds to ifiity, its distributio coverges to that of a costat radom variable. Thus the first coditio is satisfied. I the wireless etwork the umber of S-D lies passig through each cell is O ) a) ad the arrival process for each S-D pair is a idepedet Poisso process with rate T ) = Θ / ) a). Therefore a appropriate choice of costats guaratees that the total traffic at each server is less tha, its service capacity, due to Coditio 2 beig satisfied. Usig the product form for the queue size distributio i equilibrium, it follows that the average queue size at a queue with total traffic λ < ad uit mea service is of the form c 3 λ/ λ) where c 3 is some costat. By Little s law this implies that the average delay at each server is bouded above by a costat idepedet of. We summarize the above discussio i the lemma below. Lemma 4: For the cotiuous-time ope etwork N C with customer routes as described above the average delay at each server is bouded above by a costat idepedet of. Packet Schedulig i N D usig N C : However we caot use this PL policy i the discrete time etwork N D because of the followig reasos: ) Due to the discrete time ature of the etwork N D, a packet that is geerated at time t becomes eligible for service i.e. ext hop trasmissio) oly at time t.
2) A complete packet has to be trasmitted i a time-slot, i.e. fractios of the packets caot be trasmitted. This meas that a preemptive type of service like PL is ot allowed. To address these problems for N D, we preset a cetralized schedulig policy derived from emulatig i parallel, the cotiuous-time etwork N C with PL queue maagemet at each server. The exogeous arrivals i both N C ad N D are the same. Let a packet arrive i N C at some server at time a C ad i N D at the same server at time a D. The it is served i N D usig a LIFO policy with the arrival time cosidered to be a C istead of a D. Clearly such a policy ca be implemeted if ad oly if a D a C for every packet at each server, i.e., each packet arrives before its scheduled departure time. Let d C ad d D be the departure times of a packet from some server i N C ad N D respectively. The this is the same as sayig that d D d C for each packet i every busy cycle of each server i N C. I what follows, we show that for all packets i ay busy cycle of ay server, the departures i N D occur at or before the departures i N C. Lemma 5: Let a packet depart i N C from some server at time d C ad i N D at time d D, the d D d C. Proof: Fix a server ad a particular busy cycle of N C. Let it cosist of packets umbered,..., k with arrivals at times a... a k ad departures at times d,..., d k. Let the arrival times of these packets i N D be A,..., A k ad departures be at times D,..., D k. By assumig that A i a i for i =,..., k, we eed to show that D i d i for i =,..., k. Clearly this holds for k = sice D = A + a i + = d. Now suppose it holds for all busy cycles of legth k ad cosider ay busy cycle of k + packets. If a < a 2, the because of the LIFO policy i N D based o times a i, we have D = a + a + k + = d. The last equality holds sice i N C, the PL service policy dictates that the first packet of the busy cycle is the last to depart. Ad the remaiig packets would have departures times as for a busy cycle of legth k. Otherwise if a = a 2 the the LIFO policy i N D based o arrival times a i results i D = a + k + = d ad the packets umbered 2,..., k depart exactly as if they belog to a busy cycle of legth k. This completes the proof by iductio. Thus we have show that it is possible to use LIFO i N D based o the arrival times i N C istead of the actual arrival times i N D. We are ow ready to prove Theorem 2. Proof: of Theorem 2) Packets reach their destiatio with fiite average delay, which shows that the throughput is just the rate at which each source seds its data. This proves that the throughput T ) = Θ / ) a). Next we compute the average packet delay D). Lemma 5 also holds for the fial departure of each packet from the etwork. Therefore if DD i is the delay of a packet of route i i N D i.e. S-D pair i i the wireless etwork) ad DC i is the delay of the correspodig packet i N C the D i D Di C +. Hece takig expectatios it follows that E[D i D ] E[Di C ] +, i. Therefore delay averaged over all routes is give by D) = 2 E[D i D] 2 E[D i C] +. ) Sice each hop i the wireless etwork covers a distace of a) ) Θ, the umber of hops per packet for S-D pair i is Θ d i / ) a) where d i is the legth of S-D lie i. Now D i C is the delay for a packet of route i, which is equal to the sum of the delays alog all queues o its route. But from Lemma 4, the average delay at each server is bouded above by some costat idepedet of. Therefore from ), we obtai that D) 2 E[d i ] c 2 + = Θ / ) a) a) sice 2 E[d i]/ = Θ). The followig theorem shows that the trade-off provided by Scheme Π is optimal. The proof follows easily from the coverse for the fluid model Theorem 2 i []). Theorem 3: Let the average delay be bouded above by D). The the achievable throughput T ) for ay scheme scales as O D)/). III. CONCLUSION The optimal throughput-delay trade-off for static radom wireless etworks was determied i [] usig a fluid model where the packet size eeded to scale dow with the umber of odes i the etwork. I this paper, we imposed the costrait that the packet size remais costat ad established that the throughput-delay trade-off remais uchaged. This also provides a justificatio for the simplifyig fluid assumptio made i [] sice it does ot affect the essetial etwork dyamics. The ext atural issue to address would be the mobile radom etwork model, itroduced i [2], for which the trade-off was studied usig a fluid model i []. REFERENCES [] A. El Gamal, J. Mamme, B. Prabhakar, D. Shah, Throughput-Delay Trade-off i Wireless Networks, IEEE INFOCOM, Hog Kog, 2004. [2] M. Grossglauser ad D. Tse, Mobility Icreases the Capacity of Adhoc Wireless Networks, IEEE INFOCOM, Achorage, Alaska, pp.360-369, 200. [3] P. Gupta ad P. R. Kumar, The Capacity of Wireless Networks, IEEE Tras. o Iformatio Theory, 462), pp. 388-404, March 2000. [4] F. P. Kelly, Reversibility ad Stochastic Networks, Wiley, 979. [5] S. R. Kulkari ad P. Viswaath, A Determiistic Approach to Throughput Scalig i Wireless Networks, IEEE Tras. o Iformatio Theory, 506), pp. 04-049, 2004. [6] R. Motwai ad P. Raghava, Radomized algorithms, Cambridge Uiv. Press, 995. [7] M. J. Neely ad E. Modiao, Capacity ad Delay Tradeoffs for Ad-Hoc Mobile Networks, IEEE Trasactios o Iformatio Theory, vol. 5, o. 6, pp. 97-937, Jue 2005. [8] R. W. Wolff, Stochastic Modelig ad the Theory of Queues, Pretice- Hall, 988.