Queuing network models for delay analysis of multihop wireless ad hoc networks q

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1 Available olie at Ad Hoc Networks 7 (2009) Queuig etwork models for delay aalysis of multihop wireless ad hoc etworks q Nabhedra Bisik, Alhussei A. Abouzeid * Resselaer Polytechic Istitute, 0 8th Street, Troy, NY 280, Uited States Received 23 December 2006; received i revised form 5 November 2007; accepted 0 December 2007 Available olie 23 December 2007 Abstract I this paper we aalyze the average ed-to-ed delay ad maximum achievable per-ode throughput i radom access multihop wireless ad hoc etworks with statioary odes. We preset a aalytical model that takes ito accout the umber of odes, the radom packet arrival process, the extet of locality of traffic, ad the back off ad collisio avoidace mechaisms of radom access MAC. We model radom access multihop wireless etworks as ope G/G/ queuig etworks ad use the diffusio approximatio i order to evaluate closed form expressios for the average ed-to-ed delay. The mea service time of odes is evaluated ad used to obtai the maximum achievable per-ode throughput. The aalytical results obtaied here from the queuig etwork aalysis are discussed with regard to similarities ad differeces from the well established iformatio-theoretic results o throughput ad delay scalig laws i ad hoc etworks. We also ivestigate the extet of deviatio of delay ad throughput i a real world etwork from the aalytical results preseted i this paper. We coduct extesive simulatios i order to verify the aalytical results ad also compare them agaist NS-2 simulatios. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Ad hoc etworks; Queuig etworks; Throughput delay aalysis; Modelig ad performace evaluatio; Capacity; Delay; Diffusio approximatio. Itroductio A multihop wireless ad hoc etwork is a collectio of odes that commuicate with each other without ay established ifrastructure or cetralized q Prelimiary versios of this work have bee preseted at the Iteratioal Wireless Commuicatios ad Mobile Computig Coferece (IWCMC), July 3 6, 2006, Vacouver, Caada ad Iteratioal Workshop o Wireless Ad Hoc ad Sesor Networks (IWWAN), Jue 28 30, 2006, New York, USA (ivited). * Correspodig author. addresses: bisi@rpi.edu (N. Bisik), abouzeid@ecse. rpi.edu (A.A. Abouzeid). cotrol. The trasmissio power of a ode is limited, thus the packets may have to be forwarded by several itermediate odes before they reach their destiatios. Hece each ode may be a source, destiatio ad relay. The wireless medium is shared ad scarce, therefore ad hoc etworks require a efficiet MAC protocol []. Sice ad hoc etworks lack ifrastructure ad cetralized cotrol, MAC protocols for ad hoc etworks should be distributed, such as radom access MAC protocols, e.g. MACA [8] ad MACAW []. The delay ad throughput of wireless ad hoc etworks deped o the umber of odes, the trasmissio rage of the odes, the etwork traffic patter ad the /$ - see frot matter Ó 2008 Elsevier B.V. All rights reserved. doi:0.06/j.adhoc

2 80 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) behavior of the MAC protocol. I this paper we ivestigate how the ed-to-ed delay ad maximum achievable throughput i a radom access based MAC multihop wireless etwork with statioary odes deped o the umber of odes, trasmissio rage ad traffic patter. We propose a queuig etwork model for delay aalysis of radom access multihop wireless ad hoc etworks. The queuig etwork model proposed i this paper is uique i that it allows us to derive closed form expressios for the average ed-to-ed delay ad maximum achievable throughput. The packet delay is defied as the time take by a packet to reach its destiatio ode after it is geerated. The average ed-to-ed delay is the expectatio of the packet delay over all packets ad all possible etwork topologies. Our aalysis takes ito accout the queuig delays at source ad itermediate odes. The packets are assumed to have a fixed size ad radom arrival process. Moreover we also characterize how the average ed-to-ed delay ad maximum achievable throughput vary with the degree of locality of traffic. The primary purpose of this study is ot to accurately model the performace of specific stadard protocols like IEEE 802. MAC (eve though the results do provide a good match with NS-2 simulatios) but to gai isights ito the queuig delays ad maximum achievable throughput i radom access multihop wireless ad hoc etworks. Several studies have focused o fidig the maximum achievable throughput ad characterizig capacity-delay tradeoffs i wireless ad hoc etworks e.g. [6,7,,2]. I[7] it is show that for a wireless etwork with statioaryp ffiffiffiffiffiffiffiffiffiffiffiffiffi odes, the per-ode capacity scales as HðW = log Þ. I [], the authors use simulatios i order to study the depedece of per-ode capacity o IEEE 802. MAC iteractios ad traffic patter for various topologies like sigle cell, chai, uiform lattice ad radom etwork. A estimate of the expressios for oe-hop capacity ad upper boud of per-ode throughput is obtaied usig the simulatio results. I [6], the authors characterize the delay-throughput tradeoffs i wireless etworks with statioary The asymptotic otatios used i this paper have the followig meaigs: f ðþ ¼HðgðÞÞ ) 9c ; c 2 ; o > 0s:t: c gðþ 6 f ðþ 6 c 2 gðþ8 P 0. f ðþ¼oðgðþþ ) 9c; 0 > 0s:t: 0 6 f ðþ 6 cgðþ 8 P 0. f ðþ ¼oðgðÞÞ ) 9c; 0 > 0s:t: 0 6 f ðþ < cgðþ8 P 0. f ðþ ¼xðgðÞÞ ) 9c; 0 > 0s:t: 0 6 cgðþ < f ðþ8 P 0. ad mobile odes. It is show that for a etwork with statioary odes, the average delay ad throughput are related by DðÞ ¼HðT ðþþ, where DðÞ ad T ðþ are the average ed-to-ed delay ad throughput, respectively. However the delay is defied as the time take by a packet to reach the destiatio after it has left the source. Also, accordig to the etwork model, the packet size scales with the throughput. Uder these assumptios the delay is simply proportioal to the average umber of hops betwee a source destiatio pair. i.e. I their model, there is o delay due to queuig. If, more realistically, the packet size is assumed to be costat ad the delay is defied as time take by a packet to reach the destiatio after its arrival at the source, there would be queuig delays at the source ad itermediate odes. Several recet studies have proposed queuig models for performace evaluatio of the IEEE 802. MAC. A fiite queuig model is proposed ad used i [8] for evaluatig the packet blockig probability ad MAC queuig delays i a Basic Service Set. A queuig model for performace evaluatio of IEEE 802. MAC based WLAN i the presece of HTTP traffic is proposed i [3]. I [4] the service time of a ode, i IEEE 802. MAC based wireless ad hoc etwork, is modeled as a Markov modulated geeral arrival process. The resultig M/MMGI//K queuig model is used for delay aalysis over a sigle hop i the etwork. A aalytical model for evaluatig closed form expressio for the average queuig delay over a sigle hop i IEEE 802. based wireless etworks is preseted i [7]. I[5], the authors use queuig theoretic approach i order to calculate the mea packet delay, maximum throughput ad collisio probability for a elemetary four ode etwork with hidde odes ad exted the results to liear wireless etworks. It is worth otig that oe of the prior works [3 5,7,8] exteds to a geeral two dimesioal wireless etwork. We propose a detailed aalytic model for multihop wireless ad hoc etworks based o ope G/G/ queuig etworks. We first evaluate the mea ad secod momet of service time over a sigle hop by takig ito accout the back off ad collisio avoidace mechaisms of IEEE 802. MAC. We the use the diffusio approximatio for solvig ope queuig etworks i order to derive closed form expressio for the average ed-to-ed packet delay. Usig the average service time of the odes we obtai a expressio for the maximum achievable throughput. We preset detailed discussios o how the maximum

3 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) achievable throughput obtaied from our model compares with the per-ode capacity of Gupta Kumar s model. The mai results of this paper are:. The average ed-to-ed packet delay for our model is DðÞ ¼ q, where q is the utilizatio kð ^qþ factor of a ode, k is the packet arrival rate at the odes ad ^q is a variable whose value depeds o first ad secod momets of iterarrival ad service times. 2. The maximum achievable throughput i a multi- hop wireless ad hoc etwork is k max ¼ o srðþ 2 where s is the expected umber of hops betwee a source destiatio pair ad rðþ is the trasmissio radius of the odes. 3. Whe the parameters of our etwork model are comparable to the Gupta Kumar s model [7], k max ¼ o pffiffiffiffiffiffiffiffi W. log The aalytical results are verified agaist extesive simulatios ad umerical computatios. We also perform NS-2 simulatios ad discuss how the aalytical results compare with the delay results obtaied for some of the established wireless protocols. The rest of the paper is orgaized as follows. I Sectio 2 we briefly describe the well kow diffusio approximatio for solvig ope G/G/ queuig etworks. Sectio 3 presets a detailed descriptio of the etwork model. Delay ad throughput aalysis of multihop wireless etworks is preseted i Sectio 4. I Sectio 5 we preset ituitive iterpretatios of the aalytical results ad ivestigate how the results deviate from delay ad throughput i realistic etworks. The compariso of the aalytical ad simulatio results is preseted i Sectio 6. Fially we preset cocludig remarks i Sectio Diffusio approximatio method 2.. About the diffusio approximatio ad its accuracy, The diffusio approximatio was proposed by Kobayashi [9] i 970s as a techique to solve o-product form queuig etworks. The discrete valued queuig process is replaced with a cotiuous path Markov process with ifiitesimal icremets. The cotiuous Markov process is called diffusio process ad its probability distributio is described by a diffusio equatio. I a queuig etwork, the diffusio equatio accouts for the iteractio betwee queuig statios by usig variace covariace matrices for arrival ad service processes at each statio. Usig appropriate boudary coditios, the diffusio equatio may be solved for a queuig statio ad the overall probability distributio of the state of a etwork is represeted by the product of the states of the idividual queues. I other words, the diffusio process approximates a o-product form etwork as a product form etwork by takig ito accout the iteractios betwee queuig statios i the diffusio approximatio. This makes the diffusio approximatio a powerful tool for evaluatig mea closed form results for o-product etworks which caot be accurately aalyzed by usig other existig aalytical techiques. The fact that the iteractio betwee the queues of the etwork are take ito accout i the form of variace covariace matrices makes the diffusio approximatio a much more accurate approximatio tha the expoetial approximatio, where the statios are replaced with a server with expoetial arrival ad service process of the same meas. The compariso of diffusio approximatio with expoetial approximatio i terms of accuracy is studied i [6]. It is show that for a ope etwork, the error i diffusio approximatio lies betwee (.5%, 5%) ad (%, 6%) uder light ad heavy load coditios. O the other had, the error of simplistic expoetial approximatio lies betwee (30%, 65%) ad (30%, 00%) for the same light ad heavy coditios. The error of diffusio approximatio is much smaller tha the expoetial approximatio ad is practically egligible for high load coditios. This is because i heavy load coditios, the approximatio of discrete queuig process with cotiuous Markov process becomes more accurate. Although solvig the diffusio equatio is quite ivolved, the ed result is ot very complex. Thus we get icreased accuracy with little icrease i complexity. I the rest of this sectio we preset a road map for solvig a arbitrary ope queuig etwork usig diffusio approximatio. The advatage of usig the diffusio approximatio i this work is that it allows us to obtai closed form expressios for the average ed-to-ed delay Diffusio approximatio steps I this sectio we briefly describe how the diffusio approximatio is used to solve a ope G/G/ queuig etwork. (Please see [3] for details.)

4 82 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) Cosider a ope queuig etwork with service statios, umbered from to. The exteral arrival of a job is a reewal process with a average iterarrival time of =k e ad the coefficiet of variace of iter-arrival time equals c A. The mea ad coefficiet of variace of the service time at a statio i are deoted by =l i ad c Bi, respectively. The visit ratio of a statio i a queuig etwork is defied as the average umber of times a job is forwarded by (i.e. visits) the statio. The visit ratio of statio i, deoted by e i, is give by e i ¼ p 0i ðþþ Xj¼ p ji ðþe j ; j¼ ðþ where p 0i deotes the probability that a job eters the queuig etwork from statio i ad p ji deotes the probability that a job is routed to statio i after completig its service at statio j. There are two sources of job arrivals at a statio: the jobs that are geerated at the statio ad the jobs that are forwarded to the statio by other statios. The resultig arrival rate is termed the effective arrival rate at a statio. The effective arrival rate at the statio i, deoted by k i is give by k i ¼ k e e i : ð2þ The utilizatio factor of statio i, deoted by q i, is give by q i ¼ k i =l i : ð3þ The squared coefficiet of variace of the iterarrival time at a statio i, deoted by c 2 Ai, is approximated usig c 2 Ai ¼ þ X c 2 Bj p 2 ji e j e i ; ð4þ j¼0 where c 2 B0 ¼ c2 A. Accordig to the diffusio approximatio, the approximate expressio for the probability that the umber of jobs at statio i equals k, deoted by ^p i ðkþ, is ^p i ðkþ ¼ q i k ¼ 0; ð5þ q i ð ^q i Þ^q k i k > 0; where ^q i ¼ exp 2ð q iþ c 2 Ai q : ð6þ i þ c 2 Bi The mea umber of jobs at a statio i, deoted by K i,is K i ¼ q i =ð ^q i Þ: ð7þ 3. Queuig etwork model I this sectio we preset the etwork model ad develop a queuig etwork model for multihop wireless etworks. We also derive expressios for the parameters of the queuig etwork model. 3.. The etwork model The etwork cosists of þ odes, umbered to þ, that are distributed uiformly ad idepedetly over a torus of uit area. We assume a torus area i order to avoid complicatios i the aalysis caused by edge effects. Each ode is assumed to have a equal trasmissio rage, deoted by rðþ. Let r ij deote the distace betwee odes i ad j. Nodes i ad j are said to be eighbors if they ca directly commuicate with each other, i.e. if r ij 6 rðþ. Let NðiÞ deote the set of odes that are eighbors of ode i. All the eighbors of a ode lie o a disc of area AðÞ ¼prðÞ 2 cetered at the ode. The area AðÞ is termed the commuicatio area of a ode. The commuicatio area is chose such that the etwork is coected which esures that NðiÞ /8i. The trasmissio rate of each ode equals W bits/s. We cosider deploymet over uit area so that also equals the ode desity of the etwork. Thus the results of this paper idicate how delay ad maximum achievable throughput scale with ode desity. Characterizig delay ad capacity i terms of ode desity is more apprehesible sice it makes the results idepedet of the area over which the etwork is deployed. For a etwork of odes, each with commuicatio radius R, deployed over area S, our results ca be directly applied by cosiderig a equivalet etwork deploymet of uit area. This ca be doe by simply settig p the parameter rðþ of our model equal to R= ffiffiffi S. We use a special case of the Protocol Model of iterferece described i [7]. If ode i trasmits to ode j the the trasmissio will be successful oly if (a) r ij 6 rðþ ad (b) r kj > rðþ for every other ode k i; j that trasmits simultaeously with ode i. I other words, ode i ca successfully trasmit a packet to ode j oly if i is a eighbor of j ad o other eighbor of j is trasmittig cocurretly with i. (This is equivalet to settig D = 0 i the Protocol Model i [7].) The traffic model for the etwork may be described as follows. Each ode i the etwork could be a source, destiatio ad/or relay of

5 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) packets. Each ode geerates packets with rate k packets/s. The delay aalysis is possible for ay packet geeratio process as log as the mea ad SCV of packet iter-arrival time are kow. For the sake of simplicity, we assume i our model that the packet geeratio process at each ode is a i.i.d. Poisso process. The size of each packet is costat ad equals L bits. Whe a ode receives a packet from ay of its eighbors, it either forwards the packet to its eighbors with probability ð pðþþ or absorbs the packet with probability pðþ. The probability pðþ is referred to as absorptio probability. I other words, the absorptio probability is the probability that a ode is the destiatio of a packet give that the ode has received the packet from its eighbors. Whe a ode forwards a packet, each of its eighbors is equally likely to receive the packet. The advatage of such a model is that we ca cotrol the locality of the traffic by varyig the parameter pðþ. The traffic is highly localized if pðþ is large while a small value of pðþ implies ulocalized traffic. This would help us to characterize the effect of the locality of the traffic o the average delay ad maximum achievable throughput. For example, suppose that ode j i Fig. receives a packet from i. The probability that ode j is the destiatio of the packet is pðþ. The probability that ode j forwards the packet to oe of its eighbors is ð pðþþ. Suppose ode j forwards the packet, the the probability that the packet is forwarded to ode k is ¼. jnðjþj 4 We assume that each ode i the etwork has ifiite buffers which meas that o packets are dropped i the etwork. The packets are served by the odes o first come first serve basis. Multihop wireless ad hoc etworks ca be modeled as a queuig etwork as show i Fig. 2a. The statios of the queuig etwork correspod to the odes of the wireless etwork. The forwardig probabilities i the queuig etwork, deoted by p ij, correspod to the probability that a packet that is trasmitted by ode i eters the ode j s queue. Fig. 2b shows a represetatio of a ode i the ad hoc etwork as a statio i the queuig etwork. The ed-to-ed delay i a wireless etwork equals the sum of queuig ad trasmissio delays at source ad itermediate odes. We will use the queuig etwork model show i Fig. 2a adbi order to mathematically aalyze the ed-to-ed delay. a b 3.2. Parameters of the queuig etwork model I this sectio we preset expressios for the parameters of the queuig etwork model of multihop wireless etworks. The (relatively straightforward) detailed proofs of the results are preseted i [2]. Lemma. The expected probability that a packet is forwarded from ode i to ode j, deoted by p ij ðþ, is ( pðþ ð ð AðÞÞ Þ i j; p ij ðþ ¼ ð8þ 0 i ¼ j: j i k r() Fig.. A portio of a multihop wireless ad hoc etwork. p 2 () 2 3 p 2 () p 3 () Filter for absorbig packets destied to the ode p() Packets absorbed -p() p 3 () p 4 () Packets geerated i p 5 () p 4 () p 5 () 4 FCFS Queue 5 l N(i) N(i) Xi N(i) Fig. 2. Queuig etwork model for multihop wireless ad hoc etwork. (a) Represetatio of multihop wireless ad hoc etwork as a queuig etwork. (b) Represetatio of a ode of multihop wireless ad hoc etwork as a statio i the queuig etwork.

6 84 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) Lemma 2. The expected visit ratio of ode i, deoted by e i, is give by e i ¼ ð þ ÞpðÞ 8 i: ð9þ Lemmas ad 2 idicate that the odes visit ratio ad the forwardig probabilities averaged over all possible istaces of the topologies are similar to the visit ratios ad forwardig probabilities of a average topology where each ode has a umber of eighbors equal to the average case. Thus, as a result of these two lemmas, oe may derive the remaiig set of model parameters (effective packet arrival rate ad umber of packets traversed) by cosiderig the average case topology. Applyig these results i the diffusio model will provide expressios for the average ed-to-ed delay, defied as the expectatio of the packet delay over all packets ad all possible etworks. Lemma 3. The effective packet arrival rate at a ode i, deoted by k i,is k i ¼ k=pðþ: ð0þ Lemma 4. The expected umber of hops traversed by a packet betwee its source ad destiatio, deoted by s, equals. pðþ The average queuig delay depeds upo the service time distributio of the odes. The latter depeds o the MAC protocol used by the odes. 4. Queuig aalysis The radom access MAC model Before trasmittig each packet the odes cout dow a radom back off timer. The duratio of the timer is expoetially distributed with mea =. As i IEEE 802., the timer of a ode freezes each time a iterferig eighbor starts trasmittig a packet. Whe the back off timer of a ode expires, it starts trasmittig the packet ad the back off timers of all its iterferig eighbors are immediately froze. The timers of the iterferig eighbors are resumed as soo as the trasmissio of the packet is complete. The time required to trasmit a packet from a ode to its eighbor is L=W þ T 0, where T 0 is the time required for the exchage of RTS, CTS ad ACK packets. We assume that T 0 is egligible compared to L=W, so i our aalysis we assume that the time required to trasmit a packet is L=W. The model is mathematically tractable ad at the same time captures the behavior of IEEE 802. MAC protocol Delay aalysis With the help of the followig three lemmas we determie the mea ad secod momets of the service time of odes usig the radom access MAC model. We the preset the result for ed-to-ed delay i multihop wireless etworks. The detailed proofs of the results preseted i this sectio are preseted i [2]. I this sectio we first preset a model for a radom access MAC that accouts for the back off ad collisio avoidace mechaisms of IEEE 802. MAC. We the preset the delay aalysis of multihop wireless ad hoc etworks by itegratig the MAC model with the queuig etwork model developed i Sectio The MAC model 4... Iterferig eighbors Two odes are said to be iterferig eighbors if they lie withi a distace of 2rðÞ of each other (see Fig. 3). The trasmissio of a ode would be successful if oe of the iterferig eighbors of the ode trasmits cocurretly. Also two odes may successfully trasmit at the same time if they are ot iterferig eighbors of each other. The defiitio of iterferig eighbors is similar to the defiitio give i [7]. Fig. 3. This figure shows the eighbors ad iterferig eighbors of ode i which is i the ceter of the figure.

7 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) Lemma 5. Let H i deote the umber of iterferig eighbors of a ode i. The E½H i Š¼4AðÞ; ðþ E½H 2 i Š¼4AðÞð þ 4ð ÞAðÞÞ; ð2þ where AðÞ ¼p rðþ 2. Lemma 6. Let M i deote the umber of iterferig eighbors of a ode i that have at least oe packet to trasmit. The uder steady state, E½M i Š¼q4AðÞ: ð3þ E½M 2 i Š¼q2 4AðÞð þ 4ð ÞAðÞÞ þð qþq4aðþ: ð4þ where q is the utilizatio factor of the odes. Lemma 7. Let Z i deote the umber of times the timer of a ode i is froze before its expiratio. The E½Z i Š¼4qAðÞ: ð5þ Theorem. Let X i ad X 2 i deote the mea ad secod momet of service time required to serve a packet by a ode i. The X i ¼ E½X i Š¼ þ L W L : ð6þ 4AðÞk i W X 2 i ¼ E½X 2 i Šð þ 3m þ 2m2 Þ L2 W 2 þ 2ð2m þ Þ L W þ 2 : 2 ð7þ where m ¼ E½M i Š Eq. (3) ad m 2 ¼ E½M 2 i Š Eq. (4). Proof. The time take by ode i to serve a packet, deoted by X i, is the sum of three terms: (i) the duratio of the radom back off timer ðt i Þ, (ii) the duratio for which the timer remais froze ðz i L=W Þ, ad (iii) the trasmissio time ðl=w Þ. Thus L X i ¼ t i þ Z i W þ L W : ð8þ Takig expectatio of both sides we get, E½X i Š¼E½t i ŠþE½Z i Š L W þ L W ¼ þ 4qAðÞ L W þ L W : ð9þ Substitutig q ¼ k i X i ad by rearragig, we get (6). Also from (8) we have X 2 i ¼ t i þ Z i þ L 2 W Give M i ¼ m ad T i ¼ t i ; Z i has a Poisso distri- butio. So EZ 2 i jm i ¼ m; T i ¼ t i ¼ mti þðmt i Þ 2. Usig this ad (54), we get EX 2 i jt L 2 i ¼ t i ; M i ¼ m ¼ þ W 2 m2 2 t 2 i þ 2L t W þ 3L2 W m 2 i þ L2 W : 2 Takig expectatio with respect to t i we get EX 2 i jm i ¼ m ¼ ET i EX 2 i jt i ¼ t i ; M i ¼ m ¼ðþ3mþ2m 2 Þ L2 W 2 þ 2ð2m þ Þ L W þ 2 : 2 Takig expectatio of the RHS w.r.t m, we get (7). h Corollary. The stadard deviatio of service time of a ode i, deoted by r 2 X i, is give by r 2 X i ¼ L2 m þ m W 2 2 þ r 2 m þ 2ð2m þ Þ L W þ ; 2 ð20þ where r 2 m ¼ m2 m 2. The squared coefficiet of variace of the service time at a ode i, deoted by c 2 Bi is give by r2 X i =X 2 i. Usig (4), the squared coefficiet of variace of the iter-arrival time at a ode i, deoted by c 2 Ai, is give by c 2 Ai ¼ þ Xþ c 2 Bi pðþ j¼;j i ¼ þ c 2 Bi ð pðþþ: With the kowledge of c 2 Ai ; c2 Bi ad q, we ca fid the parameter ^q as give i (6). Theorem 2. For the radom access MAC model the average ed-to-ed delay i a multihop wireless etwork, deoted by DðÞ, is give by q DðÞ ¼ i k ð ^qþ : ð2þ 4.3. Maximum achievable throughput The maximum achievable throughput, deoted by k max, is defied as the maximum packet arrival rate at each ode for which the average ed-to-ed delay remais fiite. If the packet arrival rate exceeds k max,

8 86 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) the delay would ted to ifiity. The followig corollary, that follows from Theorem, yields a relatioship betwee the maximum achievable throughput ad the etwork parameters. Corollary 2. For a multihop wireless etwork the maximum achievable throughput k max is pðþ k max ¼ þ L þ 4AðÞ : ð22þ L W W Also from (22), k max ¼ oð=saðþþ. The result of Corollary 2 re-emphasizes the importace of carefully choosig the trasmissio rages of odes. k max icreases with decrease i rðþ. However if rðþ is too small the the etwork would become discoected. Accordig to [0], the etwork pis ffiffiffiffiffiffiffiffiffiffiffiffiffiffi asymptotically coected for rðþ ¼xð log =Þ. So for a coected etwork pðþ AðÞ ¼xðlog =Þ ad k max ¼ oð Þ. cþ4logðþðl=w Þ Aother iterestig observatio is the depedece of k max o the traffic patter. k max is directly proportioal to pðþ. From(53) the expected umber of hops traversed by a packet equals =pðþ. Thus aother way of iterpretig the result is that k max is iversely proportioal to the expected umber of hops betwee a source destiatio pair. We further ivestigate how our result o the maximum achievable throughput compares with the result by Gupta Kumar o throughput capacity. Accordig to the Gupta Kumar model, the odes are distributed uiformly ad idepedetly over a sphere of uit surface area ad each source chooses a radom destiatio. Therefore the expected distace betwee a source ad the correspodig destiatio equals the expected distace betwee two poits uiformly ad idepedetly distributed o a sphere. Thus the expected distace betwee a source destiatio pair i Gupta Kumar s model is a costat (i.e. does ot vary with ), psay ffiffiffiffiffiffiffiffiffiffiffiffiffiffi s GK. The trasmissio rage i their model is xð log =Þ. Thus the expected umber of hops betwee a source destiatio pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pair i Gupta Kumar model is oð = log Þ. I order to compare our results with Gupta Kumar s results we choose our model parameters such that we have comparable average umber of hops betwee a source destiatio pair ad comparable trasmissio p rage. I our model if we choose pðþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi log =, the the expected umber of hops betwee a psource ad destiatio ode is s ¼ =pðþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = log, which is comparable p to the Gupta Kumar model. Also rðþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi log = or AðÞ ¼p log = makes the trasmissio rage of our model comparable to that of the Gupta Kumar model. So for the model parameters that are comparable to the Gupta Kumar model, the maximum achievable throughput is k max ¼ 4p p ffiffiffiffiffiffiffiffi W log L : ð23þ c þ 4p log ðl=w Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi or k max ¼ oðw = log Þ. The above discussio implies that for the similar values of parameters of the etwork model we get a boud similar to the Gupta Kumar s boud o throughput capacity, but for our model the boud is ot achievable. The reaso for the boud ot beig achievable is that i our model we cosider a radom access MAC protocol rather tha a perfect determiistic schedulig. Thus the boud becomes uachievable because some chael capacity is wasted by the odes durig cotetio for the chael. 5. Discussios I this sectio we preset a brief ituitive iterpretatio of the mea service time ad evaluate the maximum achievable throughput for multihop wireless etworks. We the discuss how the edto-ed delay scales with umber of odes i the asymptotic case, where the etwork size teds to ifiity ad the throughput of odes teds to the maximum achievable throughput. We also discuss how our aalytical results vary from those obtaied for a more pragmatic etwork model. 5.. Iterpretatio of mea service time We preset a mathematically o-rigorous, but ituitive, derivatio of mea service time of a ode for the radom access MAC model. This derivatio further elucidates the result o service time. Cosider a hypothetical two ode etwork where oe of the odes trasmits packets to the other ode. Both odes use the radom access MAC model described i Sectio 4.. I this sceario there is o cotetio for the chael ad the average service time of the trasmitter would be þ L. We refer W to þ L as the ucoteded service time. W Now cosider a ode (say ode 0) with m iterferig eighbors, umbered through m. The ode ad its iterferig eighbors use the radom access MAC model for collisio avoidace. Packets of size L j bits

9 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) rffiffiffiffiffiffiffiffiffiffiffiffiffi arrive at a rate of a j packets/s at eighbor j. From the logðþ poit of view of ode 0, the chael is available whe pðþ ¼ : ð25þ o other iterferig eighbor is trasmittig. Uder steady state, the fractio of time that the chael is available to ode 0 is P We itroduce the followig ew otatio for the m k¼ a kðl k =W Þ. So the service time of ode 0 would be the ucoteded service aalysis:. We represet the mea duratio of back off timer time scaled by the fractio of time the chael is as product of the packet trasmissio time ad a available to ode 0. Hece the service time of ode =þl 0 equals P 0 =W. We refer to P m m a k¼ kðl k =W Þ k¼ a costat a > 0, i.e. kðl k =W Þ as the cotetio term. ¼ a L ; a > 0: ð26þ W I a multihop wireless etwork, m is aalogous to the umber of iterferig eighbors ad 2. We represet the packet geeratio rate (k) as a j ¼ k i ; L j ¼ L 8j. The expected value of the cotetio term (or fractio of time chael is ot avail- ad a costat b > 0, i.e. product of maximum achievable throughput L able to a ode) is 4AðÞk i ad therefore the W =þl=w service time of a ode equals. W =L k ¼ b p 4AðÞk iðl=w Þ 4p ffiffiffiffiffiffiffiffiffiffiffiffiffi ; b > 0: ð27þ log 5.2. Asymptotic scalig of delay Although the result obtaied i Theorem 2 provides a closed form expressio of ed-to-ed delay, it does ot provide direct ituitio ito how delay scales with various etwork parameters. This is because of the complex depedece of q ad ^q o various etwork parameters. I this sectio we simplify the expressio of delay for a asymptotic case with etwork size teds to ifiity ad the packet geeratio rate teds to the maximum achievable per-ode throughput of the etwork. We use various approximatios to simplify the complex expressios. As a result we obtai a closed form boud o the average ed-to-ed delay that is easy to iterpret. The boud is very tight for the asymptotic case cosidered i this sectio. We will first preset the assumptios ad otatios that we use i the aalysis. We the cosider each of the etwork parameters whose value we calculated i Sectio 4 ad simplify the expressio for the above metioed asymptotic case.. Network size teds to ifiity, i.e. etwork size ðþ is very large or!. 2. High load coditios, i.e. the probability that a ode has a packet to trasmit is high ð0 q < Þ. 3. The odes trasmit at miimum power that guaratees asymptotic coectivity, i.e. rffiffiffiffiffiffiffiffiffiffi log rðþ ¼ : ð24þ 4. The average distace betwee a source destiatio pair is a costat, i.e. Sice we cosider the heavy load coditios, b is assumed to be close to. We ow preset the aalysis for simplificatio of the delay results. Previously we have obtaied the results for the values of the etwork parameters usig the queuig etwork model. We further simply ad approximate the results i order to make them easy to iterpret Number of active iterferig eighbors Active iterferig eighbors are the iterferig eighbors of a ode that have a packet to sed while the ode is tryig to gai access to the chael. Let M i represet the umber of iterferig eighbors of ode i. Previously we have show that the expected value of M i is give by M i ¼ 4qAðÞ ¼4pqrðÞ 2 : Substitutig rðþ from (24), we get M i ¼ 4pq logðþ: The secod momet of M i is give by M 2 i ¼ q 2 4AðÞð þ 4ð ÞAðÞÞ þð qþq4aðþ: Substitutig rðþ from (24), we have M 2 i ¼ q 2 4p logðþ þ 4p log þð qþq4p log : Sice!, we substitute equal to ad get ð28þ to be approximately

10 88 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) M 2 i q 2 4p log ð þ 4p log Þþq4p log q 2 4p log ¼ q4p log ð þ qþþq 2 4p log ð4p log Þ: Agai, sice we cosider the case where! therefore we approximate 4p log 4p log which yields M 2 i q4p log ð þ qþþðq4p log Þ 2 : The stadard deviatio of M i is give by ð29þ r 2 M i ¼ M 2 i M i 2 ¼ q4p log ð þ qþ: ð30þ Service time The stadard deviatio of the service time of ode i is give by r 2 X i ¼ L2 M W 2 i þ M 2 i þ r 2 M i þ 2ð2M i þ Þ L W þ 2 : Substitutig M i ; M 2 i, ad r2 M i from (28) (30), we get r 2 X i ¼ L2 ð4pq log þ 8pq log ð þ qþ 2 W þð4pq log Þ 2 Þþ2ð8pq log þ Þ L W þ 2 : ð3þ The squared coefficiet of variace (SCV) of the service time is give by c 2 Bi ¼ r2 X i X ¼ r 2 2 k2 X i i i q 2 ¼ k 2 L 2 4p log i þ 8p log W 2 q q þ! þð4p log Þ 2 8p log þ 2 þ! L q q 2 W þ 2 q 2 ¼ k 2 L 4p log 3L i W q W þ 4 þ 8p log L2 W 2 þ 4p log L 2 þ W q 2 þ L! : ð32þ W We kow that k i ¼ k=pðþ. Substitutig pðþ from (25), we get rffiffiffiffiffiffiffiffiffiffi k i ¼ k : ð33þ log Substitutig k i from (33) ito (32), weget c 2 Bi ¼ k2 L 4p log 3L log W q W þ 4 þ 8p log L2 W 2 þ 4p log L 2 þ W q 2 þ L! W ¼ k 2 6p L L q W W þ 4p L 2 q W 2 þ8p L2 W þ log 4p L 2 þ 2 W q 2 log þ L! : W ð34þ Substitutig ¼ a L from (26), we get W c 2 Bi ¼ k2 6p L 2 q W ð þ aþ 4p L 2 L2 þ log 2 6p2 2 q W W 2 að þ aþ L 2 þ q 2 log W 2 ¼ 4pk 2 L2 4ð þ aþ þ 2 þ 4p log W 2 q q aða þ Þ þ : ð35þ 4pq 2 log Sice we are cosiderig the case where q ;!, ad a is a costat, c 2 Bi may be approximated as c 2 Bi 6p2 k 2 L 2 log : 2 W ð36þ Arrival process for each ode (queue i the queuig etwork) The effective packet arrival rate ito a queue ðk i Þ is give by k=pðþ. Substitutig pðþ from (25) ad k from (27), we get k i ¼ b W =L 4p log : ð37þ Also accordig to the diffusio approximatio, the SCV of the packet iter-arrival time at ode i is give by c 2 Ai ¼ þðc2 Bi Þð pðþþ: ð38þ Utilizatio ratio ðqþ The utilizatio ratio of a ode is equal to k i X i, where k i is the effective packet arrival rate at a ode ad X i is the average service time of the ode. The average service time is give by X i ¼ þ L W L : 4AðÞk i W

11 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) Substitutig rðþ;, ad k i from (24), (26), ad (37), we get X i ¼ L ð þ aþ W b : ð39þ Usig k i from (37), q is give by q ¼ k i X i ¼ bðþaþ 4p log b : ð40þ The diffusio approximatio costat ð^qþ As previously stated, ^q is give by 2ð qþ ^q ¼ exp : c 2 A i q þ c 2 Bi Substitutig c 2 Ai from (38), we have 2ð qþ ^q ¼ exp : q þ c 2 Biqð pðþþ þ c 2 Bi ð pðþþ As!; pðþ!0. Thus we approximate pðþ ad therefore ^q may be expressed as 2ð qþ ^q ¼ exp : ð4þ c 2 Bið þ qþ ð qþ Ed-to-ed delay As already metioed i (2), the average ed-toed packet delay is give by q D ¼ kð ^qþ : We use the followig iequality i order to simplify the above relatio: e P 8x > 0: ð42þ x x The relatio i the above iequality is close to equality whe x is close to 0, i.e. as e x x x! 0. Usig (42), D is may be writte as D P q k log ^q : The above relatio is close to equality whe q is close to. Substitutig ^q from (4) i the above relatio we have q D P 2kð qþ ðc2 Bið þ qþ ð qþþ: Because of high load assumptio we approximate q i the umerator of the above expressio. Thus D may be expressed as D P qð þ qþ ð qþ 2k c2 Bi : Substitutig q from (37), we get D P bðþaþ 4p log b bðþaþ 4p log b þ bðþaþ 4p log b 2k c2 Bi : Sice!, we approximate b bðþaþ b 4p log ad b þ bðþaþ b. Thus we get 4p log D P þ a b 4p log b 2k c2 Bi : ð43þ Substitutig k from (27) ad c 2 Bi from (36), we get D P þ a b bw =L p 4p log b 2 4p ffiffiffiffiffiffiffiffiffiffiffiffiffi 6p 2 L2 log W 2 log : ð44þ Rearragig we the above relatio we get D P þ a rffiffiffiffiffiffiffiffiffiffi L b : ð45þ 2 W b log Or, rffiffiffiffiffiffiffiffiffiffi b D ¼ X : ð46þ b log The above result gives us more valuable isight ito how delay scales with various etwork parameters tha the result preseted i (2). For fixed etwork k size, the delay scales roughly as k max k with k. For a fixed packet geeratio rate, the ed-to-ed delay q ffiffiffiffiffiffi scales as. log 5.3. Compariso with delay ad throughput i real etworks The aalytical model i this paper is kept reasoably simple so that it is possible to obtai closed form expressios for delay ad throughput. I particular our MAC model does ot take ito accout packet collisios ad our routig model is radom walk of packets over the etwork. Thus our model deviates from the real world scearios where the packets collide due to radom access MAC with fiite collisio widows ad the packets are routed alog fixed paths dictated by routig protocols. I this subsectio we discuss how much the delay ad maximum achievable throughput i real world etworks deviate from our aalytical results Effect of packet collisios Cosider a more practical MAC model where a ode trasmits as soo as its trasmit timer expires ad the iterferig eighbors freeze their timers

12 90 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) oly whe they sese the trasmissio. For such a MAC, the trasmissio of ode i may collide with trasmissio of a iterferig eighbor if differece betwee the time istaces whe the trasmit timers of odes i ad that of iterferig eighbor expire is less tha the propagatio delay betwee the odes. Let d deote the propagatio delay betwee ode i ad its iterferig eighbor that has a packet to sed, the the probability that the trasmissio of i does ot collide with that of the iterferig eighbor equals e 2d. Sice the iterferig eighbors are located withi two hops of ode i; d 6 2r=c ¼ d, where c is velocity of electromagetic waves. Thus the probability that the trasmissio of ode i does ot collide with a iterferig ode s trasmissio is greater tha e 2d. So if ode i has I iterferig eighbors, the the probability that a trasmissio is a success is bouded by P½SuccessŠ P e 2dI : ð47þ Let P s deote the expected probability of success, averaged over all possible topologies, the P s P ð ð e 2d Þ4AðÞÞ ¼ P ðlþ s : ð48þ The expected umber of times a ode trasmits a packet before it is received successfully by its eighbor equals =P s. It is easy to see that the RHS of Eq. (9) is scaled by a factor of =P s ad mea service time may be evaluated by rearragig the resultig equatio. So for the more practical MAC model, i which packet collisios take place, the mea service time is bouded by þ L W 4AðÞk i L=W 6 X i 6 P ðlþ s þ L W 4AðÞk i L=W : ð49þ The maximum achievable throughput, evaluated usig k i X i <, is bouded by k ðlþ max ¼ 6 P ðlþ s pðþ þ L W þ 4AðÞ L W pðþ þ L W þ 4AðÞ L W 6 k max ¼ k ðuþ max : ð50þ The depedece of k ðlþ max, the lower boud of k max,o the rate of trasmit timer,, is particularly iterestig. As icreases, both P ðlþ s ad = term i the deomiator decrease. Thus there is a tradeoff i choosig the rate of the trasmit timer-a high leads to lower waitig time before trasmissio but leads to higher probability of packet collisio. Let H be the optimal value of that maximizes the lower boud of k max. Equatig dk ðlþ max =d to 0 yields that H satisfies the followig relatio ðbðþ H2 þ H Þe 2H d ð 4AðÞð e 2H d ÞÞ ¼ 8AðÞd : ð5þ where bðþ ¼L=W þ 4AðÞL=W. Closed form expressio for H caot be evaluated from the above relatio. However, by approximatig e 2Hd (high probability of success) ad solvig the resultig quadratic equatio we get sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! H ð þ 4AðÞÞL þ : 2L=W þ 4AðÞ 2AðÞW d ð52þ As expected, H decreases with icrease i the expected umber of iterferig eighbors, packet trasmissio time ad propagatio delay. Fig. 4 shows how the performace of our collisio free model compares with the boud o the delay ad maximum achievable throughput i a etwork with packet collisios. The etwork parameters for the plots are the followig: ¼ 5 0 4, qffiffiffiffiffiffi log rðþ ¼3, L ¼ Kb, W ¼ Mbps, ad qffiffiffiffiffiffi log pðxþ ¼. Fig. 4a shows that i the presece of collisio the service time of a large etwork may be up to 30% greater tha that of our aalytical model. However the error is small for smaller etwork sizes. Also the error i the maximum achievable throughput is almost egligible Effect of determiistic routig I the routig model used i this paper, a ode forwards a packet to ay of its eighbor with equal probability which spreads the traffic evely throughout the etwork. O the other had, a determiistic routig protocol routes each packet belogig to a particular flow (typically idetified by source destiatio pair) alog a determiistic path, determied usig some goodess metric. This ofte leads to a situatio where large umber of flows pass through a few odes that have good paths to may flow destiatios. This leads to creatio of routig bottleecks leadig to large queuig delays itermediate odes ad higher ed-to-ed delays. The MAC layer delays with determiistic routig are always higher tha those with radom routig. For high load situatios, this traslates to ed-to-ed delays. For low load situatios, MAC delays are low for

13 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) a Average Service Time (sec) Without Collisio Upper Boud With Collisio b Maximum Achievable Throughput (packets/sec) Without Collisio Lower Boud With Collisio Number of Nodes Number of Nodes Fig. 4. Compariso of performace of the idealistic sceario with a etwork with packet collisios. (a) Average time required to successfully trasmit a packet to the ext hop, with ad without collisio. (b) Maximum achievable throughput as a fuctio of etwork size, with ad without collisio. both types of routig, ad thus the ed-to-ed delays for determiistic routig could be less tha those for radom routig (predicted by (2)) because packets travel fewer hops. 6. Simulatios We perform the followig simulatios:. Simulatios of the model: These simulatios verify the validity of the assumptios made i the aalysis ad the accuracy of diffusio aalysis. 2. Simulatios usig shortest path routig istead of probabilistic routig: We compare the aalytical results with a practical sceario where the packets are routed alog the shortest path rather tha udergoig probabilistic routig. 3. NS simulatios: These simulatios provide compariso of the results of our aalytical model agaist results obtaied from NS simulatios that employ stadard MAC (IEEE 802.) ad routig (DSDV) protocols. The rest of this sectio presets the results for each of the above simulatio sets. 6.. Simulatios for validatig the aalytical results We use a simulator writte i C i order to simulate the model ad compare the aalytical results with the simulatio results. The simulatio settig is the followig. The etwork topology for the simulatios cosists of odes scattered radomly over a torus of uit surface area. Each ode ca commuicate with the odes withi a distace rðþ ¼. qffiffiffiffiffiffi log The radom access MAC protocol used by the odes is the same as described i Sectio 4.2. Each ode produces packets of size L ¼ Kbits at the rate of k packets/s. The trasmissio rate of each ode is W ¼ 0 6 bits/s. The probabilistic routig described i Sectio 3 is used for the simulatios. The simulatio time is 500 s. I order to esure that the etwork is i a steady state, the first 00 s of the simulatios are discarded. The average delay for a particular topology is obtaied by averagig the ed-to-ed delay of all packets produced durig the simulatio. I order to average out the effect of topology, each simulatio is repeated over several topologies. The average ed-to-ed delay ad 95% cofidece itervals are obtaied by obtaied from 35 simulatio rus. Fig. 5 shows how the average ed-to-ed delay, as obtaied from the simulatios, varies with the umber of odes for k ¼ 0:5; k ¼ 0:7 ad k ¼ :0 qffiffiffiffiffiffi log with pðþ ¼. Fig. 6 shows how the average ed-to-ed delay varies with the arrival rate (k) for qffiffiffiffiffiffi log ¼ 500; 600 ad 800 with pðþ ¼. Fig. 7 shows how the average ed-to-ed delay varies with the umber of odes for various values of absorptio probability with k ¼ packets/s. The theoretical values of the average ed-to-ed delay as obtaied from the aalytical results are plotted alogside the simulatio results i Figs Itis observed that the simulatio results agree closely with the theoretical values Compariso of results for the shortest path routig with the aalytical results I our model the packets are subjected to probabilistic routig, which is similar to a radom walk.

14 92 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) Theory, λ =.0 Simulatio, λ =.0 Theory, λ = 0.7 Simulatio, λ = 0.7 Theory, λ = 0.5 Simulatio, λ = Theory, p() =.2(log()/) 0.5 Simulatio, p() =.2(log()/) 0.5 Theory, p() = (log()/) 0.5 Simulatio, p() = (log()/) 0.5 Theory, p() = 0.8(log()/) 0.5 Simulatio, p() = 0.8(log()/) Number of odes Fig. 5. Average ed-to-ed delay vs. umber of odes for varyig arrival rates Number of odes Fig. 7. Average ed-to-ed delay vs. umber of odes for varyig absorptio probabilities Theory, = 800 Simulatio, = 800 Theory, = 600 Simulatio, = 600 Theory, = 500 Simulatio, = Arrival Rate (λ) results for the shortest path routig alog with the aalytical results obtaied from our model. The absorptio probability of our model is scaled appropriately so that the average umber of hops betwee a source destiatio pair for the shortest path routig matches with the average umber of hops traversed by a packet i the probabilistic routig model. It is observed that the for low traffic arrival rate or small etwork size, the simulatio results agree closely with the aalytical results. However for higher packet arrival rates ad large etwork size the delay obtaied from simulatio is larger tha the aalytical result. This deviatio may be explaied i the followig maer. Shortest path Fig. 6. Average ed-to-ed delay vs. arrival rate for varyig etwork size. It would be pertiet to compare the aalytical results for our model with the simulatios where the packets are routed to the destiatios alog correspodig shortest paths. The simulatio settig differs from the settig described i the previous subsectio i the followig. Whe a ew packet is geerated at a ode, a destiatio ode for the packet is chose at radom. The packet is routed to the destiatio ode alog the shortest path. I order to route the packets alog shortest paths, each ode maitais a routig table. The routig tables are costructed usig Bellma Ford algorithm. Fig. 8 shows the simulatio Theory, λ =.2 Simulatio, λ =.2 Theory, λ =.0 Simulatio, λ =.0 Theory, λ = 0.8 Simulatio, λ = 0.8 Theory, λ = 0.6 Simulatio, λ = Number of odes Fig. 8. Compariso of aalytical results with simulatio results for the shortest path routig.

15 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) routig ievitably leads to a situatio where some liks i the etwork carry more traffic tha the average traffic i the etwork. These are the liks that lie o shortest path of may source destiatio pairs. These liks are referred to as routig hotspots. Whe the traffic load is high, such routig hotspots become bottleeck ad the packet routed through these liks experiece much larger delays. I cotrast, probabilistic routig esures that the traffic is spread uiformly over the etwork ad thus the average ed-to-ed for the aalytical model icreases less rapidly. As the umber of odes i the etwork icreases, it is accompaied by icrease i traffic i the etwork leadig to high delays o the bottleeck liks. Aother factor for the deviatio is the followig. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sice the commuicatio radius is set equal to log pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi =, the diameter of etwork is approximately = log. For small, the umber of hops traversed by a packet are small therefore routig is more similar to radom walk. However as icreases, the diameter of the etwork icreases ad packets traverse more hops betwee a source destiatio pair. For this case radom walk poorly approximates shortest path routig Compariso agaist results from NS simulatios As metioed earlier, the aim of the delay aalysis preseted i this paper is to capture the effect of radom access MAC ad queuig delays o the average ed-to-ed delay ad maximum achievable throughput of multihop wireless etworks. Our model icludes a idealistic radom access MAC ad probabilistic routig model. The MAC model assumes that the trasmissio timers of all iterferig eighbors of a ode freeze as soo as the ode starts trasmittig. This precludes the possibility of ay packet collisio at the iteded receiver (except for the results i Sectio 5.3.). However IEEE 802., which is the de facto MAC protocol of wireless etworks ad simulated i NS, is ot free from collisios. The RTS packets trasmitted by a ode may collide with a trasmissio at the iteded receiver which would the prevet the ode from grabbig the chael. (This is more likely i etworks with large cotetio widow, e.g. log propagatio delays.) Also whe a ode, i a ad hoc IEEE 802. etwork, starts trasmittig, the trasmissio timers of oly a subset of iterferig eighbors are froze. This subset icludes oly the eighbors of the trasmitter ad the iteded receiver. I this sectio we compare the aalytical results for our model with the average delay obtaied from NS-2 simulatios, that use IEEE 802. as MAC ad DSDV for routig. The purpose of this compariso is to uderstad how the ed-to-ed delay i a etwork based o specific established protocols would differ from the results obtaied from our geeric model, give that our model icludes some simplifyig assumptios that does ot capture the actual protocols (ad their iteractios). The NS simulatio set-up is as follows. The etwork cosists of odes that are uiformly ad idepedetly distributed over a 500 m 500 m area. This deploymet area is chose to represet a realistic etwork deploymet. A expoetial traffic source is attached to each ode, which produces packets of legth 000 bytes at the rate of k packets per secod. Each ode chooses a radom destiatio ad the traffic is routed to the destiatio usig routes maitaied by DSDV. 2 UDP is used as the trasport layer protocol i order to avoid delays due to cogestio avoidace mechaisms of TCP. The receive threshold 3 of the odes is setp such ffiffiffiffiffiffiffiffiffiffiffiffiffiffi that each ode withi a distace of 500 log = meters from a trasmitter is able to liste to the trasmissio, i absece pof ay iterferece. This correspods to rðþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi log = for our model where odes are deployed over uit area. The IEEE 802. MAC ad free space propagatio models are used for the simulatios. I order to compare the simulatio results with the aalytical results we set the values of the parameters of the aalytical model such that they are comparable to that of the simulatio. This is doe i the followig maer. We obtai the values of the average duratio of the IEEE 802. backoff timer ð 0 Þ ad the trasmissio duratio ðt 0 0 Þ4 for each simulatio settig. We use ¼ 0 ad L ¼ T 0 W 0 i the aalytical results. This esures that the average backoff duratio of MAC protocol ad the trasmissio time of the aalytical model is same as that of the simulatios. We used the default value for the parameter CSThresh i NS which is :5 0, while the RXThresh was set such that the trasmis- 2 Sice the odes are statioary, DSDV is the ideal routig protocol. 3 If the sigal stregth at a receiver is below receive threshold, the receiver caot detect the sigal. Please see [5] for details. 4 T 0 0 icludes the time take for the exchage of RTS, CTS, data ad ACK packets i.e. T 0 0 ¼ T RTS þ T SIFS þ T CTS þ T SIFS þ T DATA þ T SIFS þ T ACK.

16 94 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) p sio radius is 500 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi log =. This meas that for the simulatios the RXThresh varied from 5:6 0 9 to : We perform 35 simulatios for each sceario i order to obtai the mea ad 95% cofidece iterval for each poit. The simulatio time for each simulatio ru equals 2500 s. Fig. 9 shows the plots of average ed-to-ed delay, obtaied from NS-2 simulatios ad aalytical model, as a fuctio of umber of odes. We observe that for lightly loaded coditios (small k ad/or ) the simulatio results are less tha the aalytical results. This is because i IEEE 802. oly the trasmissio timers of the eighbors of the trasmitter ad receiver are froze durig a trasmissio while i our model we assume that the timers of all iterferig eighbors are froze. Thus the umber of times the trasmissio timer of a ode is froze i IEEE 802. is smaller tha that i our model. So the average time i which the trasmissio timer of a ode expires is less i IEEE 802. leadig to lesser delays. However the simulatio results closely follow the tred idicated by the aalytical results. As the traffic size or load i the etwork icreases the average ed-to-ed delay of the simulatio becomes larger tha the aalytical results. This is due to the followig three reasos: (i) The umber of collisios of RTS/CTS packets icreases. As umber of odes i the etwork icreases, the umber of odes cotedig for chael icreases hece icreasig the chaces of packet collisio. As the packet geeratio rate icreases, Model, λ = 0. NS simulatio, λ = 0. Model, λ = 0.2 NS simulatio, λ = Number of odes Fig. 9. Average ed-to-ed delay vs. umber of odes for NS simulatios Model, λ = 0.5 NS simulatio, λ = Number of odes Fig. 0. Average ed-to-ed delay vs. umber of odes for NS simulatios. Due to icreased cogestio, packet collisios, ad cotrol overheads, for higher packet geeratio rate the NS simulatios further deviate from the theoretical results. the odes attempt to trasmit more ofte which icreases the chaces of packet collisio. (ii) Due to formatio of routig hotspots as a result of shortest path routig. (iii) Due to collisio of data packets with routig cotrol packets. The routig update packet which are ot preceded by virtual carrier sese ad RTS CTS exchage [4] which makes routig packets vulerable to collisio. As the umber of odes i the etwork the umber of routig update packets that trasmitted icreases, leadig to icreased chaces of packet collisios ad higher delays. Fig. 0 shows how delay obtaied from NS simulatios varies with the umber of odes for k ¼ 0:5 packets/s. It is observed that for higher etwork sizes ad traffic arrival rate the simulatio results differ substatially from the theoretical results. The icreased deviatio with icrease i traffic is due to the three reaso metioed previously. These results idicate that the delay ad capacity i real etworks should be expected to be worse tha the results obtaied i this paper. This deviatio is the impact of issues associated with the existig MAC ad routig protocols such as poor load balacig, high collisio rates, ad protocol overheads. So although our aalytical results closely approximate delay i real etworks for the light load sceario, there is cosiderable deviatio for the high load sceario. Developig accurate results for the high load sceario will require more detailed modelig of the impact of specific routig ad MAC algorithms i ad hoc etworks.

17 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) Coclusio ad future work Characterizatio of capacity ad delay i ad hoc etworks has bee focus of cosiderable research. However capacity ad delay of etworks based o radom access MAC, like IEEE 802., have ot bee substatially studied. I this paper we preseted delay aalysis of radom access MAC multihop wireless ad hoc etworks. We derived closed form expressios for the average ed-to-ed delay ad maximum achievable throughput. We showed that, for comparable etwork parameters, the upper boud o maximum achievable throughput is of the same order as the Gupta Kumar s boud. However for the radom access MAC the boud is ot achievable. The aalytical results are verified usig simulatios. The NS-2 simulatios idicate that uder heavy load the performace of the stadard wireless protocols is worse tha the performace predicted by our model. The results ad framework preseted i this paper leads to several veues for future research. Our curret directios iclude the delay aalysis ad characterizatio of the maximum achievable throughput for hierarchical etworks, may to oe commuicatio scearios, wireless etworks with sleepig odes ad wireless etworks with other medium access cotrol algorithms. Ackowledgemet This work was fuded i part by the Natioal Sciece Foudatio uder Grats CNS ad CNS Appedix A. Omitted proofs Proof of Lemma. Let P½i! jš deote the probability that a packet forwarded by ode i eters the queue at ode j. We defie b j;k ij ¼ P½i! jjj 2 NðiÞ; jnðiþj ¼ kš, b j ij ¼ P½i! jjj 2 NðiÞŠ ad aj;k i ¼ P½j NðiÞj ¼ kjj 2 NðiÞŠ. Thus b j;k ij ¼ k pðþ ð P½jabsorbs the packetšþ ¼ : k Sice the odes are uiformly ad idepedetly distributed over a uit area, the probability that a ode is i eighborhood of the ode i equals AðÞ. Hece P½j 2 NðiÞŠ ¼ AðÞ ad a j;k i ¼ k ð AðÞÞ k AðÞ k : Therefore, h b j ij ¼ E bj ij i¼ X b j;k ij aj;k i k¼ pðþ ¼ AðÞ ð ð AðÞÞ Þ: Also accordig to the model ode i caot forward a packet to ode j uless j 2 N ðiþ. Hece E½P½i! jšjj R NðiÞŠ ¼ 0. So the expected forwardig probability is give by p ij ðþ ¼b j ij pðþ P½j 2 NðiÞŠ ¼ ð ð AðÞÞ Þ: Proof of Lemma 2. The visit ratio of a ode i the queuig etwork is give by (). Takig expectatio of both sides of the equatio we have, e i ¼ j¼þ þ þ X p ji ðþe j : j¼ Each ode of the wireless etwork is similar, thus from symmetry e i ¼ e j 8i; j. Also p ij ¼ pðþ ð ð AðÞÞ Þ. Sice i our model AðÞ is chose such that the etwork is coected with high probability, therefore ð ð AðÞÞ Þ ad hece p ij ðþ pðþ. From symmetry e i ¼ j¼þ þ þ X j¼;j i pðþ e i : By rearragig the above equatio we get (9). Proof of Lemma 3. The packet arrival process at each ode is a i.i.d. Poisso process with rate k. So the total exteral arrival rate, deoted by k e, equals ð þ Þk. Accordig to (2), k i ¼ k e e i. Substitutig e i from (9) ad k e we get (0). Proof of Lemma 4. Let s deote the umber of hops betwee a source ad destiatio, the P ½s ¼ kš ¼ð pðþþ k pðþk P. Thus, s ¼ E½sŠ ¼ X k¼ k ð pðþþ k pðþ ¼ pðþ : ð53þ Proof of Lemma 5. Sice the odes are uiformly distributed over a uit area, the probability that a ode is a iterferig eighbor of ode i equals pð2rðþþ 2. Thus the probability that H i ¼ h is give by P½H i ¼ hš ¼ ð4prðþ 2 Þ h ð 4prðÞ 2 Þ ð hþ : h

18 96 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) Thus H i has a biomial distributio. () ad (2) are the first ad secod momet of the biomial distributio. Proof of Lemma 6. Let the umber of iterferig eighbors of ode i be H i. Let Y j ; 6 j 6 H i,be a idicator radom variable associated with ode j, idicatig whether uder steady state ode j has a packet to trasmit or ot. (Y j ¼ if ode j has a packet to trasmit, Y j ¼ 0 if ode j has o packet to trasmit). Usig (5) PðY j ¼ Þ ¼q j, where q j is the utilizatio factor of ode j. By symmetry q j ¼ q8j. M i is equal to P H i j¼ Y j. The expected value of M i equals " # E½M i Š¼E H i ½E½M i jh i ¼ hšš ¼ E H i X h j¼ E½Y j Š ¼ qe½h i Š: Substitutig (), we get (3). Similarly the expected value of M 2 i, give " H i ¼ h, is give by!! E½M 2 i jh X # h X h i ¼ hš ¼E Y j Y k : j¼ k¼ Sice Y j is idepedet of Y k, we get E½M 2 i jh i ¼ hš ¼ Xh j¼ X h k¼;k j E½Y j ŠE½Y k Šþ Xh j¼ E½Y 2 j Š ) E½M 2 i Š¼q2 E½H 2 i Šþð qþqe½h iš: Substitutig () ad (2), we get (4). Proof of Lemma 7. Let T i deote the duratio of the back off timer of ode i. Durig a trasmissio epoch M i may ot remai costat. I order to simplify the aalysis we assume that M i remais costat throughout a trasmissio epoch of ode i. The timer of ode i is froze each time a timer of ay of the iterferig eighbors of i expires. The timer of each ode has a expoetial distributio. Thus the probability that Z i ¼ z, give that M i ¼ m ad T i ¼ t i,is P½Z i ¼ zjt i ¼ t i ; M i ¼ mš ¼ e mti ðmt i Þ z =z! ) E½Z i jt i ¼ t i ; M i ¼ mš ¼mt i ) E½Z i jm i ¼ mš ¼ me½t i Š¼m ) E½Z i Š¼E½M i Š: Substitutig E½M i Š from (3), we get (5). ð54þ Proof of Theorem 2. Let D i deote the average delay at a ode i. Accordig to Little s Law, D i ¼ K i =k i, where K i is the average umber of packets i the queue of ode i. Substitutig K i from (7) we get D i ¼ K i =k i ¼ q=ðk i ð ^qþþ: By symmetry the average delay at each ode is same. Thus the average ed-to-ed delay equals the product of the average umber of hops traversed by a packet ad the average delay at each ode. Hece DðÞ ¼s D i which leads to (2). Proof of Corollary 2. From (6) the utilizatio factor of a ode, q i, is give by q i ¼ k i þ L W : ð55þ L 4AðÞk i W For the average delay to be fiite q i must be strictly less tha. Thus the followig iequality must be satisfied to esure fiite delay. k i þ L W < : L 4AðÞk i W Substitutig c ¼ þ L ; k W i ¼ k ad rearragig, we pðþ get pðþ k < : ð56þ c þ 4AðÞ L W Thus the maximum achievable throughput k max is pðþ. Also c > 0, thus k cþ4aðþw L max < pðþw. So for a fixed AðÞL packet size L ad trasmissio rate W ; k max ¼ o pðþ AðÞ Refereces. Substitutig pðþ ¼, k s max ¼ o saðþ. [] V. Bharghava, A. Demers, S. Sheker, L. Zhag, MACAW: a media access protocol for wireless LANs, i: Proceedigs of the Coferece o Commuicatios Architectures, Protocols ad Applicatios, ACM Press, 994, pp [2] N. Bisik, A.A. Abouzeid, Queuig etwork models for delay aalysis of multihop wireless ad hoc etworks, Techical Report, ECSE Departmet, Available at: [3] G. Bolch, S. Greier, H. de Meer, K.S. Trivedi, Queuig Networks ad Markov Chais, Joh Wiley ad Sos, 998, pp (Chapter 0). [4] J.-M. Choi, J. So, Y.-B. Ko, Numerical aalysis of IEEE 802. broadcast scheme i multihop wireless ad hoc etworks, i: ICOIN 05: Proceedigs of the Iteratioal Coferece o Iformatio Networkig 2005, pp. 0. [5] K. Fall, K. Varadha, The NS-2 user maual. [6] A.E. Gamal, J. Mamme, B. Prabhakar, D. Shah, Throughput-delay trade-off i wireless etworks, i: Proceedigs of IEEE INFOCOM (INFOCOM 04). IEEE, March [7] P. Gupta, P.R. Kumar, Capacity of wireless etworks, IEEE Tras. o Iformatio Theory, March 2000, pp

19 N. Bisik, A.A. Abouzeid / Ad Hoc Networks 7 (2009) [8] P. Kar, MACA: a ew chael access methods for packet radio, i: Proceedigs of the Nieth Computer Networkig Coferece, September, 990, pp [9] H. Kobayashi, Applicatio of the diffusio approximatio to queueig etworks I: Equilibrium queue distributios, J. ACM 2 (2) (974) [0] A. Kumar, D. Majuath, J. Kuri, Commuicatio Networkig A Aalytical Approach, Morga Kaufma Publishers, 2004 (Chapter 8). [] J. Li, C. Blake, D.S.D. Couto, H.I. Lee, R. Morris, Capacity of ad hoc wireless etworks, i: MobiCom 0: Proceedigs of the Seveth Aual Iteratioal Coferece o Mobile Computig ad Networkig, ACM Press, New York, NY, USA, 200, pp [2] X. Li, N.B. Sheroff, Advaces i Pervasive Computig ad Networkig, Spriger Sciece, New York, NY, 2004 (Chapter 2). [3] D. Mioradi, A.A. Kherai, E. Altma, A queueig model for HTTP traffic over IEEE 802. WLANs, i: Proceedigs of 6th ITC Specialist Semiar o Performace Evaluatio of Wireless ad Mobile Systems, August [4] M. Ozdemir, A.B. McDoald, A M/MMGI//K queuig model for IEEE 802. ad hoc etworks, i: Proceedigs of the First ACM Iteratioal Workshop o Performace Evaluatio of Wireless Ad Hoc, Sesor, ad Ubiquitous Networks, ACM Press, 2004, pp. 07. [5] S. Ray, D. Starobiski, J.B. Carruthers, Performace of wireless etworks with hidde odes: A queuig-theoretic aalysis, Comput. Commu. 28 (0) (2005) [6] M. Reiser, H. Kobayashi, Accuracy of the diffusio approximatio for some queueig systems, IBM J. Res. Dev. 8 (974) [7] O. Tickoo, B. Sikdar, A queueig model for fiite load IEEE 802. radom access MAC, i: Proceedigs of IEEE ICC, Paris, Frace, Jue [8] G. Zeg, H. Zhu, I. Chlamtac, A ovel queueig model for 802. wireless LANs, i: Proceedigs of WNCG Wireless Networkig Symposium, Alhussei A. Abouzeid received the B.S. degree with hoors from Cairo Uiversity, Cairo, Egypt i 993, ad the M.S. ad Ph.D. degrees from Uiversity of Washigto, Seattle, WA i Jue 999 ad July 200, respectively, all i electrical egieerig. From 994 util 997, he served as a Project Maager i the Middle East Regioal Office of Alcatel telecom. From 998 to 200, he was a Research Assistat i the Departmet of Electrical Egieerig, Uiversity of Washigto, Seattle, WA. His research collaboratios durig this period icluded Hoeywell (previously AlliedSigal), Redmod, WA, HRL, Malibu, CA ad Microsoft Research, Redmod, WA. I August 200 he joied Resselaer Polytechic Istitute, Troy, NY, where he is curretly a Associate Professor of Electrical, Computer, ad Systems Egieerig. His research iterests iclude stochastic modelig of trasport protocols, modelig active queue maagemet i the Iteret, ad iformatio extractio, routig, ad self-cofiguratio i large-scale variable topology etworks, icludig wireless sesor etworks ad overlay (P2P) etworks. His teachig cotributios iclude the developmet of a graduate-level wireless etworks course at RPI, with a particular focus o wireless ad hoc etworks. He received the Faculty Early Career Developmet Award (CAREER) from NSF i He is a member of IEEE ad ACM, ad, sice Jauary 2006, a Associate Editor of Elsevier Computer Networks joural. Nabhedra Bisik received the B.Tech. i electrical egieerig from Idia Istitute of Techology, Kapur i 2003, ad the Ph.D. degree i electrical egieerig from Resselaer Polytechic Istitute, Troy, NY, i He is curretly with Itel, Portlad, OR. His research iterests iclude modelig ad aalysis of wireless ad hoc etworks, pheomea coverage problems i wireless sesor etworks ad resource discovery i dyamic etworks.