On the Throughput-Delay Trade-off in Georouting Networks

Transcription

1 22 Proceedigs IEEE INFOCOM O the Throughput-Delay Trade-off i Georoutig Networks Philippe Jacquet Alcatel Lucet Bell Labs Villarceaux, Frace philippe.jacquet@alcatel-lucet.com Salma Malik INRIA Rocquecourt, Frace salma.malik@iria.fr Berard Mas Macquarie Uiversity Sydey, Australia berard.mas@mq.edu.au Aloso Silva UC Berkeley Berkeley, CA, USA asilva@eecs.berkeley.edu Abstract We study the scalig properties of a georoutig scheme i a wireless multi-hop etwork of mobile odes. Our aim is to icrease the etwork capacity quasi liearly with while keepig the average delay bouded. I our model, mobile odes move accordig to a i.i.d. radom walk with velocity v ad trasmit packets to radomly chose destiatios. The average packet delivery delay of our scheme is of order /v ad it achieves the etwork capacity of order. This logloglog shows a practical throughput-delay trade-off, i particular whe compared with the semial result of Gupta ad Kumar which shows etwork capacity of order /log ad egligible delay ad the groudbreakig result of Grossglauser ad Tse which achieves etwork capacity of order but with a average delay of order /v. The foudatio of our improved capacity ad delay trade-off relies o the fact that we use a mobility model that cotais free space motio, a model that we cosider more realistic tha classic browia motios. We cofirm the geerality of our aalytical results usig simulatios uder various iterferece models. I. INTRODUCTION Gupta ad Kumar [7] studied the capacity of wireless etworks cosistig of radomly located odes which are immobile. They showed that if each source ode has a radomly chose destiatio ode, the useful etwork capacity is of order C /log where is the umber of odes ad C is the omial capacity of each ode. However, if the odes are mobile ad follow i.i.d. ergodic motios i a square area, Grossglauser ad Tse [6] showed that the etwork capacity ca rise to O(C by usig the mobility of the odes. Note that i this case, a source ode relays its packet to a radom mobile relay ode which trasmits this packet to its destiatio ode oly whe they come close together, i.e., at a distace of order /. Therefore, the time it takes to deliver a packet to its destiatio would be of order L/v where v is the average speed of the odes ad L is the legth of the fixed square area where odes are deployed. I cotrast, i Gupta ad Kumar s result [7], the packet delivery delay teds to be egligible, although the etwork capacity drops by a factor of log. I this article, we aim to maximize the capacity of mobile etworks while keepig the mea packet delivery delay We recall the followig otatio: (i f( = O(g( meas that there exists a costat c ad a iteger N such that f( cg( for > N. (ii f( = Θ(g( meas that there exists two costats c ad c 2 ad a iteger N such that c g( f( c 2 g( for > N. Gupta & Kumar [7] Network Capacity ( O log Delivery Delay egligible Grossglauser & Tse O( ( O v [6] Our work ( O logloglog O ( v TABLE I: Network Capacity vs. Delivery Delay Trade-off. bouded with icreasig umber of odes. For relayig packets towards their destiatios, mobile odes use our proposed georoutig strategy, called the Costraied Relative Bearig (CRB scheme. We show that, i a radom walk mobility model, this strategy achieves a etwork capacity of order logloglog C with a time to delivery of order L/v. Our mai cotributio is summarized i Table I. Note that i radom walk mobility models, odes have free space motio ad move i straight lies with costat speed. This mobility model is a subclass of the free space motio mobility model. Therefore, we ca also exted our result to mobility models where the average free space distace l is o zero. Cosider a example of a urba area etwork i a fixed square area of legth L with umber of odes = 6, omial badwidth C = kbps ad delay per store-adforward operatio of ms. The average packet delivery delay for Gupta ad Kumar s case would be aroud oe secod but with a etwork capacity of Mbps. I the case of Grossglauser ad Tse, the etwork capacity would icrease to about Gbps but if the straight lie crossig time L/v is about oe hour (e.g., with cars as mobile odes, the time to delivery would be aroud oe moth. However, our model of usig mobility of odes alog with the proposed CRB scheme, to relay packets to their destiatios, would lead to a etwork capacity of Gbps with time to delivery of about oe hour. This article is orgaized as follows. We first summarize some importat related works ad results i Sectio II. We discuss the models of our etwork ad CRB scheme i Sectios III ad IV respectively. The aalysis of capacity ad delay ca be foud i Sectio V ad we cofirm this /2/$3. 22 IEEE 765

2 aalysis usig simulatios i Sectio VI. We also discuss a few extesios of our work i Sectio VII ad cocludig remarks ca be foud i Sectio VIII. II. RELATED WORKS The mai differece betwee the proposed models i the works of Gupta ad Kumar [7] ad Grossglauser ad Tse [6] is that i the former case, odes are static ad packets are trasmitted betwee odes like hot potatoes, while i the latter case, odes are mobile ad relays are allowed to carry buffered packets while they move. Both strategies are based o the followig model: if p is the trasmissio rate of each ode, i.e., the proportio of time each ode is active ad trasmittig, the radius of efficiet trasmissio is give by r L κ p whe approaches ifiity for some costat κ > which depeds o the protocol, iterferece model, etc. I the cotext of [7], the umber of relays a packet has to traverse to reach its destiatio is h = O(/r. Cosequetly, p C must be divided by h to get the useful capacity: p C/h = O(C p. I order to esure coectivity i the etwork, so that every source is able to commuicate with its radomly chose destiatio, p must satisfy the limit p O(/log. This leads to Gupta ad Kumar s maximum capacity of O(C /log with hot potatoes routig. I cotrast, i the cotext of [6], the etwork does ot eed to be coected sice the packets are mostly carried i the buffer of a mobile relay. Therefore there is o limit o p other tha the requiremet that it must be smaller tha some ε < that depeds o the protocol ad some other physical parameters. Thus r is O(/. I Grossglauser ad Tse s model, the source trasmits the packet to the closest mobile relay or keeps it util it fids oe. This mobile relay delivers the packet to the destiatio whe it comes withi rage of the destiatio ode. Such a packet delivery requires a trasmissio phase which also icludes retries ad ackowledgemets so that the packet delivery ca be evetually guarateed. The proposed model of [6] requires a GPS-like positioig system ad the kowledge of the effective rage r. The estimate of r could be achieved via a periodic beacoig from every ode, where each beaco cotais the positio coordiates of the ode, so that a ode kows the typical distace for a successful receptio. However, the relay caot rely o beacoig i order to detect whe it is i the receptio rage of the destiatio. The reaso is that a ode stays i the receptio rage of aother ode for a short time period of order r /v = / ad this caot be detected via a periodic beacoig with bouded frequecy sice p = O( (the frequecy of periodic beacoig should be of O(. We may also assume that the destiatio ode is fixed ad its cartesia coordiates are kow by the mobile relay. Otherwise, if the destiatio ode is mobile, there would be a requiremet for this ode to track its ew coordiates ad dissemiate this iformatio i the wireless etwork as i [4], [9]. It is also iterestig to ote that Diggavi, Grossglauser, ad Tse [3] showed that a costat throughput per sourcedestiatio pair is feasible eve with a more restricted mobility model. Fraceschetti et al. [4] proved that there is o gap betwee the capacity of radomly located ad arbitrarily located odes. Throughput ad delay trade-offs have appeared i [5], [8] where delay of multi-hop routig is reduced by icreasig the coverage radius of each trasmissio, at the expese of reducig the umber of simultaeous trasmissios the etwork ca support. We will show that, i our work, we have a delay of O(/v ad throughput per source-destiatio pair of O( logloglog. If we take the otatio of a( i [5], [8] to measure the average distace traveled toward the destiatio betwee two cosecutive emissios of the same packet, the we will show that our scheme yields a( = Θ(/log. If we compare with the result of [5], [8], we should have a throughput of Θ( log but our scheme log delivers a higher throughput by a factor greater tha. I fact, if l is the average free space distace of the radom walk, the our scheme yields ( a( = Θ. The apparet l cotradictio comes from the fact that the +log authors i [5], [8] cosider a mobility model based o browia motio. This correspods to havig l = ad, i this case, our scheme would be equivalet to the hot potatoes routig of [7] with a( = Θ(r. Let us poit out that the browia motio mobility is a importat yet worst case model ad it is ot realistic for real world situatios such as urba area mobile etworks. I the sectio devoted to geeralizatios, we exted our result to fit a more geeral mobility model where mobile odes follow fractal trajectories with l = l = Θ(/log ad the throughput of our scheme remais of Θ( logloglog. O the practical side, may protocols have bee proposed for wireless multi-hop etworks. These protocols may be classified i topology-based ad positio-based protocols. Topology-based protocols [2], [8], [7] eed to maitai iformatio o routes potetially or curretly i use, so they do ot work effectively i eviromets with high frequecy of topology chages. For this reaso, there has bee a icreasig iterest i positio-based routig protocols. I these protocols, a ode eeds to kow its ow positio, the oe-hop eighbors positios, ad the destiatio ode s positio. These protocols do ot eed cotrol packets to maitai lik states or to update routig tables. Examples of such protocols ca be foud i [], [9], [] [3], [5], [6], [2]. I cotrast to our work, they do ot aalyze the trade-off betwee the capacity ad the delay of the etwork uder these protocols ad their scalig properties. III. NETWORK AND MOBILITY SETTINGS We cosider a etwork of mobile odes with their iitial positios uiformly distributed over the etwork area. Each mobile ode trasmits packets to a radomly chose fixed ode, called its destiatio ode, which is also radomly located i the etwork area. We assume that mobile odes are aware of their ow cartesia coordiates, e.g., by usig GPS or from the iitial positio, a mobile ode could use the kowledge of its motio vector to compute its cartesia coordiates at ay give time. 766

3 Iitially we cosider that oly mobile odes participate i the relay process to deliver packet to its destiatio ode. The case where the fixed odes may also participate i the relay process is discussed i Sectio VII. A mobile ode should be aware of the cartesia coordiates of the destiatio ode of a packet it carries. Ideed it ca be assumed that this iformatio is icluded i all packets or is relayed with the packets. Hece our model oly requires that a source or relay ode is aware of the cartesia coordiates of the destiatio ode which is assumed fixed. Note that if the destiatio ode is mobile, a mechaism to dissemiate its updated cartesia coordiates i the etwork ca be used, e.g., [4], [9]. However, this is outside the scope of this paper as we particularly focus o the throughput-delay tradeoff. With the available iformatio, a mobile relay ca determie: - its headig vector, which is the motio vector whe its speed is o zero, - its bearig vector, which is the vector betwee its positio ad the positio of a packet s destiatio; ad, - the relative bearig agle, i.e., the absolute agle betwee its headig ad bearig vectors. I the example of Fig., ode A is carryig a packet for ode D. This figure also shows the headig vector of mobile ode A ad its bearig vector ad relative bearig agle for destiatio ode D. Note that a mobile relay may carry packets for multiple destiatios but ca easily determie the bearig vector ad relative bearig agle for each destiatio ode. IV. MODEL OF CRB SCHEME I this sectio, we will preset the parameters ad specificatios of the model of our georoutig scheme. A. Parameters We defie the parameters θ c, called the carry agle, ad θ e, called the emissio agle. Each mobile ode carries a packet to its destiatio ode as log as its relative bearig agle, θ, is smaller tha θ c which is strictly smaller tha π/2. Whe this coditio is ot satisfied, the packet is trasmitted to the ext relay. B. Model Specificatio With Radio Rage Awareess I the followig descriptio, we iitially assume that each ode is aware of the effective rage of trasmissio r. This meas that there is a periodic beacoig that allows this estimate to be made. I Sectio IV-C, we will ivestigate how to specify our model without a estimate of the effective rage r. Assume that ode A is carryig a packet for ode D. The velocity of ode A is deoted by v(a. - If ode A is withi rage of ode D, it trasmits the packet to D; otherwise, - if the relative bearig agle is smaller tha θ c, ode A cotiues to carry the packet; otherwise, - ode A trasmits the packet to a radom eighbor mobile ode iside the coe of agle θ e, with bearig vector as the axis, ad the forgets the packet. Mobile ode A Headig vector θ (relative bearig agle θ e Bearig vector Destiatio ode D Fig. : Figurative represetatio of our model. Ufilled circles represet the potetial mobile relays for packet trasmitted by ode A for ode D. I order to better uderstad the model of our georoutig scheme, cosider the example i Fig.. Assume that ode A is out of rage of ode D ad, because of that, it caot deliver the packet directly. Now, if θ < θ c, ode A will cotiue to carry the packet for ode D. Otherwise, it trasmits the packet to oe of the radom mobile relays, represeted by ufilled circles i the figure. Trasmissio procedure: To trasmit the packet towards aother mobile ode, ode A shall proceed as follows: - it first trasmits a Call-to-Receive packet cotaiig the positios of odes A ad D; - a radom mobile odeb which receives this Call-to-Receive packet ca compute the agle (AB,AD. If this agle is smaller tha θ e, it replies with a Accept-to-Receive packet cotaiig a idetifier of ode B; - ode A seds the packet to the first mobile ode which replied with a Accept-to-Receive packet. The first ode which seds its Accept-to-Receive packet otifies the other receivers of the Call-to-Receive packet, to cacel their trasmissios of Accept-to-Receive packets. There may be more tha oe (but fiite trasmissios of Accept-to- Receive packets i case two or more receivers are at distace greater tha r from each other. Note that this procedure does ot eed ay beacoig or periodic trasmissio of hello packets. The back-off time of odes, trasmittig their Accept-to-Receive packet, ca also be tued i order to favor the distace or displacemet towards D, depedig o ay additioal optioal specificatios. C. Model Specificatio Without Radio Rage Awareess The estimatio of r would require that the odes employ a periodic beacoig mechaism. If such a mechaism is ot available or desirable, the CRB scheme relies o the sigal to iterferece plus oise ratio (SINR for trasmittig packets to their destiatios or radom mobile relays. I other words, a mobile ode ca relay a packet to its destiatio ode or aother mobile ode oly if the SINR at the receiver is above a give threshold. Note that i this case, the specificatio of the trasmissio procedure is also modified so that it termiates whe the fial 767

4 destiatio receives the packet. To trasmit the packet towards its destiatio ode or aother mobile ode, ode A shall proceed as follows: - it first trasmits a Call-to-Receive packet cotaiig the positios of odes A ad D; - if ode D receives this packet, it respods immediately with a Accept-to-Receive packet with highest priority. Node A, o receivig this packet, relays the packet to ode D; otherwise, - the procedure of selectig a radom mobile ode, as the ext relay, is similar to the procedure described i Sectio IV-B. A radom mobile ode B, which receives the Callto-Receive packet, computes the agle (AB, AD. If this agle is smaller tha θ e, it respods with a Accept-to- Receive packet; - ode A relays the packet to the first mobile ode which set its Accept-to-Receive packet successfully. The first ode which trasmits its Accept-to-Receive packet also makes the other receivers to cacel their trasmissios of Accept-to- Receive packets. V. PERFORMANCE ANALYSIS I our georoutig scheme, the average trasmissio rate of each mobile ode is p = O(/loglog. We will also show that the umber of trasmissios per packet is of O(log ad this would lead to a useful etwork capacity of logloglog O(C. Note that our scheme could lead to packet loss because of trasmissio failure but we show that the probability of this packet loss is iverse power of log ad it teds to zero as approaches ifiity. We assume that the etwork area is a square area ad without loss of geerality we assume that it is a square uit area. The mobile odes move accordig to i.i.d. radom walk: from a uiformly distributed iitial positio, the odes move i a straight lie with a certai speed ad radomly chage directio. The speed is radomly distributed i a iterval [v mi,v max ] with v mi >. To simplify the aalysis, we assume that v mi = v max = v. We also assume that each ode chages its directio with a Poisso poit process of rate τ. Whe a mobile ode hits the border of the etwork, it simply bouces like a billiard ball. This leads to the isotropic property (Jacquet et al. []: at ay give time the mobile odes are uiform distributed i the square ad move i uiformly selected directio idepedetly of their positio. The radius of efficiet trasmissio r is derived from the value of p ad is give by r =, for some β >. Therefore, the average umber of eighbors of a arbitrary ode at a arbitrary time is βloglog. I order to keep the average cumulated load fiite, the odes have a β loglog π average trasmissio rate of p = βloglog. Therefore, the actual desity of simultaeous trasmitters is A. Methodology The parameters of iterest are the followig: βloglog. - The delay D (r of deliverig a packet to the destiatio whe the packet is geerated i a mobile ode at distace r from its destiatio ode. - The average umber of timesf (r the packet chages relay before reachig its destiatio whe it has bee geerated i a mobile ode at distace r from its destiatio ode. I order to exhibit the actual performace of our proposed CRB scheme, we aim to derive a upper-boud o the parameters D (r ad F (r. I the ext two sub-sectios, we assume w.l.o.g. that there is always a relay ode, to receive the packet, i the emissio coe (as the ode desity ad agle, θ e, are sufficietly large whe a relay chage must occur. B. Delivery Delay We igore the queueig delays which ca become apparet whe several packets could be i competitio i the same relay to be trasmitted at the same time. Theorem : r D (r vcos(θ c + F ( (r t f v r, ( v cos(θ c where t f is the store ad forward delay quatity. Proof: We ca write D (r as D (r F (rt f + r F (rr vcos(θ c where F (rt f is the average total delay of store ad forward operatios ad r F(rr vcos(θ c is a upper boud of the delay whe mobile relays carry a packet at costat speed v with a relative bearig agle always smaller tha θ c. Rearragig this equatio leads to (. Remark : If r is greater tha t f v, secod factor i ( becomes egative which gives the upper boud of r D (r vcos(θ. However, if r c is less tha t f v, D (r will also iclude a O(log factor because of F (r. C. Number of Relay Chages There are two evets that trigger relay chages. Relay chage due to tur, i.e., the mobile ode, carryig the packet, chages its headig vector such that the relative bearig agle becomes greater tha θ c. 2 Relay chage due to pass over, i.e., the mobile ode keeps its trajectory ad the relative bearig agle becomes greater tha θ c. Cosider a packet geerated at distace r from its destiatio. Let F(r t be the average umber of relay chages due to tur. Equivaletly, let F(r p be the average umber of relay chages due to pass over. Therefore, we have F (r = F(r t + F(r p ad we expect that the mai cotributio of O(log i F (r will come from F(r. p Number of Relay Chages Due to Tur: We prove the followig theorem: Theorem 2: We have the boud Proof: F t (r π θ c θ c τ vcos(θ c r., 768

5 Mobile ode A at positio 2 Mobile ode A at positio θ < θ c θ > θ c r Destiatio ode D Fig. 2: Figurative descriptio of relay chage due to tur. At positio, θ < θ c ad ode A carries the packet for ode D. At positio 2, ode A chages its headig vector ad must trasmit the packet. We cosider the case i Fig. 2 ad assume that a mobile ode is carryig a packet to its destiatio located at distace r. The ode chages its directio with Poisso rate τ. Whe the ode chages its directio, it may keep a directio that stays withi agle θ c with the bearig vector ad this will ot trigger a relay chage. This occurs with probability θc π. Otherwise, the packet must chage relay. But the ew relay may have relative bearig agle greater tha θ c which would result i a immediate ew relay chage. Therefore, at each directio chage, there is a average of π θc θ c relays. Multiplied by D (r this gives our upper-boud of F(r. t Note that we have ot cosidered the tur due to bouces o the borders of square map. But it is easy to see via straightforward geometric cosideratios that they caot actually geerate a relay chage. 2 Number of Relay Chages Due to Pass Over: We prove the followig theorem: Theorem 3: We have the boud F(r p πta(θ ( c r θc 2 log. r Proof: Here we cosider the case of Fig. 3. We assume that a mobile ode at distace r, from its destiatio, has a relative bearig agle equal to θ. If it keeps its trajectory (i.e., does ot tur, it will eed to trasmit to a ew relay whe it passes over the destiatio, i.e., whe it arrives at a distace of ρ(θ,r = si(θ si(θ c r from the destiatio. The fuctio of θ ρ(θ,r is bijective from [,θ c ] to [,r]. For x [,r] let ρ (x,r be its iverse. Assume that r is the distace to the destiatio whe the relay receives the packet or just after a tur. Thus the agleθ is uiformly distributed o[,θ c ], i.e., with a costat probability desity θ c. The probability desity of the pass over evet at x < r (assumig o directio chage is therefore θ c x ρ (x,r = si(θ c θ c cos(ρ (x,rr = ta(ρ (x,r θ c x. Sice ρ (x,r θ c, the poit process where the packet would eed a relay chage due to pass over is upper bouded by a Poisso poit process o the iterval [r,r] ad of itesity equal to ta(θc θ c x for x [r,r]. Mobile ode A at positio Mobile ode A at positio 2 θ < θ c θ = θ c r ρ(θ, r Destiatio ode D Fig. 3: Figurative descriptio of relay chage due to pass over. At positio, θ < θ c ad ode A carries the packet for ode D. At positio 2, ode A has the same headig vector but θ = θ c ad it must trasmit the packet. A relay chage due to pass over correspods to a average of π θ c relays which is higher tha the average umber of relays i case of relay chage due to tur. The reaso of this higher umber is the fact that i case of relay chage due to tur, a mobile relay may still move i a directio that stays withi the carry agle θ c ad cotiue to carry the packet. Neglectig the decremet of distace durig each trasmissio phase r F(r p πta(θ c dx = r θc 2 x = πta(θ ( c r θc 2 log. r We have thus F (r π θ c θ c τ cos(θ c v r + πta(θ ( c r θc 2 log r Therefore we have a mai cotributio of O(log relay chages that comes from log(/r. The result holds because we assume that there is always a receiver i each relay chage. I the ext sub-sectio we remove this coditio to establish a result with high probability. D. Number of Relay Chages With High Probability of Success I the previous subsectio we assumed that there is always a receivig relay i the emissio coe at each relay chage ad we said that the relay chage is always successful. The case with failed relay chage would itroduce additioal complicatios. For example oe could use the fixed relays if the packet caot be delivered to a mobile relay. Ayhow, to simplify the preset cotributio, we will show that with high probability, i.e., with probability approachig oe whe approaches ifiity, every relay chage succeeds. Theorem 4: With high probability o arbitrary packets, all relay chages succeed for this packet ad are i average umber F (r ad the delay is D (r. Proof: We use a modified stochastic system to cope with failed relay chages. The modificatio is the followig: whe there is o relay i the emissio coe durig a relay chage a decoy mobile relay is created i the emissio coe that will receive the packet. Each decoy relay is used oly for oe packet ad disappear after use. Notice that the modified system is ot a practical scheme i a practical etwork. The aalysis i the previous sectio still holds ad i particular F (r is ow the average ucoditioal umber of relay chages (icludig. 769

6 those via decoy relays for ay packet startig at distace r from destiatio. Let P (r be the probability that a packet startig at distace r has a failed relay chage. The probability that a relay chage fails is equal to ( θ e r 2 e θ2 e r2 = (log β θe π. Therefore the average umber of failed relay chagese (r F (r(log β θe π which teds to zero whe β θe π >, sicef (r = O(log. The fial result comes sice P (r E (r. VI. SIMULATIONS We performed simulatios with CRB georoutig scheme uder two cotexts: a simplified cotext where the etwork is modeled uder uit disk model; 2 a realistic cotext where the etwork operates uder slotted ALOHA ad a realistic SINR iterferece model is cosidered. The simulatios of CRB scheme are stressed to the poit that the motio timigs are ot so large compared to slot times. A. Uder Disk Graph Model I this sectio, we cosider a etwork of mobile odes. We assume that all odes have the same radio rage give by β loglog r =. π Each mobile ode moves accordig to a i.i.d. radom walk mobility model, i.e., it starts from a uiformly distributed iitial positio, moves i straight lie with costat speed ad uiformly selected directio ad reflects o the borders of the square area (like billiard balls. I the ext sectio (Sectio VI-B, we will further explore the effect of iterferece o the simulatios, but for the momet we oly cosider a source mobile ode ad its radomly located destiatio ode which is fixed. We adopt the disk graph model of iterferece, i.e., two odes are coected or they ca exchage iformatio if the distace betwee them is smaller tha a certai threshold (called radio rage, otherwise, they are discoected. A mobile ode relays the packet oly if the relative bearig agle, i.e., the absolute agle made by the headig vector ad the bearig vector, becomes greater tha θ c. Otherwise, it cotiues to carry the packet. Simulatio parameters ad assumptios: The purpose of our simulatios is to verify the scalig behavior of average delay ad umber of hops per packet with icreasig umber of odes i the etwork. Therefore, the umber of mobile odes,, i the etwork is varied from to two millio odes. The values of other parameters, which remai costat, ad do ot impact the scalig behavior are listed as follows. (i Parameters of CRB scheme, θ c ad θ e, are take to be π/6. (ii The speed of all mobile odes is costat, i.e.,.5 uit distace per slot. (iii All mobile odes chage their directio accordig to a Poisso poit process with mea equal to slots. (iv The value of costat factor β is assumed to be equal to 4. Average delay per packet (slots Average umber of hops per packet e+5.e+6.5e+6 2.e+6 Number of mobile odes Fig. 4: Average delay per packet. 5.e+5.e+6.5e+6 2.e+6 Number of mobile odes Fig. 5: Average umber of hops per packet. 2 Results: We have evaluated the followig parameters. (i Average delay per packet. (ii Average umber of hops per packet. We cosidered the Mote Carlo Method with simulatios. The delay of a packet is computed from the time whe its processig started at its source mobile ode util it reaches its destiatio ode. Figure 4 shows the average delay per packet with a icreasig umber of odes. We otice that as icreases, the average delay per packet appears to approach a costat upper boud which ca be computed from (. Figure 5 shows the average umber of hops per packet with icreasig values of. B. With slotted ALOHA uder SINR iterferece model I this sectio, we will preset the simulatios of CRB georoutig scheme with a trasmissio model which does ot rely o the estimate ofr ad is based o the required miimal SINR threshold. Trasmissio model: Our trasmissio model is as follows. Let P i be the trasmit power of ode i ad γ ij be the chael gai from ode i to ode j such that the received power at ode j is P i γ ij. The trasmissio from ode i to ode j is successful oly if the followig coditio is satisfied P i γ ij N + k i P kγ kj > K, where K is the desired miimum SINR threshold for successfully receivig the packet at the destiatio ad N is the backgroud oise power. For ow, we igore multi-path fadig or shadowig effects ad assume that the chael gai from 77

7 ode i to ode j is give by γ ij = z i z j α, where α > 2 is the atteuatio coefficiet ad z i is the locatio of ode i. 2 Simulatios uder SINR iterferece model: For the theoretical aalysis i Sectio V, we have assumed that the effective rage of successful trasmissio is r = β loglog, π which requires that the mobile odes have a average trasmissio rate ofp = β/loglog. I other words, if the mobile odes emit packets at the give average rate, the average distace of successful trasmissio uder SINR iterferece model is of O(r ad the results from theoretical aalysis are applicable as well. We assume that time is slotted ad mobile odes determie their relative bearig agles at the begiig of a slot. We also assume that all odes are sychroized ad simultaeous trasmitters i each slot emit a Call-to-Receive packet at the begiig of the slot. Moreover, we also assume that fixed odes do ot emit ay packet except, maybe, a Accept-to- Receive packet i respose to a trasmissio by a mobile ode. I our simulatio eviromet, mobile odes start from a uiformly distributed iitial positio ad move idepedetly i straight lies ad i radomly selected directios. They also chage their directio radomly at a rate which is a Poisso poit process. Each mobile ode seds packets towards a uique destiatio (fixed ode, ad all destiatios odes are also uiformly distributed i the etwork area. I order to keep load i the etwork fiite, the packet geeratio rate at a ode, ρ, should be of O(p /X where X is the expected umber of trasmissios per packet. From the theoretical aalysis, we kow that X = O(log( β 2 +c, where c is a costat if θ c is o-varyig. I our simulatios uder SINR iterferece model, we assume that the kowledge of r is ot available ad mobile odes use miimal SINR threshold for successfully receivig a packet. We also assume that each mobile ode geerate packets, destied for its uique fixed destiatio ode, at a uiform rate give by ρ = β log( β 2 loglog, (2 for some β > ad β 2 >. We igored the value of costat c ad have observed that the simulatio results are asymptotically correct because, with icreasig, value of c should be isigificat as compared to the O(log(/β 2 factor. 3 Simulatio parameters ad assumptios: The purpose of our simulatios is to verify the scalig properties of etwork capacity, delay ad umber of trasmissios per packet with icreasig umber of odes i the etwork. The umber of mobile odes,, i the etwork is varied from 25 odes to, odes. All odes use uiform uit omial trasmit power ad the backgroud oise power N is assumed to be egligible. The values of other parameters are listed as follows. (i Parameters of CRB scheme, θ c ad θ e, are take to be π/6. (ii The speed of all mobile odes is costat, i.e.,. uit distace per slot. (iii All mobile odes chage their directio idepedetly ad radomly accordig to a Poisso poit process with mea equal to slots. (iv The values of costat factors β ad β 2 are assumed equal to 5 ad respectively. (v SINR threshold, K, is assumed equal to. (vi Atteuatio coefficiet, α, is assumed equal to 2.5. I our simulatios, we make the followig assumptios. (i Each mobile ode geerates a ifiite umber of packets, at rate ρ, for its respective destiatio ode. (ii A mobile ode may carry, i its buffer, its ow packets as well as the packets relayed from other mobile odes. Therefore, it may have more tha oe packet i its buffer which it must trasmit because their respective relative bearig agles are greater tha θ c. I such a case, it first trasmits the packet which is furthest from its destiatio. 4 Results: We have examied the followig parameters. (i Throughput capacity per ode, λ. (ii Average umber of hops, h, ad trasmissio attempts, t, per packet. (iii Average delay per packet. The throughput capacity per ode, λ, is the average umber of packets arrivig at their destiatios per slot per mobile ode. With icreasig, throughput capacity per ode should follow the followig relatio λ = η β log( β 2 loglog, (3 for some < η < which depeds o K, α ad protocol parameters. Note that the values of these costats do ot affect the asymptotic behavior of λ which is also observed i our simulatio results. I order to verify the asymptotic character of simulated packet geeratio rate ad throughput capacity, we have aalyzed the parameters m ρ ad m λ which are give by ( m ρ = ρ (β log β ( 2 m λ = λ (β log β 2 log log( log log( From the defiitio of ρ i (2, the value of m ρ should be costat at whereas, with icreasig, value of m λ should coverge to the costat η. From Fig. 6, value of η is foud to be approximately equal to.45. Figure 7 shows the simulated ad theoretical packet geeratio rate, ρ, ad throughput capacity, λ, i the etwork. The theoretical values of ρ ad λ are computed from (2 ad (3. Figure 8 shows the average umber of hops, h, ad trasmissio attempts, t, per packet. The value of t is slightly higher tha the value of h because of the possibility that a successful receiver may ot be foud i each trasmissio phase, i.e., i the coe of trasmissio formed with θ e. With icreasig, h ad t are expected to grow i O(log(/β 2. To verify this character i simulatio results, we examie the., 77

8 2.5.5 m λ m ρ Average delay per packet (slots Number of mobile odes Fig. 6: Verificatio of etwork throughput capacity with plots of m λ ad m ρ. Average umber of packets per slot Number of mobile odes λ ρ Fig. 7: Simulated (solid lies ad theoretical (dotted lies etwork throughput capacity, λ, ad etwork packet geeratio rate, ρ. Average per packet Number of mobile odes Fig. 8: Average umber of hops, h, ad trasmissio attempts, t, per packet h Number of mobile odes Fig. 9: Verificatio of umber of hops ad trasmissio attempts with plots of m h ad m t. parameters m h ad m t give by m h = h log( β 2, m t = t log( β 2. t m h m t Number of mobile odes Fig. : Average delay per packet. If the values of h ad t are i O(log(/β 2, the values of m h ad m t should approach a costat value which is the case i Fig. 9. The delay of a packet is computed from the time whe its processig started at its source mobile ode util the time it arrives at its destiatio ode. Figure shows the average delay per packet. As the umber of mobile odes icrease, the average delay appears to approach a costat value. It ca be observed that whe is small, the average umber of hops per packet is almost of O( which also meas that the average delay per packet is of O( ad the etwork throughput capacity is of O(η: although, i simulatio results, it is bouded by the etwork packet geeratio rate logloglog which is of O(. This ca be observed i Fig. 7, 8 ad. The reaso is that whe is small, the umber of simultaeous trasmissios i the etwork is also small ad packets ca be delivered by the mobile odes, directly to their destiatio odes, i O( hops. As icreases, umber of simultaeous trasmitters icrease ad cosequetly the effective trasmissio rage of each trasmitter shriks. Therefore, the domiat factor i the umber of trasmissios per packet comes from the fact that a mobile relay has to be close to the destiatio, to deliver a packet. Accordig to theoretical aalysis, h ad t grow i O(log which is also observed i the simulatio results. Simulatios also show that, asymptotically, etwork throughput capacity is of O( logloglog ad average delay per packet is of O(/v which complies with our theoretical aalysis. VII. EXTENSIONS AND GENERAL MOBILITY MODELS I our discussio, we primarily focussed o the capacitydelay tradeoff ad thus for the iitial sake of clarity assumed that the fixed odes ca oly receive packets destied for them. We could also cosider a slight variatio i the specificatio of the model of CRB scheme such that the fixed odes also participate i the routig of packets to their destiatio odes. For example, durig a trasmissio phase, if a packet caot be trasmitted to its destiatio ode or relayed to a radom mobile eighbor i the coe of trasmissio, it ca be relayed to a fixed ode. This fixed ode must emit this packet immediately to its destiatio ode or to ay mobile relay i the eighborhood. Note that this will also help icrease the coectivity of the etwork. 772

9 T distace T 2 T T 2 T 3 T 4T5 T 6 T 3 T 7 T 4,...,T 7 Fig. : Illustratio of self-similar trajectories i urba areas. The coditio about i.i.d. radom walks ca be relaxed ad the result about the expected umber of relay chages will still be valid. I other words, the i.i.d. radom walk model ca be see as a worst case compared to realistic mobility models. If the mobile relays move like cars i a urba area, the we ca expect that their mobility model will sigificatly depart from the radom walk. Ideed cars move toward physical destiatios ad i their jourey o the streets toward their destiatio, their headig after each tur is positively correlated with the headig before the tur. This implies that the probability that a relay chage is eeded after a tur is smaller tha it would be uder a radom walk model, where headigs before ad after tur are ot correlated. Furthermore o a street, the headigs are positively correlated (cosider Mahatta oe-way streets ad i this case a relay chage due to a pass over will have more chaces to arrive o a relay with good headig (oe half istead of θ c /π. Agai this would lead to less relay chages due to pass over. The result still holds if we assume that the tur rate τ depeds o ad τ = τ = O(log. I this case, the mobility model would fit eve better for the realistic mobility of a urba area. Ideed the trajectories of cars should be fractal or self-similar, showig more frequet turs whe cars are close to their physical destiatio (differet tha packet destiatio or whe leavig their parkig lot. I this case, the overall tur rate teds to be i O(log with a coefficiet depedig o the Hurst parameter of the trajectory. This would lead to the same estimate of O(log relay chage per packet. Figure illustrates a self-similar trajectory i a urba area. It shows a two-dimesioal trajectory (upper half ad its traveled distace (lower half. The successive turs are idicated by T,...,T 7. The trajectory after ay tur T i looks like a reduced copy of the origial trajectory. The CRB scheme may eed some adaptatio to cope with some uusual street cofiguratios, e.g., to replace the cartesia distace with the Mahatta distace i the street map. VIII. CONCLUSIONS We have examied asymptotic capacity ad delay i mobile etworks with a georoutig scheme, called CRB, for commuicatio betwee source ad destiatio odes. Our results show that CRB allows to achieve the etwork capacity of O( logloglog with packet delivery delay of O( ad trasmissios per packet of O(log. It is oticeable that this scheme does ot eed ay sophisticated overhead for implemetatio. However, i this case, the mobile odes must be aware of their positio via a GPS system, for example. We have show the asymptotic performace via aalytical aalysis uder a uit disk graph model with radom i.i.d. walks. The aalytical results have bee cofirmed by simulatios ad i particular uder ALOHA with SINR iterferece model. We have see that the performace of CRB ca be maitaied eve with o i.i.d. radom walks, the latter beig a worst case sceario. However, this latter result would require that the mobile odes stay withi same headig for O(/log time. A ext step would be to aalyze the performace of this scheme o real traffic traces i urba areas. REFERENCES [] S. Basagi, I. Chlamtac, V. R. Syrotiuk, ad B. A. Woodward. A distace routig effect algorithm for mobility (DREAM. I MOBICOM, pages 76 84, 998. [2] T. Clause ad P. Jacquet. Optimized lik state routig protocol (OLSR, IETF-RFC I RFC 3626, Uited States, 23. [3] S. Diggavi, M. Grossglauser, ad D. Tse. Eve oe-dimesioal mobility icreases ad hoc wireless capacity. I IEEE Iteratioal Symposium o Iformatio Theory, page 352, 22. [4] M. Fraceschetti, O. Dousse, D. N. C. Tse, ad P. Thira. Closig the gap i the capacity of wireless etworks via percolatio theory. 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