Time-Space Opportunistic Routing in Wireless Ad Hoc Networks, Algorithms and Performance

Transcription

1 Time-Space Opportuistic Routig i Wireless Ad Hoc Networks, Algorithms ad Performace Fraçois Baccelli, Bartlomiej Blaszczyszy, Paul Muhlethaler To cite this versio: Fraçois Baccelli, Bartlomiej Blaszczyszy, Paul Muhlethaler. Time-Space Opportuistic Routig i Wireless Ad Hoc Networks, Algorithms ad Performace. The Computer Joural, Oxford Uiversity Press (UK), 2009, 53, pp < /comjl/bxp049>. <iria > HAL Id: iria Submitted o 4 May 2009 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 Time-Space Opportuistic Routig i Wireless Ad Hoc Networks Algorithms ad Performace Fraçois Baccelli 1, Bartłomiej Błaszczyszy 2 ad Paul Mühlethaler 3 Abstract I classical routig strategies for wireless adhoc (mobile or mesh) etworks packets are trasmitted o a pre-defied route that is usually obtaied by a shortest path routig protocol. I this paper we review some recet ideas cocerig a ew routig techique which is opportuistic i the sese that each packet at each hop o its (specific) route from a origi to a destiatio takes advatage of the actual patter of odes that captured its recet (re)trasmissio i order to choose the ext relay. The paper focuses both o the distributed algorithms allowig such a routig techique to work ad o the evaluatio of the gai i performace it brigs compared to classical mechaisms. O the algorithmic side, we show that it is possible to implemet this opportuistic techique i such a way that the curret trasmitter of a give packet does ot eed to kow its ext relay a priori, but the odes that capture this trasmissio (if ay) perform a self selectio procedure to chose the packet relay ode ad ackowledge the trasmitter. We also show that this routig techique works well with various medium access protocols (such as Aloha, CSMA, TDMA). Fially, we show that the above relay self selectio procedure ca be optimized i the sese that it is the ode that optimizes some give utility criterio (e.g. miimize the remaiig distace to the fial destiatio) which is chose as the relay. The performace evaluatio part is based o stochastic geometry ad combies simulatio a aalytical models. The mai result is that such opportuistic schemes very sigificatly outperform classical routig schemes whe properly optimized ad provided at least a small umber of odes i the etwork kow their geographical positios exactly. I. INTRODUCTION Routig is the process of selectig paths i a etwork alog which to sed etwork traffic. I packet switchig 1 INRIA/ENS, 45 rue d Ulm, Paris FRANCE; Fracois.Baccelli@es.fr, 2 INRIA/ENS ad Math. Ist., Uiv. of Wrocław, 45 rue d Ulm, Paris FRANCE; Bartek.Blaszczyszy@es.fr, 3 INRIA, Le Chesay, FRANCE; Paul.Muhlethaler@iria.fr etworks routig directs packet forwardig the trasit of logically addressed packets from their source toward their ultimate destiatio through itermediate odes. Prior to this, i such etworks the odes usually exchage cotrol packets cotaiig the etwork topology iformatio that allow each ode to fid its ext relay towards ay destiatio i the coected part of the etwork. Oce the paths (routes) are established i the etwork, aother part of the data commuicatio protocol, called Medium Access Cotrol (MAC) layer, is resposible for movig data packets o their paths by orgaizig simultaeous trasmissios i the etwork. I wireless ad hoc etworks routig is cofroted with relatively frequet chages i the etwork topology. Ideed, i mobile ad hoc etworks the odes may go o ad off, as well as chage their geographical locatios. Besides, the variability of radio chael coditios (so called fadig) makes the etwork topology vary eve i etworks where the geographic patter of odes is relatively static (such as i mesh etworks). May studies have bee carried out to cope with this problem. Existig solutios are frequetly subdivided ito two classes: reactive protocols ad proactive protocols. Proactive protocols are mostly based o existig routig protocols developed for wired etworks. The emphasis i these protocols is usually put o reducig the cotrol overhead as they have to be ru more ofte to follow the varyig etwork topology. Reactive protocols, o the other had, use routes which are built o demad. A source ode wishig to obtai a route to a destiatio ode floods the etwork with a request packet. Whe the diffusio of this packet reaches the destiatio, the route ca be established. I this paper, which surveys ad complemets two recet coferece papers [1], [2] of the authors, we cosider aother class of routig strategies where the relay ca be defied at each hop of each packet, depedig o the local cofiguratio of simultaeous trasmitters. I cotrast to wired etworks, this cofiguratio essetially determies the feasibility of trasmissios

3 2 o idividual liks i wireless etworks. This strategy, which we call opportuistic routig, ad which merges the fuctioality of routig ad MAC layer, has already bee show beeficial i the sese that it usually offers a smaller delay to carry a packet from origi to destiatio compared to classical routig schemes. I this paper we describe also a very efficiet way to implemet opportuistic routig utilizig a relay self selectio techique. I this procedure, the curret emitter of a give packet does ot eed to kow its ext relay a priori, but the odes that capture this trasmissio (if ay) perform a self selectio to chose the uique packet relay ode ad ackowledge the emitter. This techique will be show compatible with various MAC protocols implemeted i wireless etworks as e.g. CSMA or Aloha. Oe of the goals of this papers is also to evaluate the performace of opportuistic routig. For this, we itroduce a realistic model to carry out simulatios which allow for a extesive compariso of shortest path ad opportuistic routig. Our umerical results reveal some iterestig properties related to the joitly optimal tuig of the Aloha MAC ad opportuistic routig. Last but ot least, we propose a mathematical framework based o the theory of stochastic geometry that allows us to cofirm ad further study the properties of the opportuistic routig revealed by simulatios. Stochastic geometry, which is ow a rich brach of applied probability itrisically related to the theory of poit processes, allows oe to study radom pheomea o the plae or i higher dimesio. Whe applied to commuicatio etworks, it provides a atural way of defiig ad computig macroscopic properties of such etworks, by some averagig over all potetial geometrical patters for the odes, i the same way as queuig theory provides averaged respose times or cogestio over all potetial arrival patters withi a give parametric class. I the poit-to-poit routig case, the mai geometric objects are the (log) paths from a give source ode to a destiatio ode, where the relay odes are picked form some realizatio of a homogeeous Poisso poit process of the plae. The paper is orgaized as follows. Sectio II reviews the existig routig mechaisms ragig from covetioal routig algorithms to more recet schemes such geographic routig ad opportuistic routig. Sectio III describes the optimized self selectio scheme. This self selectio which uses sigalig bursts ad short slots of carrier sesig ca be see as a improved CSMA scheme. Sectio IV describes the model for the performace evaluatio of opportuistic routig. This model is used for simulatios as well as for the mathemati- O D Fig. 1. Left: Shortest path (smallest umber of hops) from O to D, with eighborhoods defied by discs of fixed radius (maximum trasmissio rage). Right: Local greedy geographical routig maximizig the progressio towards the destiatio (the abscissa i the directio towards; gree solid lie) or miimizig the remaiig distace to destiatio (dashed red lie) with the same trasmissio rages; these two geometric criteria may give differet relays close to the destiatio. cal aalysis. Sectio V presets the mai observatios obtaied by simulatio. The simulatios are carried out both with Aloha ad CSMA. Sectio VI provides a mathematical framework for the aalysis of opportuistic routig. This framework allows oe to better uderstad a few observatios obtaied i Sectio V. II. FROM SHORTEST-PATH TO OPPORTUNISTIC ROUTING FOR WIRELESS NETWORKS STATE OF THE ART Routig protocols are distributed algorithms that fid routes for all pairs of origi ad destiatio odes (O-D pairs). Usually i a multi-hop etwork, oce a route has bee foud for a O-D pair, all the packets of this O-D pair follow this route as log as the etwork topology remais uchaged. A. Covetioal proactive routig I proactive protocols such as OLSR [3] the computatio of routes is based o the exchage of cotrol packets set by the routig protocol. Usig the topology iformatio carried i the cotrol packets, each ode ca fid its ext relay towards ay destiatio i the (coected part of the) etwork. The most promiet algorithm that builds the shortest routes (with the smallest umber of hops) is Dijkstra s algorithm [4]. Figure 1 (Left) depicts the shortest path from O to D assumig that the eighborhood of a ode is idetified via some maximum trasmissio rage parameter: eighbors of a ode are all the odes at a distace smaller tha this parameter. A importat problem i covetioal proactive routig is that the covergece time of a Dijkstra-like algorithm for fidig routes, as well as the routig state of each ode (the ext relay for ay destiatio i the etwork), icreases cosiderably as soo as the etwork has a large umber of odes. O D

4 3 B. Reactive routig Oe way to reduce the routig state of odes is to build routes o demad as i AODV [5]: a source ode wishig to obtai a route to a destiatio ode floods the etwork with a request packet. Whe the diffusio of this request packet reaches the destiatio, the backtrackig of its tree allows the required route to be established. However this solutio does ot essetially reduce the complexity of the algorithm, which has to be ru each time a source is lookig for a destiatio. C. Local geographic routig Copig with scalability problems has bee the primary goal of geographic routig [6]. A reductio i complexity, however, comes at the cost of kowig the positios of the odes that are used to determie the routes to the destiatios. More precisely, i geographic routig, istead of ruig a algorithm to fid some globally optimal routes (e.g. the shortest oes) over the whole etwork, the successive hops of a path are costructed icremetally makig local, greedy choices of the ext relays accordig to the geographical locatios of the eighbourig odes. For istace, Takagi ad Kleirock i 1984 [7] proposed to choose the ext relay i such a way that it maximizes the (geometric) progressio towards the destiatio: the ode with the largest abscissa towards the destiatio is chose. Alteratively, oe ca miimize the remaiig distace to destiatio, ad the eighborig ode that is closest to the destiatio serves as the ext relay; see [8], [9]. Figure 1 (Right) depicts the paths produced by the local greedy routig with these two geometric criteria. I all the routig schemes that we describe above, if a route is established betwee a origi ad a destiatio (proactively or o demad, via a global or a greedy search algorithm) the all the packets of this give O-D pair flow are set through the same relays. This task is carried out at the MAC layer. D. Opportuistic routig Reactive ad local geographic routig has paved the way for a ew type of routig techique i which the routes are ot costructed proactively i the etwork ad where the relays of a give O-D flow are ot fixed i advace. I this techique, called opportuistic routig, the relays are chose dyamically at each hop of each packet, amog the odes which have received the packet trasmissio. This choice ca be also optimized by takig the geographical locatios of receivers ito accout (see e.g. [10], [11], [12], [13], cf. Figure 2). It has bee show that this strategy, which ivolves both Fig. 2. Opportuistic routig. Clouds aroud each trasmitter depict odes that capture (receive correctly) the packet. For the topleft trasmissio we show the relay that maximizes progress to the destiatio, which will hece be selected as the ext relay for this packet. the routig ad the MAC layer of the etwork is able to reduce the mea time required to carry a packet from the origi to the destiatio compared to the shortest path routig techique; see [6], [1]). Note that this metric is more fudametal tha that of the umber of hops i the route (optimized by the shortest path route), which does ot iclude the time which is wasted i usuccessful attempts to make a particular hop. E. Performace compariso [6], [1] show that opportuistic routig ca reduce the mea delay required to carry a packet from the origi to the destiatio compared to shortest path routig. To address this questio, as i [1], we use a Sigal to Iterferece Ratio criterio for successful packet receptio. This model is justified by may used modulatio techiques ad has a iformatio theoretic basis. We also assume that the locatios of etwork odes are the poits of some homogeeous Poisso poit process. We carry out simulatios to compare opportuistic routig with shortest path routig, both combied with Aloha or CSMA. F. Implemetatio of opportuistic routig Opportuistic routig does however come with several techical difficulties which are discussed below. 1) Relay self selectio: The major difficulty is how to let the trasmitter of a give packet kow about its curret receivers ad chose a optimal oe as its relay. This problem ca be solved usig a relay self selectio techique. I this case, the trasmitter of the packet does ot kow its ext relay a priori, but the odes that capture this trasmissio (if ay) perform a self selectio techique to chose the uique packet relay ode ad ackowledge the trasmitter. To the best of the authors kowledge, the idea of self selectio of relays i opportuistic routig was first preseted i [11] ad [12]. The cotributio preseted i [13] also uses this

5 4 idea. The relay self selectio techique proposed i the preset article has already bee described i [14] i the sectio o implemetatio issues, but the primary focus of [14] was the optimizatio of the slotted Aloha MAC i the cotext of opportuistic routig. Our relay self selectio procedure is optimized i the sese that it is the ode that optimizes some give geometric utility criterio (e.g. miimize the remaiig distace to the destiatio) which is chose as the relay. 2) MAC ad routig iterplay: Aother difficulty ivolved i opportuistic routig cosists i mergig the fuctios of two, traditioally separated, etwork layers. I particular, we have to kow whether this techique ca be used with various existig MAC solutios. The techiques preseted i [11] ad [13] assume a IEEE type MAC where the ackowledgmet scheme is modified to allow for the relay selectio. O the other had, [12], [14] assume slotted Aloha. I the preset article we show how the relay self selectio scheme ca be used with various MAC techiques: these schemes may be cotrolled access schemes such as Time Divisio Multiple Access (TDMA) or radom access schemes such as Aloha or CSMA. 3) Node positioig: As already stated, geographic routig, which is used i our self selectio procedure, requires kowledge of the odes geographic positios. Nevertheless, we will show that it is sufficiet that a small umber of odes i the etwork kow their positios exactly, e.g. usig GPS, ad provide this iformatio to the remaiig odes, for the proposed techique to work well ad outperform covetioal routig techiques. III. THE OPTIMIZED RELAY SELF SELECTION SCHEME VIA SIGNALING BURSTS I opportuistic routig with relay self selectio the trasmitter of the packet does ot kow its ext relay a priori, but there is a self selectio of this relay amog the odes that capture this trasmissio. I wireless commuicatios a simple way of electig a wier ad lettig it trasmit is to use a backoff mechaism. Suppose that the receivers of the tagged packet pick idepedetly radom times before tryig to forward it ad that the receiver with the smallest delay iitiates the trasmissio. Other packet holders hearig this trasmissio resig ad discard the packet. Implemetig this mechaism would lead to a radom choice of oe of the odes amog the curret receivers (holders) of the packet as its relay. Of course it is atural to prefer a self selectio mechaism that elects the relay i some locally optimal maer; e.g. the oe that maximizes the packet s progressio towards the destiatio or miimizes the remaiig distace to the destiatio. A. Preferetial backoff More geerally, let us assume that each receiver ca objectively evaluate its rak o some uiversal scale. 3 The problem is thus the followig: how ca we select the packet receiver with the highest rak through a distributed algorithm. The requiremet is that whe this algorithm rus o a ode, it oly kows the rak of this ode. Oe solutio is to assig backoff times accordig to the ode s rak: the higher the rak, the shorter the delay. This would make the optimal receiver the first ode that starts forwardig the packet. However, a reasoable self selectio mechaism must prevet all the other odes that participate i the selectio process from relayig it. This requires that the retrasmissio of the packet by the best relay be heard by the other potetial relays a coditio that caot be completely guarateed sice the potetial relays may be far from each other. Also the radio coditios (icludig iterferig sigals) may chage whe the backoff time has elapsed. I additio to this problem, it is uclear whether the liear selectio of a backoff techique would be sufficietly powerful to discrimiate betwee the potetial relays. I particular, if the etwork is dese, oe may have to foresee a large backoff widow i order to accommodate a large umber of potetial relays. Fially, the self selectio mechaism must ackowledge the previous trasmitter of the packet. B. The sigalig bursts I order to cope with the above requiremets we propose a more powerful techique to elect a wier, usig sigalig bursts with logarithmic codig of the rak [15]. This techique, which was first itroduced for the HiPERLAN type 1 stadard [16], assumes that after the origial packet trasmissio ad before its relayig, i the so-called active sigalig phase, each ode that has captured the packet trasmits a ackowledgmet made up of a short sigalig burst. This ackowledgmet has two goals: first it allows the best relay to be selected ad secod it allows the seder to kow that the packet has bee received ad will be relayed by some ode. The burst is composed of a sequece of itervals of the same legth i which a give receiver ca either trasmit 3 For example, to optimize the progressio towards the destiatio, the rak of a receiver ca be take equal to the abscissa of its locatio i the coordiates system origiated at the trasmitter, with the x axis poitig to the destiatio. I the other geographic criterio evoked i Sectio II-C, which aims at miimizig the remaiig distace to the destiatio, the rak could be mius the distace to the destiatio ode.

6 5 Fig. 3. Structure of the ackowledgmet packet to select the best relay towards the destiatio. or liste (see Figure 3). I order to describe the structure of the burst, let us represet it by the biary sequece, where 0 deotes a listeig iterval ad 1 deotes a trasmissio iterval. Each ode participatig i the self selectio process computes this biary sequece (ad thus determies the form of its burst) as follows. The first d bits ecode the rak of the ode i base 2 (we recall that the self selectio should desigate oe ode with the highest rak as the relay). Optioally oe may implemet the ext r bits selected at radom to discrimiate betwee odes havig (almost) the same rak. Fially the last bit is always set to 1. This bit, as we will see, provides the ackowledgmet of the successful self selectio of the relay. After havig computed the form of their bursts, the odes start trasmittig them simultaeously applyig the followig rule: if a give ode detects a sigal from aother ode durig ay of its listeig itervals, it quits the selectio process; i.e. it stops trasmittig durig the etire remaiig part of the active sigalig phase (cf. Figure 4). It ca easily be checked that if all the odes participatig i the self selectio process remai i their commuicatio regios, the at the ed of the first d bits of the burst, the oly odes (if ay) which stay i the competitio will have equal rak, the highest amog all the participatig odes. This stems from the costructio of the sigalig bursts: the detectio of a trasmissio durig a listeig iterval implies that a better relay is takig part i the competitio. A example of a relay selectio operatio is show i Figure 4, which correspods to the relay selectio aroud O as show i Figure 2. Three odes have captured the trasmissio set by O: odes A, B ad C. Node A has the highest rak ad thus is selected as the best relay usig the active sigalig scheme. The ext r bits of the burst radomly select oe of the odes with the highest rak, if there are more tha oe. Fially, this uique wier (if ay) of the self selectio process will trasmit at the last iterval of the Fig. 4. scheme Example of the best relay usig the ackowledgmet burst. Thus, if the previous trasmitter (the ode that set the packet for which the relay is to be selected) caot detect a sigal i this iterval it ifers that its packet has ot bee received or that the selectio process betwee potetial relays has failed. I this situatio, it has to retrasmit the packet. C. Some implemetatio issues Let us ow discuss the real circumstaces, i which the above ideal self selectio process may fail. Probably the most importat of these is iterferece from other trasmitters i the etwork which are ot participatig i the self selectio process. To cope with this problem, a spreadig techique ca be used: a uique (CDMA-like) biary code of much higher frequecy ca be provided i the previous trasmissio of the packet, to be used by all the receivers durig the active sigalig burst. All the odes participatig i this self-selectio will modulate their bursts (biary multiply) before trasmittig. This will protect the commuicatio i this active sigalig burst from other ogoig commuicatios. Note that we do ot suggest usig this code for the subsequet data packet retrasmissio by the elected best relay; the give MAC used will take care of it. Aother problem is how to determie d i order for the sigalig burst to be able to correctly discrimiate betwee odes. Let us assume, for example, that the rak is some geometric distace; e.g. progressio. The d = 13 will allow distaces up to about 8 km to be coded with the precisio of 1m. Whether this is a sufficiet tuig depeds o the maximal trasmissio rage i the give etwork. D. Relay self selectio ad multiple access schemes The relay self selectio techique that we have described above ca operate with various access schemes: both with cotrolled schemes (i which access is grated by the protocol i such a way that there are o collisios) ad with radom access schemes, sice the protocol

7 6 icorporates a ackowledgmet mechaism. We discuss some possible choices below. 1) TDMA: I this protocol time slots are assiged to etwork odes i such a maer that there are o collisios. It is, however, difficult to use such a scheme i ad hoc etworks sice attributig time slots i these dyamic etworks is extremely complex. 2) CSMA/CA: These protocols have bee widely used i wireless etworks. I these radom access protocols, the chael is sesed prior to ay trasmissio to be sure that it is ot used. Above a give threshold, called the carrier sese threshold, the chael is assumed to be occupied whereas below this threshold, the chael is assumed to be free. Whe a collisio occurs, a simple backoff techique is used to schedule the re-trasmissio of the packet. These CSMA/CA protocols form the basis of the IEEE stadard, which, however, adds a additioal MAC ackowledgmet set just after the ed of the received packet. The relay self selectio techique protocol proposed i this article ca operate with the CSMA/CA techique of the IEEE stadard with this ackowledgmet modified (replaced by) the active sigalig phase described i the previous sectio. 3) MACs with RTS/CTS: Our relay self selectio techique could also be adapted to other protocols such as MACAW [17], MACA/PR [18], DBTMA [19] etc., which use a Request To Sed/Clear To Sed (RTS/CTS) exchage before the actual trasmissio of the data. I these protocols, the RTS packet should be set to all eighbors ad the CTS packet should ecompass the active sigalig phase. (Thus, i this case, the relay self selectio will take place before the receptio of the packet i questio.) 4) Aloha: Our relay self selectio ca also work with o-slotted Aloha. However, it would be more beeficial i the case of a slotted Aloha protocol i which, at each time slot, each ode with packets to be set tosses a coi with a bias p (for heads) ad accesses the chael whe gettig heads. I fact, the use of a slotted structure allows the throughput of the Aloha protocol to be improved by a factor of 2 (see Chapter 4 of [20]). I additio, the slotted structure also improves the efficiecy of the relay self selectio techique. The combiatio of the slotted Aloha with the relay self selectio protocol has bee aalyzed i [12], [14], [1] ad may very iterestig properties have bee show, especially cocerig etwork scalig. IV. PERFORMANCE EVALUATION MODEL We ow describe our model for the performace evaluatio of the opportuistic routig. This model will be used for simulatios as well as for the mathematical aalysis. A. Network architecture We cosider etworks formed of odes radomly distributed o the plae. Specifically, odes are assumed to be sampled accordig to some homogeeous Poisso poit process with itesity λ. I our simulatios, we use a fiite plaar etwork o the square [0,1000] m [0,1000] m. The locatios of the odes do ot chage with time slots, but mobility is take ito accout i the radio chael model (see model M3 i Sectio IV-C below). I our simulatios, the default optio is λ = 10 3 odes /m 2. B. O-D pairs ad backgroud traffic I the simulatios O-D pairs are selected o opposite parts of the etwork, as show i Figure 5, with a distace of about 1130 m from each other. This represets a moderate distace (approx. 9 hops away for a trasmissio rage of 140 m.). For a fixed O-D pair, ad for a give set of etwork odes sampled accordig to a Poisso poit process, a basic simulatio experimet allows oe to get a sample of the ed-to-ed trasmissio of oe packet of the tagged O-D pair flow, assumig some give physical (radio), MAC ad routig model that will be described below (cf Sectios IV-C IV-F). I this ed-to-ed trasmissio we track the route selected for this packet, the trasmissio attempts at each relay ode ad the ed-toed delay. For the sake of simplicity, i the simulatios: the tagged packets of the O-D pair are treated as higher priority packets at each ode. We should of course add a queueig delay to accout for the competitio with cross traffic, but uder atural homogeeous traffic ad stability assumptios, this would amout to addig a delay with the same law at each ode, ad should hece ot chage the mai coclusios of the compariso study. all odes are assumed to always have packets to trasmit, ad they always trasmit wheever authorized by the MAC; these trasmissios allow us to take the backgroud traffic ito accout through the iterferece they create at each time slot, ad i tur, determie which odes capture the tagged packet trasmissio. We repeat a large umber of such basic experimets to evaluate meas. We cosider both packets set from O to D for the same ad for differet etwork samples. Note that eve if we track oly the packets of the tagged O-D pair, the cross-traffic is take ito accout

8 7 via the iterferece experieced by the tagged packets due to other trasmitters i each time slot. We still assume that each ode always has a packet to trasmit ad that iterferece plays a importat role i determiig a successful receptio. C. Radio chael models The power used by all the trasmitters is assumed to be equal to some costat S = 1. We use the followig simplified power atteuatio fuctio l(r) = (Ar) β for some costats A > 0 ad β > 2, which gives the fractio of the emitted power that is received at distace r from the trasmitter. Eve if this fuctio has a pole at the origi, it is reasoable ad commoly used if the desity λ of poits is ot too large or, equivaletly, the poits are ot too close to each other. I the simulatios, we take a path-loss expoet of β = 3. I certai models, i additio to the above atteuatio fuctio, we assume that the received powers are multiplicatively modified by some locatio ad possibly time depedet radom path-loss factors. The followig 3 scearios will be cosidered. (M1): Path loss factors are costat equal to 1. This assumptio might correspod to a very slow chael fadig ad/or codig which allows for empirical averagig over fadig effects durig packet trasmissio (e.g. based o symbol iterleavig). (M2): Path loss factors are positio depedet; they are sampled idepedetly for each trasmitter-receiver pair ad stay costat for all time slots of the simulatio. This correspods to a slow fadig or shadowig effect. (M3): Path loss factors are positio ad time depedet; they are sampled idepedetly for each time slot ad each trasmitter-receiver pair. This might correspod to user mobility ad will be the default optio i the simulatios below. For models M2 ad M3, we assume a Rayleigh fadig, where path-loss factors are expoetial radom variables with parameter 1 (see e.g. [21, p. 50 ad 501]). The model also icludes a thermal oise idepedet of everythig else with power deoted by W. I the simulatios, the default optio is W = 0. D. Capture model Let us suppose that some statio trasmits durig a give time slot. We assume that it ca successfully trasmit to a give receiver of this time slot if the SINR ratio at this receiver is ot less tha some fixed threshold T. By the SINR we mea the ratio betwee the power received from the give trasmitter (atteuated ad modified by the path-loss factor) ad the sum of powers received from all other trasmitters of the give time slot, icludig the power W of the thermal oise; see (6.1) for the correspodig formula. I the simulatios, the default value is T = 10. E. MAC models We will oly cosider slotted MAC scearios; i.e. the time is divided ito equal slots. I each slot, the first part is dedicated to the trasmissio of the data set by the iitial source or repeated by the itermediate odes, ad the secod part of the slot is dedicated to the ackowledgmet packet which is set by potetial relays to elect the best relay ad to ackowledge the receptio of the packet; cf. Figure 3. However, oly the receptio of the data part i each time slot will be simulated ad the odes havig successfully received the data will be idetified (cf. Sectio IV-D). I this study we assume that the self-selectio procedure perfectly desigates the best (accordig to a give criterio) relay ode amog them. 1) Aloha: The first simulatios preseted i this paper assume a slotted Aloha. I this model, at each time slot each ode tosses a coi idepedetly of everythig else. The odes tossig heads are the trasmitters of this time slot; the other odes are the receivers. This model will also be the basis of the mathematical aalysis based o stochastic geometry. The mai parameter of this model is the probability of tossig heads, deoted by p, which is referred to as the medium access probability. Nodes which are ot authorized to trasmit at a give time slot are cosidered as potetial receivers at this time slot. 2) CSMA/CA: Next, we preset simulatios usig a CMSA/CA protocol. To simplify the simulatio, we simulate a slotted CSMA/CA system meaig that at each time slot, trasmittig odes are selected amog the odes with a pedig packet accordig to the CSMA/CA rule. More precisely, i each time slot the odes with a pedig packet try to access the chael i a radom order ad succeed oly if they satisfy the CSMA/CA rule; i.e. if the detected sigal is below the carrier sese threshold 4. This carrier sese threshold is thus the key parameter of the CSMA/CA simulatio. Nodes which are ot authorized to trasmit at a give time slot are cosidered as potetial receivers at this time slot. I this simple model we also eglect the collisio widow i which trasmitters ca start their trasmissios without sesig each other. 4 I [22] it is show that this model is a good approximatio of a real CSMA whe the packets are of the same legth ad if we also cosider the overhead iduced by the backoff algorithm.

9 8 F. Routig We ow describe two routig strategies: covetioal shortest-path ad opportuistic routig where the latter aims at miimizig the remaiig distace to the destiatio at each hop. 1) Shortest path routig: By this we uderstad routig alog the routes with the least umber of hops as foud by Dijkstra s algorithm [4]. For each give etwork, this amouts to fidig paths of miimal weight betwee a give origi O ad destiatio D (cf. Sectio IV-B) i a graph with edges betwee all pairs of odes ad where the weight of the edge betwee odes x ad y is 1 if x y R ad otherwise, where R is the maximum trasmissio rage ad is cosidered as a parameter of this routig protocol. This shortest path is used to route all packets of this O-D pair. A give MAC scheme (Aloha or CSMA/CA) is the used to let the tagged packets progress from O to D alog this path Time-space opportuistic radial path, M3 Time-space opportuistic radial path, M1 Shortest path Nodes 2) Opportuistic radial routig: It should be recalled that i opportuistic routig, the ext hop o the route to the destiatio is ot kow a priori ad the routig algorithm should be described together with the MAC. Cosider a tagged packet of the O-D pair flow located at some curret ode A. Util A is the destiatio D do: 1. Util A is selected by the MAC to trasmit, ed-to-ed delay++; 2. Whe A is selected by the MAC to trasmit do: 2.1. All the odes which are selected by the MAC to trasmit are trasmitters, the remaiig odes are receivers; 2.2. The set of trasmitters together with the fadig variables at that time slot determie the iterferece everywhere at this time slot; 2.3. The set of receivers S which satisfy the SINR capture coditio at this time slot receive the tagged packet successfully; 2.4. Amog the odes of S {A}, the earest to the destiatio, say B, is the ext relay; 2.5. The other odes of S discard the tagged packet; 2.6. ed-to-ed delay++; 2.7. if A B the umber-of-hops++; 3. A := B. A more formal descriptio of this routig protocol, as well as a proof of its covergece (the fact that it delivers Fig. 5. Samples of routig paths with opportuistic radial routig (with ad without fadig) ad with a shortest path algorithm (for Aloha MAC). the packet to the destiatio i a fiite umber of time slots) is preseted i Sectio VI uder the Aloha MAC assumptio. Figure 5 gives three examples of radial paths obtaied by simulatio for differet radio chael models. The path that is the closest to the segmet joiig the origi to the destiatio ode is obtaied with a shortest path routig algorithm. The secod path movig farther away from this segmet correspods to the time-space opportuistic radial routig strategy uder the M1 model. The third path, which allows oe to search for relays very far from the trasmitter correspods to time-space opportuistic radial routig i the presece of fadig (here uder the M3 assumptios). G. Nodes positioig 1) Perfect positioig: Note that opportuistic routig requires that the odes kow their geographical positios. I our simulatios we will assume first that all the odes have perfect kowledge of their positios. The odes ca acquire such kowledge usig, e.g. GPS. Uder this assumptio we will compare opportuistic routig with optimized self selectio of relays to covetioal routig based o the shortest path algorithm. 2) A simple localizatio algorithm: As the assumptio that each etwork ode kows its exact positio may be cosidered too demadig i practice, we will also study the performace of our relay self selectio algorithm with a weaker assumptio, amely, that oly a fractio of the etwork odes have perfect kowledge

10 R=140 R=160 R=180 R= Delay Trasmissio probability Fig. 7. Shortest path routig algorithm: ed-to-ed delay versus p for various trasmissio rages time-space opportuistic radial routig, model M1 time-space opportuistic radial routig, model M2 time-space opportuistic radial routig, model M3 shortest path routig Delay Fig. 6. Samples of routig paths with shortest path ad opportuistic radial routig obtaied uder differet assumptios for ode positioig assumptios (for slotted Aloha MAC). of their positios (e.g. are equipped with GPS). The other odes will use these so-called auto-localized odes as achors to estimate their ow positios. A simple localizatio algorithm ca approximate the positio of a ode as the baryceter of its auto-localized eighbors (see [23]). Three examples of paths obtaied by simulatio are give i Figure 6. The path which is the closest to the segmet joiig the origi to the destiatio ode is obtaied with a shortest path routig algorithm combied with the slotted Aloha as the access scheme. The secod path movig farther away from this segmet correspods to the relay self selectio mechaism usig the slotted Aloha. I this path, all the etwork odes have perfect kowledge of their positios. The third path, which is a path o the right of the direct lie betwee the source ode ad the destiatio ode, correspods to the relay self selectio mechaism usig the slotted Aloha but with oly 10% of the etwork odes havig perfect kowledge of their positio. The other odes compute their positios usig the simple localizatio algorithm; see [23]. I this case, at first glace oe might assume that the relay self selectio would ot perform as well as shortest path routig. However, as we will see, this is ot the case. We have to bear i mid that the importat metric is the ed-to-ed delay betwee O ad D ad ot the umber of hops Trasmissio probability Fig. 8. Compariso betwee routig strategies: ed-to-ed delay versus trasmissio probability p. H. Performace metrics For a give tagged packet of the O-D pair ad a give etwork of odes we cosider: the ed-to-ed delay, defied as the umber of time slots it takes for this packet to go from O to D, the umber of hops made by this packet from O to D, the average local delay (delay per hop) defied as the ratio ed-to-ed-delay/umber of hops. I. Averagig ad Cofidece Itervals I order to calculate the meas of the above performace characteristics, we average over 80 differet etworks coectig a give O-D pair ad for each etwork we average over 5 packets for the O-D pair. The results are always preseted with cofidece itervals correspodig to a cofidece level of 95%. Note that some of these cofidece itervals are small ad ca oly be see whe zoomig i o the correspodig plots. A. Aloha V. SIMULATION RESULTS 1) Mea Ed-to-Ed Delay: For shortest path routig, the maximum trasmissio rage parameter R (recall

11 10 from Sectio IV-F1 that this is a parameter of Dijkstra s algorithm) has first bee optimized i order to make the compariso fair. The ed-to-ed delays for various values of R ad of the trasmissio probability p are preseted i Figure 7. We see that the best delay is obtaied with p = ad with R = 140 m. This value, which is our default value for shortest path routig i what follows, is actually the smallest value of the trasmissio rage which coects the etwork with high probability i this case. I Figure 8, we compare the shortest path algorithm ad our time-space opportuistic routig. I this figure we give the mea ed-to-ed delay as a fuctio of the trasmissio probability p uder differet fadig scearios. Here is the mai observatio of the paper. Observatio 5.1: The algorithm based o time-space diversity sigificatly outperforms the covetioal shortest path routig strategy: the average delay of a packet is at least two ad a half times smaller for this strategy tha for Dijkstra s algorithm. We also see that the discrepacy betwee the covetioal shortest path routig strategy ad time-space opportuistic routig becomes much larger for a large p. Moreover, the performace of opportuistic routig is much less sesitive to a suboptimal choice of the parameter p. Figure 9, which refies Figure 8 for opportuistic routig strategies, shows that: Observatio 5.2: Lettig time-space opportuistic routig take advatage of the varyig fadig (e.g. due to mobility) is beeficial i terms of mea ed-to-ed delays. The aalysis of the simulatio results shows that opportuistic routig i the presece of fadig (M2 ad M3) performs roughly four times better i terms of ed-toed delay tha opportuistic routig i the absece of fadig (M1), see Figure 9. Opportuistic routig with slow fadig (M2) or with fast fadig (M3) offers similar performace. Oly very log simulatios (ot preseted here) show that opportuistic routig i M3 leads to slightly shorter delays tha i M2. Here is the secod most importat observatio of this paper. Figure 10, which plots the mea ed-to-ed delay for the M3 time-space opportuistic routig, shows that: Observatio 5.3: There is a optimal value of p that miimizes the mea ed-to-ed delay of the time-space opportuistic routig algorithm, ad that this optimal value p seems to be the same for all values of the ode desity λ. Similar observatios (ot preseted here) hold for the M1 M2 models described i Sectio IV-C. Delay time-space opportuistic radial routig, model M1 time-space opportuistic radial routig, model M2 time-space opportuistic radial routig, model M Trasmissio probability Fig. 9. Effect of fadig o time-space opportuistic radial routig: ed-to-ed delay versus trasmissio probability p. Delay time-space opportuistic routig, model M3 \lambda= \lambda=0.001 \lambda=0.015 \lambda= Trasmissio probability Fig. 10. Time-space opportuistic radial routig; ed-to-ed delay versus p for various values of the ode desity. I Figure 11 we see that : Observatio 5.4: The mea ed-to-ed delay of the time-space opportuistic routig algorithm is of the order of λ where λ is the ode desity. The matchig is excellet for opportuistic routig i M3 (ad i M2 although it is ot show i Figure 11), for opportuistic routig i M1 there is rough matchig. The discrepacy see for small λ may be caused by side effects. 2) Mea Number of Hops, Mea Local Delays: Figure 12 gives the average umber of hops to reach the destiatio for the two routig strategies with p varyig from to Observatio 5.5: I the case without fadig M1, for small values of p, the time-space opportuistic path is shorter (has a smaller mea umber of hops) tha the Dijkstra shortest path, whereas it is loger for large values of p. I the presece of fadig, time-space opportuistic routig offers shorter paths tha Dijkstra type routig for p ad slightly larger paths tha Dijkstra type routig for p > We also observe that for time-space opportuistic routig, the mea umber of hops to reach the destiatio icreases with p. This ca be easily uderstood sice whe p icreases, the time-space diversity decreases ad

12 time-space oppportuistic routig, model M1 time-space oppportuistic routig, model M3 square root approximatio for M1 square root approximatio for M3 % GPS 20% 10% 7% 5% 4% 3% 2% 1% % Delay exc 7.7% 6.1% 5.8% 4.3% 4.8% 7.7% 13.8% 39% Fig. 14. Percetage of packets with delay exceeded versus percetage of GPS odes (for slotted Aloha) Delay Node desity lambda Fig. 11. Ed to ed delay of time-space opportuistic radial routig algorithms (with ad without fadig) versus ode desity λ, compariso with λ. Number of hops time space oppportuistic radial routig, model M1 time space oppportuistic radial routig, model M3 shortest path routig Trasmissio probability Fig. 12. Mea umber of hops from origi to destiatio with timespace opportuistic radial routig (with ad without fadig) ad with a shortest path routig algorithm. Local delay time space oppportuistic radial routig, model M1 time space oppportuistic radial routig, model M3 shortest path routig Trasmissio probability Fig. 13. Mea local delay for time-space opportuistic radial routig (with ad without fadig) ad with a shortest path routig algorithm. thus the umber of hops to reach the destiatio teds to icrease. Figure 13 studies the mea local delay for the same three scearios as above. Observatio 5.6: I time-space opportuistic routig, for each p, the mea delay per hop is much smaller tha the delay per hop for Dijkstra s algorithm. This explais why the average delay is smaller for timespace opportuistic routig tha for Dijkstra s algorithm eve if the umber of hops may be larger. 3) Impact of imperfect ode positioig: Figure 8 provides the mea ed-to-ed delay of the relay self selectio algorithm whe oly 10% of the etwork odes have GPS iformatio. Figure 8 actually shows that the performace of the optimized relay self selectio is ot sigificatly affected by the lack of precisio iduced by the simple localizatio algorithm whe p is less tha For higher values, usig the localizatio algorithm leads to a icrease i the ed-to-ed delay ad to sigificat packet loss due to a excessive delay (5 % of packets with a delay exceedig ). However, if we use the localizatio algorithm ad if we maitai a small value of p, the ed-to-ed delay will be approximately 3000 whereas the mea delay is 2500 for the relay self selectio with p = Eve with 10% of GPS odes, the relay self selectio mechaism sigificatly outperforms the shortest path routig scheme i terms of the ed-to-ed delay. Last but ot least, we ca also observe i Figure 8 that the tuig of the Aloha parameter p is much easier with relay self selectio tha with the shortest path algorithm. I Figure 14, whe p = 0.01, we preset the percetage of packets ed-to-ed delay of which exceeds slots for various percetages of odes havig GPS iformatio. We see that whe more tha 3% of the odes have GPS iformatio, less tha 10% of the packets have delays exceedig The performace of the relay self selectio mechaism seems to be quite isesitive to the percetage of odes havig GPS iformatio ad thus to a lack of precisio i kowledge of the ode positios. 4) Impact of the path-loss expoet: Figure 15 shows the gai i terms of mea ed-to-ed delay of the relay self selectio algorithm over covetioal shortest path routig, as a fuctio of the path-loss expoet. I order to perform a fair compariso of both techiques, we chose the p which optimizes the mea ed-to-ed delay. We observe that this gai varies from 2.8 for β = 3 to early 4 for β = 5. I this study we have ot cosidered the effect of fadig. However it is easy to take such a effect ito accout with simple models which are ot preseted i this paper for reasos of space. I these fadig-aware models aother importat additioal gai (up to 4) ca be obtaied. Actually, the fadig effect adds spatial diversity which is used by the relay self selectio mechaism to improve its performace.

13 Delay improvemet with self relay electio versus shortest path ratio of the mea delay, model M Ed-to-ed delay versus ode desity self relay electio with CSMA, model M1 self relay electio with Aloha, model M Delay improvemet Delay Decay factor Node desity Fig. 15. Gai i average delay with relay self selectio versus shortest path versus path loss expoet β (for slotted Aloha). Fig. 17. Compariso of the mea ed-to-ed delay versus ode desity of the self selectio mechaism used with a CSMA/CA protocol. Delay Ed-to-ed delay versus carrier sese threshold self relay selectio with CSMA, model M1 shortest path with CSMA e-08 1e e-07 2e e-07 Carrier sese threshold Fig. 16. Mea ed-to-ed delay versus the defer threshold of the CSMA access protocol used as access scheme, compariso betwee the relay self selectio ad the shortest path routig algorithms. B. CSMA/CA We compare the traditioal routig strategy based o a shortest path Dijkstra algorithm ad the routig with self relay selectio described i Sectio III. Both routig schemes use the slotted CSMA protocol as the access scheme. We cosider the ed-to-ed delay of these two schemes, ad the results of these simulatios are preseted i Figure 16. It ca be see that, here agai, the relay self selectio mechaism sigificatly outperforms the shortest path algorithm. I terms of edto-ed access delay, the obtaied gai is aroud 1.5. As the Aloha case, we ca observe that the tuig of the carrier sese threshold is much easier with the relay self selectio tha with the shortest path algorithm. I Figure 17 we compare the ed-to-ed delay of the relay self selectio mechaism used with Aloha ad CSMA/CA for various values of the ode desity. Both protocols are optimized w.r.t. the trasmissio probability p for Aloha ad w.r.t. the carrier sese threshold for CSMA. We ote that CSMA/CA offers much smaller ed-to-ed delays, the gai is aroud a factor 0.5, early 0.45 for a ode desity of The gai i performace of CSMA over Aloha is cofirmed i Figure 18 where we ca see that the actual mea umber of hops to reach the destiatio is smaller for CSMA tha for Aloha. Sice CSMA offers a better exclusio area aroud the trasmitter the packet ca go farther towards the destiatio. Noetheless the improvemet i performace of CSMA show i Figures 17 ad 18 comes at the price of a optimizatio w.r.t. the carrier sese threshold which is ot idepedet of the ode desity. This is show i Figure 19. The optimizatios for λ = , λ = ad λ = lead to very differet values of the carrier sese threshold. This is i cotrast to the Observatio 5.3, which says that the optimal Aloha MAP p does ot deped o the desity of odes. The last remark is i lie with a remark made i [14] where we showed that CSMA with a fixed carrier sese threshold offers a maximum throughput 0(1). This throughput does ot scale with λ whereas for Aloha scheme it scales as 0( λ); i.e. accordig to Gupta ad Kumar s well kow law [24]. VI. MATHEMATICAL ANALYSIS I this sectio we will study opportuistic routig uder the Aloha MAC assumptio usig the theory of poit processes. I particular we will show how to optimally tue the MAC parameters so as to miimize the average umber of time slots required to carry a typical packet from origi to destiatio o log paths. We show that this optimizatio is idepedet of the etwork desity.