On the Forwarding Area of Contention-Based Geographic Forwarding for Ad Hoc and Sensor Networks

Transcription

1 On the Fowading Aea of Contention-Based Geogaphic Fowading fo Ad Hoc and Senso Netwoks Dazhi Chen Depatment of EECS Syacuse Univesity Syacuse, NY Jing Deng Depatment of CS Univesity of New Oleans New Oleans, LA Pamod K. Vashney Depatment of EECS Syacuse Univesity Syacuse, NY Abstact Contention-based Geogaphic Fowading ) is a state-fee communication paadigm fo infomation delivey in multihop ad hoc and senso netwoks. A pioi selection of the fowading aea impacts its oveall netwok pefomance and the design of the potocol as well. In this wok, we study the fundamental poblem of defining the fowading aea apioifo and detemine its impact on the pefomance. We model without void i.e., absence of a next-hop node in the fowading aea) handling as a -step fowading stategy. Based on this model and given a andom distibution of netwok nodes, we develop a geneal mathematical analysis technique to evaluate the pefomance of with diffeent fowading aeas, in tems of the pefomance metic aveage single-hop packet pogess. Futhe, we intoduce two state-fee void handling schemes, i.e., active exploation and passive paticipation, fo and study thei pefomance in depth. Ou theoetical analysis and numeically evaluated esults, validated by extensive simulations, povide a guideline egading the selection of specific fowading aeas fo the design of a pactical potocol. It also seves as a geneal pefomance evaluation famewok fo the existing potocols. I. INTRODUCTION In ecent yeas, a communication paadigm, based on the distibuted selection of a next-hop node via contention, has been poposed fo infomation delivey in multi-hop wieless ad hoc and senso netwoks [] []. In this paadigm, the geogaphic location infomation of the nodes is assumed available, eithe by apioiconfiguation o by the use of the Global Positioning System GPS) eceive o though self-configuing localization mechanisms such as [], []. We call this paadigm Contention-based Geogaphic Fowading ). Diffeent fom the topology-based communication paadigm, which equies a outing potocol such as Dynamic Souce Routing DSR) [] to collect the topology infomation, takes advantage of the geogaphic infomation to allow the data packets to gadually appoach and eventually aive at the intended destination. Compaed with the taditional geogaphic communication paadigm, allows next-hop candidate nodes to contend fo the data fowading task and a specific next-hop node is decided at the tansmission time. The contention mechanism eliminates the need to peiodically tansmit beacons, which ae used to update the location infomation of neighbos, and the need to select a next-hop node in advance at the sende. Thus, nodes employing only need to know thei own locations and that of the packet destination in ode to communicate with each othe. Futhemoe, the centalized selection of a next-hop node at the sende is eplaced by the distibuted selection of a next-hop node among the potential eceives. Since equies neithe the topology infomation no the neighbo location infomation, it is egaded as a state-fee communication paadigm. Its state-fee natue makes obust to high netwok dynamics and fequent topology changes, scalable to lage-scale node deployment, adaptable to dense node densities, and applicable to data-centic applications and esouceconstained netwoks. Futhemoe, contention among the potential eceives does not equie significantly moe A souce node can obtain the location infomation of a data packet destination eithe fom a location sevice such as geogaphical location sevice GLS) [] o via apioiconfiguation. This infomation is encoded into the data packet heade. Any intemediate nodes can diectly access it late. ---//$. C) IEEE

2 ovehead than contention esolution at the MAC laye. Due to these benefits, has ecently attacted a significant inteest fom the eseach community as a favoed candidate fo infomation delivey in Vehicula Ad Hoc Netwoks VANETs) [], Mobile Ad Hoc Netwoks MANETs) [], Wieless Senso Netwoks WSNs) [], [], [], and Wieless Senso and Actuato Netwoks WSANs). While diffeent potocols adopt diffeent appoaches [] [], the basic idea emains the same given as follows: i) a pedefined fowading aea and nodes that eside in the aea become next-hop candidate nodes and contend with one anothe fo the data fowading task; ii) a distibuted contention abitation and esolution scheme in ode to effectively establish a single nexthop node in the fowading aea; iii) the establishment of next-hop node selection citeia in ode to attain the desied netwok pefomance in an efficient way; iv) an effective mechanism to handle a void. Except the GeRaF potocol that uses an aea-based and sende-assisted contention abitation and esolution mechanism [], all othe potocols employ a timebased and eceive-ovehead contention abitation and esolution mechanism [], [] []. One of the basic pocesses in the schemes poposed in [], [] [] is the following: fist, the sende boadcasts an RTS packet to all its neighbos. Second, the neighbos esiding in the fowading aea independently set up times in ode to contend fo the data fowading task. A candidate node that has the ealiest timeout is abitated to win the contention. Thid, the winning node etuns a CTS packet to establish itself as the only next-hop node, othe nodes in the fowading aea that ovehea the CTS packet dop out of the contention. Finally, a nomal DATA and ACK exchange between two specific nodes follows. The above pocess can be easily integated into a contention-based MAC potocol such as the IEEE. Distibuted Coodinato Function DCF) []. Such coss-laye design educes the potocol ovehead and the data fowading delay [], []. A fundamental poblem in is the definition of the fowading aea, since it influences how much pogess a packet makes towads a destination on a single hop and theeby has a significant impact on the oveall netwok pefomance of. The fowading aea also affects the design of the contention esolution mechanism and The absence of a next-hop node in the fowading aea. This poblem is also temed as holes o local minimum phenomenon in the liteatue []. the possibility of having a void, the event that no nexthop candidate nodes ae pesent in the fowading aea. In this wok, we appoach the poblem of defining the fowading aea fo analytically and though extensive simulations. We concentate on in wieless ad hoc netwoks with andomly distibuted nodes, whee any pai of nodes may communicate with each othe, eithe diectly o though multiple hops. To the best of ou knowledge, ou wok is the fist study of fom this pespective. The main contibutions of this wok include: A high-level model of without void handling is established as a -step fowading stategy. This model can be used to analyze diffeent geogaphybased potocols fom a numbe of pespectives such as evaluation of multi-hop pefomance and enegy efficiency; A geneal mathematical analysis famewok, in tems of aveage single-hop packet pogess, is developed fo the pefomance evaluation of with most egula fowading aeas of inteest; The conclusion that, maximum fowading aea, at the cost of moe complex potocol design and lowe spatial euse, always pefoms the best. Also, a -degee adian aea begins to achieve the same pefomance as maximum fowading aea with an aveage numbe of neighbos equal to while maximum communication aea equies much moe than neighbos on an aveage to do so. This esult is validated by extensive simulations and is useful fo the design of a pactical potocol; The intoduction of two state-fee void handling schemes fo and thei pefomance evaluation via extensive simulations. The est of the pape is oganized as follows. In Section II, we define and discuss thee typical fowading aeas fo that have been adopted in the liteatue [] []. A high-level abstaction of as a fowading stategy and a theoetical analysis fo the pefomance of ae pesented in Section III. Section IV povides numeical and simulation esults along with some discussion. In Section V, we intoduce two state-fee void handling schemes and investigate thei pefomance. We conclude ou wok in Section VI. While, in the est of this pape, we focus ou attention on wieless ad hoc netwoks, the esults obtained ae also applicable to wieless senso netwoks, whee the infomation sink may be at any location in the egion of inteest. Definitions fo these fowading aeas will be given in Section II.

3 Fig.. S O P o E M W Thee Typical Fowading Aeas fo II. FORWARDING AREAS IN A. Fowading Aeas With a common equiement among all fowading aeas that a packet should make a positive pogess towad the destination at evey hop and an assumption that each node has a tansmission ange in the fom of a cicle, we define thee typical fowading aeas, as shown in Fig., fo as follows: Maximum Fowading Aea ): is the ovelap egion of two cicula aeas: the tansmission cicle of the sende and the cicle that is centeed at the destination with a adius equal to the distance between the sende and the destination. It is the egion OSPW shown in Fig.. The size of this aea is vaiable and dependent on the distance between the sende and the destination. Note that this fowading aea has been adopted in the GeRaF [] and SIF [] potocols. Maximum Communication Aea ): is defined as the lagest egion within which any pai of nodes can hea each othe []. Thus, it is a cicle with a diamete equal to the tansmission ange of a node. It is the cicle with SW as its diamete shown in Fig.. The beauty of is that, as suggested by its name, this is the maximum aea within which evey pai of nodes can hea each othe, eliminating potential CTS esponse collisions unless the esponses ae sent at the same time. Note that SW should be co-linea with SD in ode to maximize the possible aea within the cicle. The size of is a constant no matte whee the destination is located. D -Degee Radian Aea ): is a adial egion that includes a -degee adian aea aound the line connecting the sende and the destination on both sides. It is the egion SEWM shown in Fig.. Note that SD is the angle bisecto fo this egion. Thesizeofisalsoaconstant.wasused in the IGF potocol []. As we mentioned ealie, a time-based and eceiveovehead potocol equies that any pai of nodes in a fowading aea should be able to hea each othe in ode to avoid multiple esponses. The and aeas satisfy such a equiement as all points in these egions ae within the communication ange of each othe. The aea, howeve, may have two nodes that ae out of thei communication anges. Thus, it is possible fo two o moe next-hop candidate nodes to collide duing the contention pocess. This issue was addessed by a technique that exploits the diffeence between caie sensing ange and tansmission ange in []. We do not conside the collisions fom the nexthop candidate nodes and the associated additional cost fo in this wok. Since the size of is lage than those of and, the pobability of void in is smallest among the thee aeas, based on the assumption of andom deployment of netwok nodes. These pobabilities ae calculated in the following subsection. B. Pobability of void We model the node distibution in the egion of inteest as a two-dimensional Poisson point pocess. Thus, the pobability that k nodes ae located within an aea of size A is given by []: P {k} λa)k e λa, ) k! whee λ is the expected numbe of nodes within a unit aea. Since the fowading aeas of and ae, π / and π /, espectively, the pobabilities of void P {k }) fo and ae P {void in } e ρ, ) P {void in } e ρ, ) whee ρ is the aveage numbe of neighbos within the tansmission ange of the sende and is given by ρ π λ. ) Note that, such a technique equies a moe complex potocol design and educes the spatial euse at the MAC laye as well.

4 Pobability of Void uppe bound) Aveage Numbe of Neighbos, ρ Fig.. Pobability of Void fo Fowading Aeas The calculation of the pobability of void fo the aea is moe complicated. Howeve, an uppe bound has been deived in [] as P {void in } e π )ρ e ρ.. ) The esults fom ), ), and ) ae plotted in Fig.. Fom the figue, it is obseved that the uppe bound on the pobability of void of is lowe than the pobability of void of and. Note that, when ρ is highe than, pobabilities of void fo all thee aeas ae close to zeo. This suggests that at least one next-hop node will be found in all thee fowading aeas. Although has a lowe pobability of void than and, a moe inteesting and challenging issue is: does that employs pefom bette than that uses o, in a andomly deployed netwok? What metics can be used to evaluate them? In the next section, we will popose a pefomance metic and pesent a theoetical analysis. III. PERFORMANCE ANALYSIS A. Pefomance Metic In ode to evaluate diffeent potocols, we popose to use the metic aveage single-hop packet pogess, which is defined as the expected value of the diffeence between the befoe-hop distance between the sende and the destination) and the afte-hop distance between the next-hop node and the destination). Such The definition of pogess in this pape is the same as that of advance in []. Howeve, note that pogess is oiginally defined, in the liteatue [], as the othogonal pojection of the line connecting the sende and the next-hop node onto the line connecting the sende and the destination. It diffes fom ou definition. a metic eflects the pogess towad the intended destination at each hop and has a diect connection with the end-to-end packet delay pefomance of in the long tem. In the following subsections, we will deive aveage single-hop packet pogess fo andomly deployed netwoks that employ with the,, and fowading aeas, espectively. B. Netwok Assumptions and Abstact Stategy We assume that a homogeneous wieless ad hoc netwok is deployed with andomly distibuted nodes, whee all nodes have the common tansmission ange in the fom of a cicle of adius. Nodes adopt the potocol to communicate with each othe and the fowading aea in is specified apioi. The node distibution can be modeled as in Equation ) and the appeaance of nodes in any two non-ovelapping aeas is independent. Due to the limited adio ange, a data packet has to be fowaded by a cetain numbe of hops via intemediate nodes befoe eaching its destination. Fo any single hop, a node within the fowading aea of a sende is called a next-hop candidate node fo the sende. We assume a pefect contention abitation and esolution scheme fo, which makes a candidate node closest to the destination in the fowading aea always win the contention and become the only next-hop node fo the sende. We also assume the availability of a collision-fee media access scheme and that each node can detemine its own location and the location of packet destination. Futhe, we conside the snapshot at the time of the fist-hop packet fowading, even if a multihop fowading is equied fo most data packets to aive at thei destinations. Given a andom node deployment, any intemediate-hop fowading can be viewed as a new fist-hop packet fowading. Thus, the fist-hop aveage packet pogess is close to the oveall aveage singlehop packet pogess in a homogeneous envionment. We assume that a souce node, S, is located at the cente of a cicle of adius x, wheex is temed the netwok ange and is the lagest possible distance between S and any destination. In othe wods, the souce node will not send any packet to nodes outside the cicle. The destination node D, to which S intends to delive a data packet, is assumed to be andomly distibuted ove the entie cicle. Note that othe citeia fo a distibuted selection of a next-hop node may be used, such as enegy [], [] and link eliability. In this wok, we focus on a distance-only citeion.

5 Based on the above assumptions, can be modeled as a -step fowading stategy given below: i) If the destination node D is located within distance fom the souce node S, D always wins the contention and the souce node S delives a data packet to D diectly. ii) If the destination node D is outside the tansmission ange of the souce node S, the data packet is always fowaded by a next-hop node that is closest to the destination D in the fowading aea. iii) If thee ae no next-hop candidate nodes in the fowading aea, the souce node S does not fowad the data packet and discads it afte a pe-detemined peiod of time duing which no candidate nodes appea in the fowading aea. Fig.. A A S v D n F Illustation of Maximum Fowading Aea ) C. Aveage Single-hop Packet Pogess Fist of all, we deive a geneal expession fo aveage fist-hop packet pogess, which has been assumed to be appoximately equal to the aveage single-hop packet pogess unde ou assumptions, fo. Let v denote the distance between the souce node S and the destination node D. Whenv, diect data delivey to the destination node D can be completed, accoding to step i) in the fowading stategy; hence, the data packet pogess towad the destination node D is v. Ifv >,an appopiate i.e., neaest to the destination in this context, accoding to step ii) in the fowading stategy) nexthop node in its fowading aea is selected fo futhe packet fowading. The data packet pogess towad the destination node D is theeby equal to v n), whee n is the distance between the next-hop node F and the destination node D coesponding to the fist hop. Denote by P the andom vaiable coesponding to the data packet pogess in a single hop. Let V and N, espectively, be the andom vaiables coesponding to v and n discussed above. I is an index andom vaiable defined as below: {, at least one node in the fowading aea; I, othewise. With the above notations, the poblem of finding the aveage single-hop packet pogess is equivalent to the In this wok, we fist conside without void handling since void handling complicates and hides the pefomance of a potocol. We will investigate void handling schemes in Section V. deivation of the expected value of P,whee V, if V ; P V N, if V > I ;, if V > I. Note that, if V > I ) is tue, no packet fowading will occu. In this case, the actual packet pogess is zeo. It can, theefoe, be veified that ) E[P ]E[P V ) V > I )] P {V ) V > I )}. ) We now poceed to deive the expectation of P given that V ) V > I )is tue. Fom [], we have: P {P >p V ) V > I )} P {p <V } +P{V N >p V > I }. P {V } +P{V > I } ) We theefoe need to detemine fou pobability expessions in ). Among them, P {p <V } and P{V } can be easily detemined below: P {p <V } p ) x U p), ) P{V } x, ) The deivation of this conditional expectation fom ) to ) is simila to that given in [], whee the authos used such a pogess to deive an optimum node tansmission ange fo ad hoc netwoks. In this wok, we futhe develop it as a pat of ou genealized evaluation famewok fo, independent of any fowading aeas.

6 whee U function is defined as: {, if p; U p), if <p. It emains to detemine P {V > I } and P {V N >p V > I }. Next, we poceed to detemine P {V > I }. Let A SD denote a geneal fowading aea. Futhe, we can divide A SD into two egions by the cicle centeed at D with adius n. The aeas of these two egions ae espectively denoted as A and A, as the shaded egions showninfig.,fig.,andfig.. P{V > I } x x P{V > I V v}dp V v) P{at least one node in A SD }dp V v) ) e λasd ) v x dv x x ve λasd dv, ) whee the lowe integation limit is eplaced with in ) due to V >. So fa, we have completed the detemination of P{V > I }. Futhe, obseve that: P{N n I V v} P{I V v}, if n v ; P{Q}, if v <n<v;, if n v, whee Q epesents the event in which no neighbos exist in A and at least one exists in A. By the assumption of independence of node appeaance in non-ovelapping egions, the above expession fo v < n < v can be futhe deived as: P{Q} P{no neighbos in A and at least one in A } P{no neighbos in A } P{at least one in A } e λa e ρa ) e λa e λasd. Asaesult, P {N <n I V v} P{I V v} P {N n I V v}, if n v ; e λa, if v <n<v; ) e λasd, if n v, whee P {I V v} e λasd has been used. Using ), we obtain: P {V N >p V > I } P {N <v p I V v} dp V v) x P {N <v p I V v} v x dv x ) x ve λa dv U p). ) This detemines P {V N >p V > I }. Note that n in ) has been substituted by v p duing the above deivation. Thus, any n associated with the expession of the A aea should be eplaced with v p wheneve necessay late. Finally, substituting ), ), ), and ) into ), we have, fo p>, P {P >p V ) V > I )} x p ve λa dv U p). x ve λasd dv The expected value of P given the validity of V ) V > I )is then equal to [, Eq..)]: E [ P V ) V > I ) ] P {P >p V ) V > I )} dp [ ] x p ve λa dv dp x x x ve λasd dv ve λa dvdp ). ) ve λasd dv Substituting ), ), and ) into ), we attain a geneal expession fo aveage single-hop data packet pogess, independent of any fowading aeas, fo netwoks with below: x ve λa dvdp E[P ] x, ) whee we have used P {V ) V > I )} P{V } +P{V > I }

7 In the case of, A ) is A ) { A ) if <p<c ; A ) if c <p<, Fig.. S A F A v n Illustation of Maximum Communication Aea ) D whee c v v ) + ) and n A )n cos +v ) ) ) nv ) n n sin cos +v ) )) ) nv ) + π π cos ) +v ) n )) v ) + sin π cos ) +v ) n ))) v ), Fig.. S A F A v n Illustation of -Degee Radian Aea ) since these two events ae mutually exclusive. Now we poceed to deive the aveage single-hop pogess of thee diffeent fowading aeas: as shown in Fig., as shown Fig., and as shown in Fig., since A is diffeent fo, and. In the case of, A ) is A ) cos + v n ) D v +n cos n + v vn + v + n)v + n ) + v n) + n v). Thus, aveage single-hop packet pogess fo can be finally detemined as: [ ] E P x whee ρ is given by ). ve ρa ) π dvdp ) x. ) n A )n cos +v ) ) ) nv ) n n sin cos +v ) )) ) nv ) + cos ) +v ) n ) v ) cos sin ) +v ) n )) v ). Theefoe, aveage single-hop packet pogess fo is [ ] E P x x c ) vdv e ρa ) π dp + e ρa ) π dp c x. In the case of, A ) is { A) if <p<c ; A ) A ) if c <p<, whee c v + v v and A) n sin v n π ) ) + π n sin sin v n π ) ) n sin sin v n π ) ). )

8 Aveage Single hop Packet Pogess m) Aveage Numbe of Neighbos, ρ Fig.. Numeical esults fo aveage single-hop packet pogess vs aveage numbe of neighbos m and x m) Hence, aveage single-hop packet pogess fo can be finally deived as: [ ] x E P x c ) vdv e ρa) π dp + e ρa ) π dp c x. IV. NUMERICAL AND SIMULATION RESULTS ) A. Numeical Results As shown in ), ), and ), the aveage singlehop packet pogess is a function of node tansmission ange, netwok ange x, and aveage numbe of neighbos ρ o nodal density λ). We pesent two sets of numeical esults as shown in Figs. and. Figue compaes the analytically obtained aveage single-hop packet pogess fo,, and, with espect to diffeent aveage numbe of neighbos. The netwok ange is fixed to be a cicle with adius metes i.e., x m) while node tansmission ange is set at m i.e., m). Note that simila esults have been obseved fo othe values of and ae not pesented hee. It can be obseved fom Fig that the aveage singlehop packet pogess appoaches with an incease in ρ, fo all thee techniques. Howeve, it does not become equal to o. This is expected: when ρ is vey small, i.e., close to, it is unlikely that the sende will find any next-hop candidate to fowad its data packets. Howeve, it is still possible that the sende and the destination ae within the communication ange of each othe. Thus, the aveage single-hop packet pogess is positive. When ρ inceases, the pobability of void deceases and the sende is able to locate a next-hop node in its fowading aea in most situations. The aveage single-hop packet pogess, theefoe, ises. As ρ becomes lage and lage, it is moe and moe likely fo the sende to locate a next-hop node neaest to the destination and fathest away fom the sende as well, i.e., the data packet pogess duing this hop is equal to node tansmission ange. As a esult, we can expect that, when ρ goes to infinity, aveage single-hop packet pogess will appoach asymptotically. Howeve, it is still possible fo a destination node to be within the ange of the sende, then the packet pogess is less than o equal to. Theefoe, the aveage single-hop packet pogess will not become equal to. Anothe inteesting obsevation fom Fig. is that the aveage single-hop packet pogess inceases quickly fo small ρ andthenslowlyafteρ exceeds a cetain theshold value, i.e., thee exists a knee in the cuves aound ρ. The explanation of this phenomenon lies in the pobability of void as shown in Fig.. Fom Fig., we can see that the pobability of void deceases quickly as ρ inceases fom to the above theshold value of ρ. This obsevation suggests that the pobability of void dominates the aveage single-hop packet pogess, especially when ρ is small and the netwok is spase. As a esult, the tend of the cuves in Fig. is invese of that of the cuves in Fig.. Ou final but most impotant obsevation fom Fig, as one of the conclusions fom this wok as well, is that, in tems of aveage single-hop packet pogess, when ρ is small, i.e., fom to aound nodes, pefoms the best, then, followed by ; when ρ is medium, i.e., between and nodes, still pefoms the best, is the second and both outpefom ; when ρ is lage, i.e., nodes up to infinity, conveges to the pefomance of and both ae bette than. It is expected that the pefomance of will also convege to that of as ρ goes up to a vey lage value i.e., much lage than ) o moe. Note that and neve out-pefom. The above obsevation can be explained as follows: when ρ is small, the size of the fowading aea dominates the aveage single-hop packet pogess since it detemines the pobability of void fo that aea; when ρ is lage enough to counteact the effect of voids, the

9 Aveage Single hop Packet Pogess m) Netwok Range m) Aveage Single hop Packet Pogess m) Netwok Range m) Aveage Single hop Packet Pogess m) Netwok Range m) a) m and ρ nodes b) m and ρ nodes c) m and ρ nodes Fig.. Numeical esults fo aveage single-hop packet pogess vs netwok ange Packet Pogess m) Packet Pogess m) Packet Pogess m) Aveage fist hop analytical) Aveage fist hop simulation) Aveage single hop simulation) Aveage Numbe of Neighbos, ρ Aveage fist hop analytical) Aveage fist hop simulation) Aveage single hop simulation) Aveage Numbe of Neighbos, ρ Aveage fist hop analytical) Aveage fist hop simulation) Aveage single hop simulation) Aveage Numbe of Neighbos, ρ a) b) c) Fig.. Simulations esults fo the validation of ou analysis m and x m) location of the fowading aea dominates the calculation of aveage single-hop packet pogess since a fowading aea which coves moe aea that is fathe away fom the sende and is neae to the destination will have bette next-hop candidate nodes and theeby attain a lage aveage single-hop packet pogess. Simila conclusions can be dawn fom Fig., which suggests that ou obsevation holds fo diffeent values of netwok ange. The above esults povide a theoetical basis fo the design of a pactical potocol based on time and eceive-ovehead capability in a andomly distibuted wieless ad hoc netwok. That is, without void handling, if ρ is known apioito be small o medium, we should select as the fowading aea; if ρ is o moe, should be selected since it has the same netwok pefomance as, and moe impotantly, a simple MAC design and a less expensive adio capability. should only be chosen when ρ is extemely high i.e., much lage than ). In Fig., we compae the aveage single-hop packet pogess with espect to diffeent values of netwok ange x. Node tansmission ange is fixed at m while ρ is set to nodes in Fig. a), nodes in Fig. b), and nodes in Fig. c). Fom Fig., we obseve that, attains an aveage single-hop packet pogess that is almost m, i.e., %, highe than and, fo ρ o ρ. Fo ρ, and achieve a.m, i.e., %, highe aveage single-hop packet pogess than. In the case of ρ, when netwok ange is small and is compaable to node tansmission ange, aveage single-hop packet pogess plummets quickly and then maintains a elatively constant value as netwok ange inceases. In the case of ρ o ρ, this tend is

10 Oveall Aveage Single hop Packet Pogess m) Aveage Numbe of Neighbos, ρ Fig.. Simulation esults fo the compaison of oveall aveage single-hop packet pogess m and x m) evesed and aveage single-hop packet pogess jumps to a highe level and stays at that level as x inceases. This esult is elated to ou -step fowading stategy and can be explained as follows: when netwok ange is small, only step i) affects the esult; howeve, when netwok ange is much lage than the node tansmission ange, step ii) and step iii) also impact the esult. When ρ, step iii) has a lage impact than step ii), so the initial value is lage than the stable value. In the case of ρ o ρ, step ii) has a lage impact than step iii) and the initial value is theeby lowe than the stable value. B. Simulation Results Simulations pogams witten in C language and un in Micosoft Visual C++.) have been pefomed to veify ou analysis. In ou simulations, the netwok nodes ae distibuted in a cicula egion accoding to a two-dimensional Poisson distibution. The cicle is centeed at, ) with adius metes while the node tansmission ange is set to metes. The souce node is fixed at, ) while destination nodes ae andomly chosen in the cicle. The souce node tansmits packets to the selected destination node in accodance with ou -step fowading stategy while specifying a fowading aea as,, o. We measued aveage fisthop packet pogess, aveage single-hop packet pogess not only the fist-hop, but also the aveage of all the hops fom the souce node to the destination) of each pai of souce and destination. All the esults pesented in Figs. and ae the aveage of uns, each of which selects destinations andomly. As shown in Fig., simulation esults match well with ou analytical esults fo aveage fist-hop packet pogess in,, and. Howeve, ou actual desied pefomance measue, namely the oveall aveage single-hop packet pogess, is slightly lowe than ou analytically deived aveage fist-hop packet pogess, as expected. The main eason is that the last-hop packet pogess, whee a packet is diectly deliveed to a destination, is often less than the packet pogess in the pevious hops, whee a packet is fowaded using a geedy appoach. Figue confims ou obsevations made using analytical esults. In fact, the effect of last-hop packet pogess is simila fo diffeent fowading aeas. Extensive simulations have been pefomed and all the esults showed the same tendency. Theefoe, ou esults on fist-hop packet pogess can be employed as the oveall single-hop packet pogess to evaluate with diffeent fowading aeas. V. THE EFFECT OF VOID HANDLING SCHEMES Thus fa, we have studied the aveage single-hop packet pogess pefomance of without any void handling schemes fo diffeent fowading aeas. We futhe investigate the effect of void handling schemes on ou pevious esults in this section. A void occus when a packet is at a node which has no immediate neighbos in the fowading aea. This may lead to the data packet being dopped by this node, even though the oiginal souce and the intended destination ae topologically connected. A void handling scheme is necessay to somehow bypass the void. Befoe a node initiates a void handling pocedue, it should fist detemine whethe the void is tempoay due to packet collisions o tempoay unavailability of next-hop nodes. Tempoay voids will disappea as nodes wake-up, collisions esolved, and mobile nodes move in. Such tempoay voids only affect the end-to-end packet delay. Note that voids can be handled by state-based void handling schemes such as peimete outing in GPSR [] and BOUNDHOLE in [], but these techniques equie nodes to collect the local topology infomation eithe poactively o on demand. Thus, we focus on appoaches that have the state-fee featue by intoducing two state-fee void handling schemes fo hee and study thei pefomance via extensive simulations. The fist state-fee void handling scheme is temed active exploation. In this scheme, when a node cannot

11 Aveage Single hop Packet Pogess m) Active Exploation + Active Exploat + Passive + Paticipation Passive Patici. Aveage Numbe of Neighbos, ρ Aveage Single hop Packet Pogess m) + Active Exploation + Active Exploat + Passive + Paticipation Passive Patici Aveage Numbe of Neighbos, ρ Aveage Single hop Packet Pogess m) + Active Exploation + Passive Paticipation Aveage Numbe of Neighbos, ρ a) b) c) Fig.. Simulation esults fo with diffeent fowading aeas m and x m) locate a next-hop node in its fowading aea, it gadually inceases its tansmission ange until a next-hop node is found in its updated fowading aea. Note that we still use location instead of topology infomation. Thus, it is a state-fee scheme. The exta costs of this scheme ae the additional delay and the equiement of inceasing a node s tansmission powe hence the tansmission ange). In this wok, we assume that a node can incease its ange as much as it needs to. This assumption may not be pactical. In pactice, howeve, if a node still does not find any next-hop candidate nodes afte extending its tansmission ange to the maximum value, the following second scheme can be employed to avoid futhe loss of data packets. The second state-fee scheme is temed passive paticipation. The main idea is that once a node cannot locate a next-hop node, it simply discads the data packet and discouages itself fom picking up any following data packets fo that destination. The node may go back to check if thee exist any next-hop nodes at a futue time. This simple stategy has a evese-popagation effect which eventually infoms othe intemediate next-hop nodes to avoid those nodes with voids fo that destination. The disadvantage of this scheme is that it cannot be well exploited unde low nodal density since most nodes and even the souce node may be discouaged, which causes moe seious netwok patition and moe packet losses than befoe. Futhe analysis of these two schemes is not pesented hee due to page limitations. The simulation expeiment has the same configuation as in the last section and simulation esults ae shown in Figs.,, and. Fom Fig., we can obseve that,, and have the same tend. That is, Aveage Single hop Packet Pogess m).... Aveage Numbe of Neighbos, ρ Fig.. Simulation esults fo plus active exploation m and x m) when ρ is small, plus active exploation pefoms the best, then without void handling, followed by plus passive exploation. As ρ inceases, the pefomance of the thee cases begins to convege. Howeve, the point of thei convegence is diffeent. Fo, it is aound nodes; fo, it is aound nodes; while fo, it is aound nodes. This can be explained as follows: with a smalle ρ, the pobability of void is lage, and in this case the void handling schemes will contibute moe to aveage packet pogess and vice vesa. Thus, diffeent pobabilities of void with espect to ρ in these thee schemes, as shown in Fig., induce diffeent convegence points. Based on Fig., we can see that active exploation is vey effective fo small values of ρ. Howeve, we want to point out that this impovement is at the cost

12 Aveage Single hop Packet Pogess m) Aveage Numbe of Neighbos, ρ Fig.. Simulation esults fo plus passive paticipation m and x m) of exta enegy consumption. In the exteme case, i.e., ρ, plus active exploation becomes a diect instead of multi-hop communication between the souce and the destination, which has the lagest aveage packet pogess but also the most enegy consumption. In Fig., we can see that that uses passive paticipation is vey ineffective and even wose than without void handling fo small values of ρ, since it can discouage most nodes and even cause moe seious netwok patitioning. Howeve, when ρ is modeate, it begins to function well. The advantage of this scheme is that it is simple, efficient, and easy to implement. Note that although handling voids cannot impove aveage single-hop packet pogess afte ρ passes a theshold value, a state-fee void handling scheme is still desiable fo highe values of ρ, since it can impove packet delivey atio and potect the netwok fom any unexpected loss of impotant data at nodes with voids. VI. CONCLUSIONS In this pape, we have modeled Contention-based Geogaphic Fowading ) without void handling as a -step fowading stategy and then constucted a geneal analytical famewok. Ou famewok povides a theoetical basis fo the selection of an effective fowading aea fo the design of a pactical potocol, in tems of aveage single-hop packet pogess. The famewok, validated by numeical esults and extensive simulations, can also seve as a pefomance evaluation technique fo the potocols that have been designed peviously. Futhemoe, we have also intoduced two state-fee void handling schemes and studied thei pefomance when employed in conjunction with via extensive simulations. In ou futue wok, it would be of inteest to execute packet-level simulations to obseve how ou analysis matches with the esults obtained fom the existing potocols. We also plan to obtain moe analytical esults in tems of othe node distibution models as well as othe pefomance metics than those used in this wok. REFERENCES [] H. Füssle, J. Widme, M. Käsemann, M. Mauve, and H. Hatenstein, Contention-Based Fowading fo Mobile Ad-Hoc Netwoks, in Ad Hoc Netwoks Jounal Elsevie), Vol.,No., Novembe. [] M. Zozi, R. R. Rao, Geogaphic Random Fowading GeRaF) fo Ad Hoc and Senso Netwoks: Multihop Pefomance, in IEEE Tansactions on Mobile Computing, vol., Oct.-Dec.. [] B. M. Blum, T. He, S. Son, and J. A. Stankovic, IGF: A Robust State-Fee Communication Potocol fo Senso Netwoks, in Technical epot CS--, CS Depatment, Univesity of Viginia,. [] H. Füssle, H. hatenstein, J. Widme, M. Mauve, and W. Effelsbeg, Contention-Based Fowading fo Steet Scenaios, in poc. of the st intenational wokshop on intelligent tanspotation WIT), Hambug, Gemany, Ma. [] D. Chen, J. Deng, and P. K. Vashney, A State-Fee Data Delivey Potocol fo Wieless Senso Netwoks, in poc. of IEEE WCNC, New Oleans, LA, USA, Mach. [] A. Savvides, C. Han, and M. B. Stivastava, Dynamic Finegained Localization in Ad-Hoc Netwoks of Sensos, in Poc. of ACM MobiCom, Rome, Italy, July. [] N. Bulusu, J. Heidemann, and D. Estin, GPS-less Low Cost Outdoo Localization Fo Vey Small Devices, IEEE Pesonal Communications Magazine, ), pp. -, Oct.. [] D. B. Johnson and D. A. Maltz, Dynamic Souce Routing in Ad Hoc Wieless Netwoks, in Mobile Computing,. [] J. Li, J. Jannotti, D. S. J. De Couto, D. R. Kage, and R. Mois, A Scalable Location Sevice fo Geogaphical Ad Hoc Routing, in Poc. of ACM MobiCom, August. [] Q. Fang, J. Gao, and L. J. Guibas, Locating and Bypassing Routing Holes in Senso Netwoks, in Poc. of IEEE Infocom, Hong Kong, Mach. [] IEEE Standad fo Wieless LAN - Medium Access Contol and Physical Laye Specification, P.,. [] J. L. Gao, Analysis of Enegy Consumption fo Ad Hoc Wieless Senso Netwoks Using a Bit-mete-pe-joule Metic, in IPN Pogess Repot, -, August. [] T. Melodia, D. Pompili, and Ian F. Akyildiz, Optimal Local Topology Knowledge fo Enegy Efficient Geogaphical Routing in Senso Netwoks, in poc. of IEEE INFOCOM, -, Hong Kong, Mach. [] J. Deng, Y. S. Han, P.-N. Chen, and P. K. Vashney, Optimum Tansmission Range fo Wieless Ad Hoc Netwoks, in poc. of IEEE WCNC, Atlanta, GA, USA, Mach. [] P. Billingsley, Pobability and Measue, New Yok, NY: John Wiley and Sons,. [] B. Kap and H. T. Kung, Geedy Peimete Stateless Routing fo Wieless Netwoks, in Poc. of ACM MobiCom, Boston, MA, August.