Load Analysis of Topology-Unaware TDMA MAC Policies for Ad-Hoc Networks

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1 Lod Anlysis of Topology-Unwre TDMA MAC Policies for Ad-Hoc Networks Konstntinos Oikonomou 1 nd Ionnis tvrkkis 2 1 INTRACOM.A., Emerging Technologies & Mrkets Deprtment, 19. Km Mrkopoulou Avenue, Pini , Athens, Greece Tel: , Fx: okon@intrcom.gr 2 University of Athens, Deprtment of Informtics & Telecommunictions Pnepistimiopolis, Ilissi 1 78, Athens, Greece Tel: , Fx: ionnis@di.uo.gr Abstrct. Medium Access Control MAC) policies in which the scheduling time slots re llocted irrespectively of the underline topology re suitble for d-hoc networks, where nodes cn enter, leve or move inside the network t ny time. Topology-unwre MAC policies, tht llocte slots deterministiclly or probbilisticlly hve been proposed in the pst nd evluted under hevy trffic ssumptions. In this pper, the hevy trffic ssumption is relxed nd the system throughput chieved by these policies is derived s function of the trffic lod. The presented nlysis estblishes the conditions nd determines the vlues of the ccess probbility for which the system throughput under the probbilistic policy is not only higher thn tht under the deterministic policy but it is lso close to the mximum chievble, provided tht the trffic lod nd the topology density of the network re known. In cse the trffic lod nd/or the topology density re not known which is commonly the cse in d-hoc networks), lterntive vlues for the ccess probbility re lso derived which, lthough not the optiml mximizing the system throughput), they do led to system throughput higher thn tht under the Deterministic Policy. imultion results for vriety of topologies with different chrcteristics support the clims nd the expecttions of the nlysis nd show the comprtive dvntge of the Probbilistic Policy over the Deterministic Policy. 1 Introduction The design of n efficient Medium Access Control MAC) protocol in d-hoc networks is chllenging. The idiosyncrsies of these networks, where no infrstructure is present nd nodes re free to enter, leve or move inside the network without prior configurtion, llow for certin choices on the MAC design depending on the prticulr environment nd therefore, it is not surprising tht severl MAC protocols hve been proposed so fr. Contention-bsed MAC protocols re widely employed in d-hoc networks, such s the CMA/CA-bsed IEEE , [1]. In ddition to the crrier sensing mechnism, MACA, [2], employs the Redy-To-end/Cler-To-end RT/CT) hndshke mechnism. This mechnism is minly introduced to void the hidden/exposed terminl problem, which cuses significnt performnce degrdtion in d-hoc networks. Other protocols bsed on vritions of the RT/CT mechnism hve been proposed s well, [], [], []. TDMA-bsed MAC protocols re lso employed, where ech node is llowed to trnsmit during specific set of TDMA scheduling time slots. -TDMA - proposed by Kleinrock nd Nelson, [6] - is cpble of providing collision-free scheduling bsed on the exploittion of noninterfering trnsmissions in the network. In generl, optiml solutions to the problem of time slot ssignment often result in NP-hrd problems, [7], [8], which re similr to the n-coloring problem in grph theory. Other collision-free scheduling schemes hve been proposed s well, [9], [10], [11], [12]. Topology-unwre TDMA scheduling schemes determine the scheduling time slots irrespectively of the underlying topology nd in prticulr, irrespectively of the scheduling time slots ssigned to neighbor nodes. The topology-unwre scheme presented in [1], exploits the mthemticl properties of polynomils with coefficients from finite Glois fields to rndomly ssign scheduling time slot sets to ech node of the network. For ech node it is gurnteed tht t lest one time slot in frme would be collision-free, [1]. Another scheme, proposed in [1], mximizes the minimum gurnteed throughput. However, both schemes employ deterministic policy to be referred to s the Deterministic Policy) for the utiliztion of the ssigned time slots tht fils to utilize non-ssigned time slots tht could result in successful trnsmissions, [1], [16], [17]. Therefore, new policy ws proposed tht probbilisticlly utilizes the non-ssigned time slots ccording to common ccess probbility p to be referred to s the Probbilistic Policy). Anlyticl results for the mximiztion of the system throughput were derived nd its dependency on the topology density ws dditionlly investigted, [17]. An importnt feture of the Probbilistic Policy is tht it is simple trnsmission policy tht cn be implemented esily, llowing for trnsmissions without the need for coordintion mong different nodes or wreness bout topologicl chnges.

2 All the forementioned works [1], [1], [1], [16], [17]) hve focused on hevy trffic conditions nd no work hs been conducted for the non-hevy trffic cse. In this pper, the probbility, 0 1, tht node hs dt for trnsmission during one time slot probbility is lso referred to s the trffic lod) is considered. Anlyticl expressions for the system throughput re derived for both the Deterministic Policy nd the Probbilistic Policy s function of. An nlysis bsed on pproximtions, whose ccurcy is lso investigted here, provides expressions for the system throughput mximiztion under the Probbilistic Policy, in cse the trffic lod nd the topology density re known. In cse the trffic lod nd/or the topology density re not known, expressions for the resulting throughput hve lso been derived; even though the system throughput under the Probbilistic Policy is not mximized, in this cse, it turns out to be higher thn tht chieved under the Deterministic Policy. Extensive simultions confirm the results of the nlysis. In ection 2, n exmple d-hoc network is described nd some key definitions re introduced. Both trnsmission policies re described in ection. In ection, nlyticl expressions for the system throughput s function of the trffic lod re derived for both policies. In ection, the conditions for the existence of n efficient rnge of vlues for the ccess probbility p vlues of the ccess probbility p under which the Probbilistic Policy outperforms the Deterministic Policy) re estblished through n pproximte nlysis. Furthermore, this nlysis determines the vlue of the ccess probbility tht mximizes the system throughput provided tht the trffic lod nd the topology density re known. In ection 6, the ccurcy of the pproximte nlysis is studied, while in ection 7 the cse when the trffic lod nd/or the topology density re not known is considered nd expressions re derived such tht the system throughput under the Probbilistic Policy, even though not mximized, is higher thn tht under the Deterministic Policy. At this point comprison is crried out between the results of the nlysis presented here for 1ndthe results of the nlysis presented in [17] hevy trffic conditions). imultion results, presented in ection 8 for vriety of topologies with different chrcteristics, support the clims nd expecttions ddressed by the results of the nlysis in the previous sections. ection 9 presents the conclusions. 2 Network Definition An d-hoc network my be viewed s time vrying multihop network nd my be described in terms of grph GV,E), where V denotes the set of nodes nd E the set of links between the nodes t given time instnce. Let X denote the number of elements in set X nd let N V denote the number of nodes in the network. Let u denote the set of neighbors of node u, u V. These re the nodes v to which direct trnsmission from node u trnsmission u v) is possible. In this work omni-directionl ntenns will be considered nd it is ssumed tht the trnsmission power is eul for ll trnsmitting nodes in the network. Let D denote the mximum number of neighbors for node; clerly u D, u V. Time is divided into time slots with fixed durtion nd collisions with other trnsmissions is considered to be the only reson for trnsmission not to be successful corrupted ). uppose tht node u wntstotrnsmittoprticulrneighbornodev in prticulr time slot i. Inorderforthe trnsmission u v to be successful uncorrupted ), two conditions should be stisfied. First, node v should not trnsmit in the prticulr time slot i, or euivlently, no trnsmission v ψ, ψ v should tke plce in time slot i. econd, no neighbor of v - except u - should trnsmit in time slot i, or euivlently, no trnsmission ζ χ, ζ v {u} nd χ ζ, should tke plce in time slot i. Conseuently, trnsmission u v is corrupted in time slot i if t lest one trnsmission χ ψ, χ v {v} {u} nd ψ χ, tkes plce in time slot i. Figure 1 depicts n exmple network of 27 nodes. For this network N 27, nd it cn be seen tht D 6.Nodesχ, whose trnsmissions corrupt trnsmission 8 1, χ 1 {1} {8}, re gry colored Fig. 1. Exmple network of 27 nodes for which N 27ndD 6.Nodesχ tht their trnsmissions corrupt trnsmission 8 1, χ 1 {1} {8} {12, 1, 1, 17, 18, 19}, re drwn gry. Trnsmission 8 1 is depicted by white rrow, while those trnsmissions tht corrupt trnsmission 8 1 re depicted by blck rrows.

3 In the event of trnsmission u v, itisimportntfornodeu to be notified bout its successful or corrupted) reception in order to retrnsmit the corresponding dt in the cse of corruption, [18]. This my be chieved by the use of retrnsmission mechnisms like the elective Repet ARQ mechnism. Another pproch is to llow prt of the end of ech time slot to be used for the trnsmission of positive cknowledgement ACK) by the receiver immeditely fter successful trnsmission), [19]. Given tht omni-directionl ntenns re used nd the trnsmission power is eul for ll trnsmitting nodes, the trnsmission corresponding to the positive cknowledgement ACK will not be corrupted by ny other trnsmission. For exmple, suppose tht trnsmission 8 1, for the network depicted in Figure 1, tkes plce in time slot i. If trnsmission 8 1 is successful nodes 12, 1, 1, 17, 18, 19 do not trnsmit), node 1 will trnsmit positive cknowledgement ACK to node 8 t the end of time slot i. Trnsmission 1 8 my be corrupted if ny of the nodes 12, 9,, 7 trnsmits t the sme time. Given tht in time slot i trnsmission 8 1 ws successful, it is obvious tht none of the forementioned nodes ws the receiver node of successful trnsmission corruption would hve tken plce due to the presence of trnsmission 8 1). Conseuently, these nodes will not trnsmit simultneously with trnsmission 1 8 t the end of time slot i) tht cknowledges the successful trnsmission 8 1 in time slot i. For the rest it will be ssumed tht the cknowledgment is received instntneously, [20]. Trnsmission Policies According to the trnsmission policy proposed in [1] nd [1], ech node u V is rndomly ssigned uniue polynomil f u of degree k with coefficients from finite Glois field of order GF )). Polynomil f u is represented s f u x) k i x i mod ), [1], where i {0, 1, 2,..., 1}; prmeters nd k re clculted bsed on N nd D, ccording to i0 the lgorithm presented either in [1] or [1]. The ccess scheme considered is TDMA scheme with frme consisted of 2 time slots. If the frme is divided into subfrmes s of size, then the time slot ssigned to node u in subfrme s, s 0, 1,..., 1) is given by f u s)mod, [1]. Conseuently, one time slot is ssigned for ech node in ech subfrme. Let Ω u bethethesetoftimeslotsssigned to node u. Given tht the number of subfrmes is, Ω u. The deterministic trnsmission policy, proposed in [1] nd [1], is the following. The Deterministic Policy: Echnodeu trnsmits in slot i only if i Ω u, provided tht it hs dt to trnsmit. The reltion between N, D, nd k is importnt in order to explin the min property of the Deterministic Policy: there exists t lest one time slot in frme over which specific trnsmission will remin uncorrupted, [1]. uppose tht two neighbor nodes u nd v hve been ssigned two uniue) polynomils f u nd f v of degree k, respectively. Given tht the roots of ech node s polynomil correspond to the ssigned time slots to ech node, k common time slots is possible to be ssigned mong two neighbor nodes. Given tht D is the mximum number of neighbor nodes of ny node, kd is the mximum number of time slots over which trnsmission from ny node is possible to become corrupted. ince the number of time slots tht node is llowed to trnsmit in frme is, if>kdor kd +1k nd D re integers) is stisfied, there will be t lest one time slot in frme in which specific trnsmission will remin uncorrupted for ny node in the network, [1]. The ssigned uniue polynomils f u to ech node u llow for the determintion of Ω u. This ssignment my be considered s similr to MAC identifiction numbers MAC IDs) nd re ssigned to ech node ccordingly: either they re included in the device or they re ssigned using control mechnism. In this work, it is ssumed node hs been ssigned uniue polynomil rndomly without tking into ccount the neighbor nodes of the node nd/or their ssigned polynomils) before the node enters the network. It is lso importnt to gurntee tht the number of uniue polynomils is enough for ll nodes in the network, or k+1 N. IfforgivenN nd, k+1 N is not stisfied, then k hs to be incresed perhps resulting to new tht stisfies >kd) until the number of uniue polynomils is sufficient. Let O i,u v be tht set of nodes χ whose trnsmissions corrupt prticulr trnsmission u v i.e. χ v {v} {u}) nd which re lso llowed to trnsmit in time slots i i.e., they re ssigned slot i or i Ω χ ). Let O c i,u v be the complementry set of nodes in v {v} {u} for which i/ Ω χ. } O i,u v {χ : χ v {v} {u},i Ω χ, 1) } O c i,u v {χ : χ v {v} {u},i / Ω χ. 2) Obviously, O i,u v + O i,u v c v. Depending on the prticulr rndom ssignment of the polynomils, it is possible tht two nodes be ssigned overlpping time slots i.e., Ω u Ω v ). Let C u v be the set of overlpping time slots between those ssigned to node u

4 nd those ssigned to ny node χ v {v} {u}. C u v Ω u χ v {v} {u} Ω χ. ) Obviously, if i C u v O i,u v > 0wheni C u v ), it is possible tht nother trnsmission will corrupt trnsmission u v in time slot i, provided tht there re dt for trnsmission. If i Ω u C u v O i,u v 0), no trnsmission corrupts trnsmission u v in time slot i. Let R u v denote the set of time slots i, i/ Ω u, over which trnsmission u v would be successful even for hevy trffic conditions 1). Euivlently, R u v contins those slots not included in set χ v {v} Ω χ.conseuently, R u v 2 χ v {v} Ω χ. ) As it hs been shown in [17], R u v k 1)D, resulting in lrge number of unused time slots especilly for k>1) tht could be used to increse the verge number of trnsmissions. In generl, the use of slots i, i R u v,my increse the verge number of successful trnsmissions, s long s R u v is determined nd time slots i R u v re used efficiently. The determintion of R u v reuires the existence of mechnism for the extrction of sets Ω χ, χ v. In ddition, the efficient use of slots in R u v by node u, reuires further coordintion nd control exchnge with neighbor nodes χ, whose trnsmissions χ ψ, withr χ ψ R u v, my utilize the sme slots in R χ ψ R u v nd corrupt either trnsmission u v or χ ψ, or both. In order to use ll non-ssigned time slots without the need for further coordintion mong the nodes, the following probbilistic trnsmission policy ws introduced in [17]. The Probbilistic Policy: Echnodeu lwys trnsmits in slot i if i Ω u nd trnsmits with probbility p in slot i if i/ Ω u, provided it hs dt to trnsmit. The Probbilistic Policy does not reuire specific topology informtion e.g., knowledge of R u v, etc.) nd, thus, induces no dditionl control overhed. The ccess probbility p is simple prmeter common for ll nodes. Under the Probbilistic Policy, ll slots i/ Ω u re potentilly utilized by node u: both, those in R u v, for given trnsmission u v, swellsthosenotinω u R u v tht my be left unutilized by neighboring nodes under non-hevy trffic conditions <1). Following the pproch described in [1], nd regrding the exmple network depicted in Figure 1,, k nd ech set Ω χ re determined for ll nodes χ in the network. In prticulr, 7,k 1 nd for trnsmission 8 1, C 8 1, R 8 1,ndΩ χ, χ 1 {1}, re determined nd depicted in Figure 2. Time slots i Ω 8 C 8 1 re depicted with light-gry colored boxes, while time slots i C 8 1 re depicted with drk-gry colored boxes. Time slots i R 8 1 re depicted with white colored boxes. Finlly, time slots i/ Ω 8 R 8 1 re depicted with mid-gry colored boxes. The upper left smll numbers in the boxes refer to the prticulr time slot in frme, while the centrl number refer to the nodes tht hve been ssigned the prticulr time slot , ,19 8, , ,17 1, , , , , 1 1 8, , 1 i i Ω C i i Ω R i i C 8 1 i i R 8 1 Fig. 2. Trnsmission sets Ω χ, χ 1 {1} {8, 12, 1, 1, 17, 18, 19}. Ω 8 {1, 9, 17, 2,, 1, 2}, C 8 1 {9, 2,, 1} C 8 1 )ndr 8 1 {2, 8, 10, 11, 19, 20, 21, 26, 28, 7, 0, 8} R ).

5 ystem Throughput In order to derive the expressions for the system throughput under both the Deterministic Policy nd the Probbilistic Policy, it is necessry to derive the expressions for the probbility of success of specific trnsmission throughput) under both policies..1 pecific Trnsmission Let P i,d,u v P i,p,u v ) denote the probbility tht trnsmission u v in slot i is successful, under the Deterministic Probbilistic) Policy. Let P D,u v P P,u v ) be the verge probbility over frme for trnsmission u v to be successful during time slot, under the Deterministic Probbilistic) Policy. Tht is, P D,u v i1 P i,d,u v P P,u v i1 P i,p,u v), where 2 is the frme size, in time slots. Under the Deterministic Policy ech node u trnsmits over time slot i with probbility, ifi Ω u.anyother node χ, forwhichi Ω χ, lso trnsmits with probbility. Conseuently,P i,d,u v 0,fori/ Ω u,ndp i,d,u v 1 ) Oi,u v,fori Ω u.giventht O i,u v 0 for those slots i Ω u C u v O i,u v > 0wheni C u v ), it is concluded tht P i,d,u v, fori Ω u C u v.asresult, P D,u v C u v + i C u v 1 ) Oi,u v 2, ) where Ω u. Under the Probbilistic Policy, if i Ω u,echnodeu trnsmits over time slot i with probbility, while if i/ Ω u ech node u trnsmits with probbility p. Anyothernodeχ, forwhichi Ω χ, trnsmits in the sme time slot i with probbility,wheresifi/ Ω χ it trnsmits with probbility p.conseuently,fori Ω u, P i,p,u v 1 ) Oi,u v 1 p) c Oi,u v, while for i/ Ω u, P i,p,u v p1 ) Oi,u v 1 p) c Oi,u v.giventht O i,u v + O c i,u v v, ) Oi,u v ) Oi,u v 1 for i Ω u, P i,p,u v 1 p 1 p) v 1, while for i / Ω u, P i,p,u v p 1 p 1 p) v.by definition, O i,u v 0,ifi Ω u C u v ) R u v,nd O i,u v > 0ifi C u v or i/ Ω u R u v.asresult,.2 ystem Throughout P P,u v ) Oi,u v 1 i C u v 1 p 2 1 p) v + C u v + p R u v 1 p) v 2 p ) Oi,u v 1 i/ R u v Ω u 1 p p) v. 6) The destintion node v of prticulr trnsmission u v, depends on the destintion of the dt nd conseuently, on the network ppliction s well s on the routing protocol. For the rest of this work it will be ssumed tht node u trnsmits to only one v, v u. For ny of the numericl or simultion results presented in the seuel, node v will be node rndomly selected out of u. The probbility of success of trnsmission verged over ll nodes in the network) under ech corresponding policy, is referred to s the system throughput nd is denoted by P D P P ) under the Deterministic Probbilistic) Policy. According to eutions ) nd 6), the system throughput P D nd P P is derived from the following eutions. P D 1 N P P 1 N + 1 N u V u V u V C u v + i C u v 1 ) Oi,u v ) Oi,u v 1 i C u v 1 p 2 2, 7) 1 p) v C u v + p R u v 2 1 p) v

6 + 1 N u V p i/ R u v Ω u ) Oi,u v 1 1 p 2 1 p) v, 8) where v u. From Eution 7) nd Eution 8) it cn be seen tht for p 0,P P P D, while for p 1,P P 1 N u V 1 ) v. The ltter is greter thn zero for 0 <<1. In Figure ) the system throughput for the network depicted in Figure 1 is illustrted for vrious vlues of p, s function of. It is obvious tht s increses, P P increses fster thn P D until mximum is ssumed. If the mximum is ssumed for <1, then s increses further, P P decreses until 1. As it cn lso be observed, those vlues of p for which P P is lrge for smll vlues of, resultinsmllp P for lrge vlues of nd vice vers. Figure b) depicts P P s function of p for different vlues of ; the sme conclusion s before my be drwn. 6\VWHP 7KURXJKXW 6\VWHP 7KURXJKXW ) D Fig.. Numericl results for the system throughput, P P, under the Probbilistic Policy ) s function of, fordifferentvlues of p p 0,p 0.1, p 0.1, p 0.2, p 0. n 1.0) nd b) s function of p, for different vlues of 0.1, 0. nd 1.0). Note tht for p 0,P P P D. From the bove discussion it is obvious tht certin vlue for p mximizes the system throughput for certin vlue of. The im of the following section is to estblish the conditions under which P P P D is stisfied nd derive the reltions between nd p for which not only P P P D, but P P is lso mximized. E ystem Throughput Mximiztion As it is difficult to derive nlyticl expressions nd estblish the conditions for the system throughput mximiztion from Eution 8), more trctble form of P P is considered. Let ˆP P denote tht vlue of P P when 1 1 p is replced by 1. Then, ˆP P 1 C u v N u V 1 p) v N u V p) v + 1 p 2 R u v ) N u V 1 p) v or, finlly, 2 ˆP P 1 N u V 1+p 1) 1 p) v. 9) ince 1 1 p 1, it is cler tht ˆP P P P. Even though ˆP P hs more trctble form thn tht of P P, it still cnnot be esily nlyzed further. It cn be seen from Eution 9) tht ˆP P corresponds to polynomil of D + 1 degree with respect to p, which is difficult or impossible to be solved in the generl cse D >1). A more trctble form of ˆP P, P P, is nlyzed insted. PP is eul to 1 1+p) N u V 1 p),where 1 N v V v. Finlly, P P is given by the following eution. P P 1+p 1) 1 p). 10) is the verge number of neighbor nodes of ech node in the network. Let /D be referred to s the topology density for prticulr network. Given tht D is known nd constnt), knowledge of is enough to determine the topology density /D nd vice vers. It is cler tht or the topology density /D) influences exponentilly the system throughput s it cn be concluded from Eution 10). ˆP P my or my not be good pproximtion of P P, while P P my or my not be good pproximtion of ˆP P.As it cn be seen in ection 6, even for those cses when ˆP P is not good pproximtion of P P,nd P P is not good pproximtion of ˆP P, the results of the following nlysis my be pplied. The following two theorems show tht there exists rnge of vlues of p of the form [0,p mx ]wherep mx,0<p mx 1, corresponds to tht vlue of p for which P P P D ) such tht P P P D, irrespectively of the vlue of. C u v +p R u 2

7 Theorem 1. There exists n efficient rnge of vlues for p such tht P P P D, irrespectively of the vlue of. Proof. Under the Deterministic Policy, is the mximum number of time slots during which trnsmission is successful nd therefore, is the mximum verge) number of successful trnsmissions in one frme. For ny trnsmission u v, ccording to Eution ), C u v + i C u v 1 ) Oi,u v. Conseuently, ccording to Eution 7), P D. It thus suffices to identify the rnge of vlues of p for which P P. In Appendix 10 it is shown tht d P P p +1)1 )+ 1 p) 1 d nd therefore, P 0 for p +1)) p 0). Given tht P P P D for p 0, the reuired rnge of vlues for p does not exist if d P P < 0 for ll p 0, 1]. d P P < 0isstisfiedifp 0 < 0 nd s it is shown in Appendix 10, p 0 < 0, if >.Giventht D eulity holds if ll nodes hve D neighbor nodes) nd kd +1 D + 1 eulity holds for k 1), it is concluded tht 1or 1. Given tht 1, > is not stisfied nd conseuently, it is not possible tht d P P < 0 for ll p 0, 1]. Conseuently, there exists rnge of vlues for p such tht P P P D, irrespectively of the vlue of. Theorem 2. The efficient rnge of vlues for p of Theorem 1 is of the from [0,p mx ],forsome0 <p mx 1. Proof. If is stisfied then, ccording to the Appendix 10, p is stisfied, where p 0. +1)) From Appendix 10 it cn be seen tht for p close to 0, d P P nd d P P < 0for1 p>p 0 d P P when > 0. Given tht d P P 0forp p 0, d P P > 0forp<p 0, 0 for only one vlue of p 0, 1) s it my be seen from Appendix 10). As result, +, PP for p 0,ndsp increses, PP increses, until p p 0.Asp increses nd p>p 0, PP decreses until p 1,where P P 1 ).If1 ) >,thenp mx 1.If1 ), there exists vlue of p p mx 1, such tht P P. If <,then d P P + > 0, for ny vlue of p. Conseuently, P P constntly increses nd therefore, p mx 1nd the mximum is ssumed for p 1. The exmple of Figure depicts three possible shpes of the curve corresponding to P P. For the cse corresponding to 1,p 0 nd p mx hve been drwn s well. For the cse corresponding to 0., it is cler the p 0 is close to 0. n mx 1. For the cse corresponding to 0.1, the mximum is ssumed for p 1n mx 1 s well. It is cler tht for different vlues of, different vlues of p correspond to P P mximiztion. Theorem estblishes the conditions nd presents the reltions for the system throughput mximiztion. P ~ P 1.0 6\VWHP7KURXJKXW p 0 p mx p Fig.. P P s function of p for different vlue of. Theorem. If < Proof. For < P P is ssumed for p 1. For +, P P is mximized for p 1.If, s it ws shown in the proof of Theorem 2, d P P +, s it ws shown in the proof of Theorem 2, d P P + Conseuently, the mximum vlue for P P is ssumed for p p 0 +, P P is mximized for p +1)). > 0. Conseuently, the mximum vlue for 0forp [0,p 0 ]nd d P P 0forp [p 0, 1]. +1)).

8 Bsed on the results of Theorem, the vlue of the ccess probbility p tht mximizes P P, denoted by p,,isgiven by the following eution. 1, if < ; + p, +1)), if. 11) + P P 6\VWHP7KURXJKXW p ~ p, P D p 0) p 0.2 p 0.1 p 1.0 p 0.1 p 0. Fig.. imultion results for the system throughput P P ) under the Probbilistic Policy s function of, for different vlues of p p 0,p 0.1, p 0.1, p 0.2, p 0., p 1.0 n p, ) nd /D 0.9. For p p,, P P is mximized but it is lso expected tht P P will hve better performnce thn tht chieved by fixed vlue of p when is not constnt. This is illustrted in Figure concerning simultion results more informtion bout the simultor cn be found in ection 8) for the network depicted in Figure 1. In this figure, P P for p p, is close to the mximum obtined by ny other curve for constnt vlues of p. The cses under which the results of this nlysis my be used to mximize P P s opposed to P P tht hs been exmined in the present section), re investigted in the following section. 6 On the Accurcy of the Approximtion How close is ˆP P to P P depends on how close is 1 1 p to 1. Obviously, 1 1 p is close to 1 for smll vlues of nd lrge vlues of p, s it cn be seen in Appendix 11. This is clerly depicted in Figure 6, concerning the network depicted in Figure 1, where P P nd P ˆ P re depicted ) s function of for different vlues of p nd b) s function of p for different vlues of. The curve corresponding to ˆP P is very close to the one corresponding to P P for smll vlues of nd/or lrge vlues of p. An interesting observtion on Figure 6 is tht those vlues of p nd, whose curves corresponding to P P nd P ˆ P re not close, hve lmost the sme shpe. This mens tht those vlues of p nd, forwhich ˆP P ssumes the mximum, re close to the vlue of p nd for which P P ssumes the mximum, s it my lso be observed from Figure 6. 6\VWHP 7KURXJKXW! D Ö! Ö! Ö!!! Ö! 6\VWHP 7KURXJKXW! Ö!!! Fig. 6. Numericl results for the system throughput P ) corresponding to P P, ˆPP nd P P, ) s function of, fordifferent vlues of p p 0.1, p 0. n 0.9) nd b) s function of p, for different vlues of 0.1, 0. nd 0.9). Note tht for p 0,P P P D. How close is ˆP P to P P depends on how close is P P ˆP P to 0. In Appendix 12 it is shown tht P P ˆP P 1+p) 1 p) 1 N u V 1 p) v 1.LetVr{ } 1 ND v V v be defined s the topology E! Ö!

9 density vrition. It is evident tht s Vr{ } increses, P P ˆP P increses exponentilly. Conseuently, PP is good pproximtion of ˆP P forrthersmllvluesofvr{ }. In the exceptionl cse of network for which ll nodes hve the sme number of neighbor nodes except perhps the prticulr node tht hs D neighbor nodes), PP ˆP P Vr{ } 0). For the cse tht ll nodes hve eul number of neighbor nodes nd eul to D, then P P ˆP P Vr{ } 0). An exmple of the ltter cse is network with uniformly distributed nodes on sphere s surfce. For the exmple network depicted in Figure 1 it cn be clculted tht Vr{ } 0.6. In Figure 6, where P P is lso depicted, it cn be seen tht the shpe of the corresponding curve is similr to tht of the curve corresponding to ˆP P. It cn lso be seen tht for smll vlues of p nd both curves re close. This holds since 1 N u V 1 p) v 1 is close to zero nd conseuently P P ˆP P is close to zero) for smll vlues of p nd. It is evident tht even though ˆP P my not be close to P P,forgiven the vlues of p for which P P is mximized re close to the vlues of p for which ˆP P is mximized. D E Fig. 7. Networks A ) nd B b) with the sme topology density vlue /D 0.9) with Network C Figure 1) but different topology density vrition Vr{ } 0.07, nd 0.16, respectively). In order to provide n exmple let the networks depicted in figures 7), 7b) nd 1 be referred to s network A, B nd C, respectively. All three networks hve the sme number of nodes N 27), the sme number of mximum neighbor nodes D 6) nd the sme topology density /D 0.9). PP 6\VWHP7KURXJKXW P P,A, Vr{ } 0.07 P P,B, Vr{ } P P,C, Vr{ } 0.16 ~, Vr{ } 0 P P p 0,C p 0,0 B p,a p 0 p Fig. 8. ystem throughput for network A, B nd C. Figure 8 depicts P P,A, P P,B nd P P,C for the three forementioned networks s well s P P s function of p nd for 1. These re simultion results nd were obtined s it is described in ection 8. Let p 0,A, p 0,B nd p 0,C those vlues of p for which P P,A, P P,B nd P P,C re mximized, respectively. It is observed from Figure 8 tht s Vr{ } increses the difference between P P for ny of the networks) nd P P increses, while the difference between the vlue of p tht mximizes P P nd the vlue of p tht mximizes P P p 0 ), increses but not drmticlly. From the previous discussion it is concluded tht the smller the vlue of nd Vr{ } nd the higher the vlue of p, the better the pproximtion of P P by P P. However, even for those cses tht P P is not too close to P P,theresults of the nlysis for P P my be used in order to chieve system throughput, P P, close to the mximum. 7 Unwreness of nd Knowledge of both nd is reuired in order to utilize the results of the nlysis presented in ection nd in prticulr, by Eution 11). If the topology density /D is known, cn be clculted D is known). In the generl cse, it is possible tht nd/or re not known nd therefore, it would be useful to derive nlyticl expressions for

10 those vlues of p tht the system throughput under the Probbilistic Policy even though it is not mximized, t lest is higher thn tht under the Deterministic Policy. For the cse for which is known but is not, the mximum topology density vlue corresponding to D) is considered nd the corresponding vlue for p, denoted by p, is given by Eution 12). For the cse for which is known but is not, hevy trffic scenrio corresponding to 1) is considered nd the corresponding vlue for p, denoted by p, is given by Eution 1). Finlly, for the cse for which both nd re not known, the corresponding vlue for p 1nd D), denoted by p, is given by Eution 1). { 1, if < D+ p ; D D+1)), if D+. 12) p 1 +1) 1). 1) p 1 D D +1) 1). 1) The following theorem shows tht for ny vlue of p derived from ny of the eutions 12), 1) nd 1), the system throughput under the Probbilistic Policy is higher thn tht under the Deterministic Policy. Theorem. For ny p eul to p, p or p, P P P D is stisfied. Proof. p, belongs to the efficient rnge of vlues for p for which P P P D is stisfied. According to Theorem 2, this rnge of vlues is of the form [0,p mx ], for some 0 <p mx 1. It is is enough to show tht p, p nd p re less or eul to p,. In order to show tht p p,, both rnges of vlues for of eutions 11) nd 12) should be compred. Given D+ tht D,.Conseuently,for< + p p, 1holds.For D+ < D+ +, p, 1nd p D D+1)) 1 is stisfied nd conseuently, p p, is lso stisfied. For, ccording to eutions 11) nd 12), the eulity D D+1)). As it cn be seen in Appendix 1, +, p, +1)) nd p D D D+1)). According to Appendix 1 D+1)), is stisfied nd conseuently, p p +1)), is lwys stisfied. In order to show tht p p, is stisfied, it is enough to show tht is stisfied. +1)) +1)) Conseuently, it is enough to show tht 1 or 1) 1 or 1) 1. The ltter is lwys stisfied nd conseuently, p p,. In order to show tht p p, is stisfied, it is enough to show tht p p is stisfied or D D+1)) +1)) or D D+1 or +1) D) D+1) ) or + 1 D D D+ D 1 D +1 or D. The ltter is lwys stisfied. For the hevy trffic scenrio 1), n pproximte nlysis ws lso presented in [17] nd boundries p 0min < p 0mx ) for the vlue of p tht mximizes the system throughput were derived. In prticulr, it ws derived tht p 0min ) 2 +1) ) ) ) +1) nd p 0mx 1. As it cn be seen in Appendix 1, p +1 < p 0 mx is lwys stisfied nd p > p 0min is stisfied if > If< then p < p 0 min.thefcttht p sometimes does not fll within the boundries provided by the nlysis in [17], is ttributed to the ccurcy of the different pproximtions considered in ech nlysis. Boundries p 0min nd p 0mx hve been derived bsed on pproximtions tht their ccurcy depends only on Vr{ }, while p s ccurcy depends dditionlly on how close is 1 1 p to 1. Given tht for this 1 cse 1, 1 p 0; tht is the worst cse. Conseuently, for those cses for which 1, p 0 min cn be better pproximtion thn p. 8 imultion Results imultion results hve been provided lredy in ection for the exmple network of 27 nodes depicted in Figure 1. In this section, networks of 100 nodes re considered for vrious vlues of D nd the topology density /D. Theim

11 is to demonstrte the pplicbility of the nlyticl results for vriety of topologies with different chrcteristics. In prticulr, four different topology ctegories re considered. The number of nodes in ech topology ctegory is set to N 100, while D is set to, 10, 1 nd 20. These four topology ctegories re denoted s DN100, D10N100, D1N100 nd D20N100, respectively. Three different topologies tht correspond to different topology density vlues /D re considered for ech topology ctegory. Figures 9, 10 nd 11 depict simultion results for the system throughput P P,for different topology density vlues. In prticulr, in Figure 9, /D is smll round 0.2), in Figure 10, /D is round 0.6, while in Figure 11, /D is high round 0.8). For ll cses the number of neighbor nodes for ech node is not the sme; this leds to nonzero vlues for the topology density vrition Vr{ }. The lgorithm presented in [1] is used to derive the sets of scheduling slots nd the system throughput is clculted verging the simultion results over 100 frmes. Time slot sets Ω χ re ssigned rndomly to ech node χ, forech prticulr topology. The prticulr ssignment is kept the sme for ech topology ctegory throughout the simultions. 6\VWHP 7KURXJKXW '1 ) ',7^ ` 6\VWHP 7KURXJKXW '1 ) ',7^ ` D E 6\VWHP 7KURXJKXW '1 ',7^ ` 6 ' 6\VWHP 7KURXJKXW '1 ',7^ ` 6 ' ) F ) G Fig. 9. ystem throughput simultion results s function of for different vlues of p p 0,p p,, p p, p p, p p nd p 1.0) smll topology density vlues /D. Figures 9, 10 nd 11 depict simultion results for the system throughput P P s function of, fordifferentvlues of p p 0,p p,, p p, p p, p p nd p 1.0). For p 0 the system throughput under the Probbilistic Policy is identicl to tht under the Deterministic Policy P P P D ). This cse is depicted throughout the simultion results for comprison resons. For p 1.0, it cn be seen tht s increses, P P increses rther fst until certin mximum vlue nd then decreses until 1.0, where P P 0. As it cn be observed from the three figures 9, 10 nd 11), s the topology density increses, the vlue of for which P P ssumes the mximum, decreses. For p p, it is evident tht the system throughput under the Probbilistic Policy is not only higher thn tht under the Deterministic Policy but it is lso close to the mximum for ll different topologies. For p p nd for smll vlues of, s it cn be seen in Figure 9, the system throughput is identicl to tht obtined for p p, nd p 1.0. This cn be concluded from Eution 11) nd Eution 12) where for < D+, p, p 1.0. For > D+ the system throughput, even though not close to tht obtined for p p,,is higher thn tht obtined under the Probbilistic Policy. As the topology density increses Figure 10 nd Figure 11) it is cler tht the system throughput obtined for p p is close to the mximum. For p p nd for smll vlues of, the system throughput is not close to tht obtined for p p, but is higher thn tht obtined under the Deterministic Policy, irrespectively of the topology density vlue. As increses the system throughput increses nd for lrge vlues of it is close to the mximum. For p p, the system throughput curve is similr to tht for p p, except tht the obtined system throughput is smller. As the topology density increses, the system throughput for both cses p p nd p p )converges. Agin, the obtined system throughout is higher thn tht obtined under the Deterministic Policy. This is n importnt observtion since p is clculted without knowledge of either or.

12 6\VWHP 7KURXJKXW '1 6\VWHP 7KURXJKXW '1 ) ',7^ ` 6 ' ) ',7^ ` D E 6\VWHP 7KURXJKXW '1 6\VWHP 7KURXJKXW '1 ) ',7^ ` ) ',7^ ` 6 ' F G Fig. 10. ystem throughput simultion results s function of for different vlues of p p 0,p p,, p p, p p, p p nd p 1.0) nd medium topology density vlues /D. In order to obtin clerer view, the system throughput for the D10N100 topology ctegory nd topology density /D 0.7 nd Vr{ } 0.172, is depicted in Figure 12 s function of p, for 0.1, 0., 0.6 nd 0.9. It is evident tht for p p,, the system throughput ssumes the mximum vlue for ny vlue of. For ny other cse p p, p p or p p) the system throughput is less thn tht obtined for p p, but definitely higher thn tht obtined under the Deterministic Policy. The ltter ws expected s direct result from Theorem. In ll cses it is evident tht the system throughput chieved under the Probbilistic Policy is higher thn chieved under the Deterministic Policy. Another fctor tht influences the system throughput is the topology density /D, s it cn be seen from the exponentil fctor of Eution 10). In Figure 1, it cn be seen tht the system throughout under the Deterministic Policy decreses slowly s /D increses, while the system throughout under the Probbilistic Policy decreses rpidly s /D increses. The only exception is for p p, where the decrement is not so rpid. It cn lso be observed tht s /D increses, the curves corresponding to p p, nd p p converge s well s those corresponding to p p nd p p. For lrge vlues of nd /D the four forementioned curves converge to the sme vlue. The reson is, ccording to eutions 11), 12), 1) nd 1), tht the vlues of p,, p, p nd p converge to the sme vlue, for close to 1 nd close to D. 9 Conclusions The system throughput under both the Deterministic Policy nd the Probbilistic Policy hs been investigted previously for hevy trffic conditions, [1], [1], [1], [16], [17]. In this work, expressions hve been derived nd nlyzed for the generl non-hevy trffic cse. The results of this nlysis estblished conditions nd provided expressions in order for the system throughput under the Probbilistic Policy to be not only higher thn tht under the Deterministic Policy, but lso be mximized, for vrious trffic lods. Expressions for the system throughput were derived for both policies s function of the trffic lod nd the ccurcy of certin nlysis pproximtions ws investigted. These derived expressions determine the vlues of the ccess probbility p for which the system throughput under the Probbilistic Policy is higher thn tht under the Deterministic Policy. For the prticulr cse for which both the trffic lod ) nd the topology density /D) re known, the system throughput is mximized for the derived vlue of ccess probbility, p p,. For the cses for which nd/or /D re not known, nlyticl expressions for the pproprite vlues of p were lso derived leding to system throughput which, even though not mximized, is higher thn tht chieved under the Deterministic Policy p p, p p nd p p).

13 6\VWHP 7KURXJKXW '1 6\VWHP 7KURXJKXW '1 ) ',7^ ` 6 ' ) ',7^ ` D E 6\VWHP 7KURXJKXW '1 ',7^ ` 6\VWHP 7KURXJKXW '1 ) ) ',7^ ` F G Fig. 11. ystem throughput simultion results s function of for different vlues of p p 0,p p,, p p, p p, p p nd p 1.0) nd high topology density vlues /D. imultions hve been conducted for vriety of different topologies with different chrcteristics nd different trffic lods tht support the clims nd the expecttions of the nlysis. The simultion results hve shown tht the chieved system throughput is mximized, provided tht nd /D re known. If the ltter is not the cse, the system throughput cn still be higher thn tht under the Deterministic Policy provided tht the results of the forementioned nlysis re pplied. The simultion results lso show how the system throughput is ffected by the topology density /D nd demonstrte the fct tht the use of the nlyticl results ssures system throughput under the Probbilistic Policy higher thn tht under the Deterministic Policy even for lrge vlues of /D. In conclusion, simple nd esily implemented trnsmission policy like the Probbilistic Policy, llows for nodes to trnsmit without ny need for coordintion mong them nd without considering the topologicl chnges; this is essentil for d-hoc networks. The results of the nlysis provided in this pper cn be used to specify n ccess probbility tht chieves system throughput close to the mximum for different trffic lods. Acknowledgement This work hs been supported in prt by the IT progrm under contrct IT BrodWy) which is prtly funded by the Europen Commission. References 1. IEEE , Wireless LAN Medium Access Control MAC) nd Physicl Lyer PHY) specifictions, Nov Drft upplement to tndrd IEEE , IEEE, New York, Jnury P. Krn, MACA- A new chnnel ccess method for pcket rdio, in ARRL/CRRL Amteur Rdio 9th Computer Networking Conference, pp. 1-10, V. Bhrghvn, A. Demers,. henker, nd L. Zhng, MACAW: A Medi Access Protocol for Wireless LAN s, Proceedings of ACM IGCOMM 9, pp , C.L. Fullmer, J.J. Grci-Lun-Aceves, Floor Acuisition Multiple Access FAMA) for Pcket-Rdio Networks, Proceedings of ACM IGCOMM 9, pp , J. Deng nd Z. J. Hs, Busy Tone Multiple Access DBTMA): A New Medium Access Control for Pcket Rdio Networks, in IEEE ICUPC 98, Florence, Itly, October -9, R. Nelson, L. Kleinrock, ptil TDMA, A collision-free Multihop Chnnel Access Protocol, IEEE Trnsctions on Communictions, Vol. COM-, No. 9, eptember A. Ephremides nd T. V. Truong, cheduling Brodcsts in Multihop Rdio Networks, IEEE Trnsctions on Communictions, 8):6-60, April 1990.

14 6\VWHP 7KURXJKXW '1 6\VWHP 7KURXJKXW '1 ) ) D E 6\VWHP 7KURXJKXW '1 6\VWHP 7KURXJKXW '1 6 ' ) ) F G Fig. 12. ystem throughput simultion results for the D10N100 topology /D 0.7 nd nd Vr{ } 0.172) s function of p for different vlues of 0.1, 0., 0.6 nd 0.9). 8. G. Wng nd N. Ansri, Optiml Brodcst cheduling in Pcket Rdio Networks Using Men Field Anneling, IEEE Journl on elected Ares in Communictions, VOL. 1, NO. 2, pp , Februry T. heprd, A Chnnel Access cheme for Lrge Dense Pcket Rdio Networks, In Proc. of ACM IGCOMM Aug. 1998). 10. L. Bo nd J. J. Grci-Lun-Aceves, A new pproch to chnnel ccess scheduling for d hoc networks, ACM Mobicom 2001, July R. Rozovsky nd P. R. Kumr, EEDEX: A MAC protocol for d hoc networks, ACM Mobihoc 01, October F. Borgonovo, A. Cpone, M. Cesn, L. Frtt, ADHOC MAC: new MAC rchitecture for d hoc networks providing efficient nd relible point-to-point nd brodcst services, to pper in WINET pecil Issue on Ad Hoc Networking. 1. I. Chlmtc nd A. Frgo, Mking Trnsmission chedules Immune to Topology Chnges in Multi-Hop Pcket Rdio Networks, IEEE/ACM Trns. on Networking, 2:2-29, J.-H. Ju nd V. O. K. Li, An Optiml Topology-Trnsprent cheduling Method in Multihop Pcket Rdio Networks, IEEE/ACM Trns. on Networking, 6:298-06, K. Oikonomou nd I. tvrkkis, A Probbilistic Topology Unwre TDMA Medium Access Control Policy for Ad-Hoc Environments, Personl Wireless Communictions PWC 200), 2-2 eptember, 200, Venice, Itly. 16. K. Oikonomou nd I. tvrkkis, Throughput Anlysis of Probbilistic Topology-Unwre TDMA MAC Policy for Ad-Hoc Networks, Qulity of Future Internet ervices QoFI), 1- October, 200, tockholm, weden. 17. K. Oikonomou nd I. tvrkkis, Anlysis of Probbilistic Topology-Unwre TDMA MAC Policy for Ad-Hoc Networks, IEEE Journl on elected Ares in Communictions JAC), pecil Issue on Qulity-of-ervice Delivery in Vrible Topology Networks. Accepted for publiction. To pper rd-th Qurter J. A. tnkovic, T. Abdelzher, C. Lu, L. h, J. Hou, Rel-Time Communiction nd Coordintion in Embedded ensor Networks, Proceedings of the IEEE, 917): , July R. Krishnn nd J.P.G. terbenz, An Evlution of the TMA Protocol s Control Chnnel Mechnism in MMWN, Technicl report, BBN Technicl Memorndum No. 1279, Dimitri Bertseks nd Robert Gllger, Dt networks, 2nd edition, Prentice-Hll, Inc., First Derivtive d P P d P P p + 1)1 )+ 1 1 p) 1. Obviously, d P P 0forp p 0). Note tht d P P +1)) 0lsoforp 1 result is not considered. It is reuired tht 0 p 0 1. p 0 0isstisfiedif 1 0or but for this cse p 1nds. p 0 1isstisfiedif 1 +1)) or 1 +1) 1) or 1 +1) 1) + or or. +1))+ +

15 6\VWHP 7KURXJKXW '1 6\VWHP 7KURXJKXW '1 ) D 6 ' ) E 6 ' 6\VWHP 7KURXJKXW '1 6\VWHP 7KURXJKXW '1 ) ) F 6 ' G 6 ' Fig. 1. ystem throughput simultion results for the D10N100 topology s function the topology density /D for different vlues of p p 0,p p,, p p, p p nd p p) nd different vlues of 0.1, 0., 0.6 nd 0.9). 11 Accurcy of p It is reuired tht p is close to 1 or 1 1 p to 0 the better the pproximtion. Obviously, g0,p) g, 1) 0 closeto0or 1 p 1 p 1 p close to 0. Let g, p) 1 p nd the closer g, p) It cn be clculted tht dg,p) d 1 p 1 p) 0. Conseuently, s increses g, p) increses. The mximum is 2 ssumed for 1, g1,p) 1. Conseuently, the smller the vlue of the better the pproximtion. It cn lso be clculted tht dg,p) 1 1 p) 0. Conseuently, s p increses g, p) decreses. The mximum 2 is ssumed for p 0, g, 0). Conseuently, the higher the vlue of p the better the pproximtion. 12 Accurcy of ˆP P P P According to Eution 9) nd Eution 10), ˆP P P P is clculted s follows. ˆP P P P 1 1+p 1) 1 p) v 1+p 1) 1 p) N u V 1+p 1) 1 1 p) v 1 p) N u V 1+p 1) 1 p) 1 1 p) v 1 N. 1 Complementry of Theorem For D+ <, + D D+1)) D D+ D+1)) D+ D u V D D D+1)) 1 will be shown tht is stisfied. ssumes the mximum vlue when the vlue for is the minimum or In order to show tht D+1)) D)+ D+1)) 1. D D+1)) D+1)) cnbewrittens D+.Conseuently, D D+1)). D D+1)) D it is enough to show tht +1)) D+1 or +1) D) +1 D +1) 1 ) or + 1 D D D+ D 1 D or D D D or 1+) D 1+) or D. The ltter is lwys stisfied.

16 1 p Versus p 0mx nd p 0min In order to show tht p < p 0mx, it is enough to show tht or > 0. The ltter is lwys stisfied. In order to show tht p > p 0min, it is enough to show tht ) 2 +1) ) ) 2 +1 ) or > 2 +1) ) or 2 +1) ) 2 +1 ) +1)) < )) > ) 2 +1 or < 1or 1 < 1 ) ) 2 +1 ) ) 2 +1) ) ) ) ) or > 1 ) ) < 1) 2 +1 or 2 < + ) or > ) 1 or < ) or 2 < + ) 2 1) or 2 < or 0 < or < 0or 1 2 ) < 2 2. Giventht1 2 < 0, it is enough tht > is stisfied. Conseuently, p > p 0 min, if > is stisfied, while if < , p < p 0 min.