Optmzaton of Schedulng n Wreless Ad-Hoc Networs Usng Matrx Games Ebrahm Karam and Savo Glsc, Senor Member IEEE Centre for Wreless Communcatons (CWC), Unversty of Oulu, P.O. Box 4500, FIN-90014, Oulu, Fnland Abstract In ths paper, we present a novel applcaton of matrx game theory for optmzaton of ln schedulng n wreless ad-hoc networs. Optmum schedulng s acheved by soft colorng of networ graphs. Conventonal colorng schemes are based on assgnment of one color to each regon or equvalently each ln s member of just one partal topology. These algorthms based on colorng are not optmal when lns are not actvated wth the same rate. Soft colorng, ntroduced n ths paper, solves ths problem and provde optmal soluton for any requested ln usage rate. To defne the game model for optmum schedulng, frst all possble components of the graph are dentfed. Components are defned as sets of the wreless lns can be actvated smultaneously wthout sufferng from mutual nterference. Then by swtchng between components wth approprate frequences (usage rate) optmum schedulng s acheved. We call ths nd of schedulng as soft colorng because any lns can be member of more than one partal topology, n dfferent tme segments. To smplfy ths problem, we model relatonshp between ln rates and components selecton frequences by a matrx game whch provdes a smple and helpful tool to smplfy and solve the problem. Ths proposed game theoretc model s solved by fcttous playng method. Smulaton results prove the effcency of the proposed technque compared to conventonal schedulng based on colorng. Index Terms- Ln schedulng, graph colorng, topology component, fcttous playng (FP), soft colorng. I. INTRODUCTION Wreless ad-hoc networs need a multple access control scheme to avod collson due to smultaneous transmssons. The most conventonal multple access scheme n wreless ad-hoc networs s tme dvson multple access (TDMA) [1, 2], although other multple access schemes le combnaton of TDMA wth code dvson multple access (CDMA) have been also used. For comparson, TDMA/CDMA s easer to synchronze but provdes lower throughputs [3, 4]. In a TDMA wreless ad-hoc networ, schedulng between lns ncreases throughput. Schedulng n wreless ad-hoc networs s smlar to channel reusng n cellular networs.e. f two lns do not nterfere wth each other or at least ther mutual nterference to sgnal rato s less than a predefned margn, they can be actvated at the same tme slot and consequently not only throughput s ncreased but also transmsson delay s reduced [5, 6]. If two lns nterfere wth each other, they are referred to as adjacent lns. Therefore for each graph one adjacency matrx s defned. In [7, 8] problem of conflct free schedulng has been smplfed through splttng networ to multple networ realzatons or partal networ topologes. Each partal topology ncludes a complete set of lns that can be actvated smultaneously and consequently to maxmze the throughput, the remaned problem s just assgnng optmum usage rates (porton of tme) to each networ realzaton. Ths approach for schedulng problem has been adopted from colorng problem n mathematcs where a graph s dvded to some regons and each regon s colored wth one assgned color. But ths approach has two defcences. Frst, colorng does not have unque soluton and for any gven ln adjacency matrx there are usually more than one mnmal colorng scheme. On the other hand, when lns are supposed to be actvated n dfferent usage rates, ln colorng does not provde optmum result. To solve both defcences we ntroduce the concept of soft colorng. Soft colorng means more than one color can be assgned to each ln n dfferent tme slots. The outcome of soft colorng n schedulng problems s dvdng topology to topology components where each component s panted by an ndvdual color and components can have non-empty ntersecton.e. one ln can be member of more than one component n dfferent tme slots. Game theory has found many applcatons n smplfy problems n mathematcs and networng [9-11]. In ths paper, we use matrx games to model relatonshp between ln usage rates and component rates. Modelng the problem by game theory not only helps us to smplfy the problem by usng domnancy concept, but also provdes fast and relable convergence.
The rest of paper s organzed as follows. In Secton II, system model and problem defnton s presented. In Secton III, problem s formulated by matrx game. Smulaton results are presented n Secton IV and fnally paper s concluded n Secton V. II. PROBLEM DEFINITION AND SYSTEM MODEL Assume a mult-source wreless ad-hoc networ ncludng N nodes. Ths networ s defned as G( V, E, ζ, s, s,, ) 1 2 sm where V s set of node wth N elements, E s set of L vrtual wreless lns, ζ s set of M sources and s s set of sns correspondng to the th source where f source s sendng uncast data sze of ts sn set s one. Wreless propagaton for ths networ assumes the followng: 1. Omn-drectonal transmsson. 2. Presence of nterference due to smultaneous transmsson. 3. TDMA as multple access scheme for dfferent hops wthout nter tme slot nterference. A. Conflct Free Operaton Assume S j as the power, n db requred at node j, for transmttng node to reach the recevng node j at dstance d wth S S d where α s attenuaton factor. j j j Accordngly, by defnton of the conflct free schedulng, any node, j, recevng the sgnal from node m, wll be nterfered by ln l j f and only f Sm S, where β, n db, s acceptable nterference margn between two lns. In other word, ln l j s adjacent to l m for any m and any, j f S S. (1) m Alternatvely, the two lns are adjacent f S S β. (2) j mj + Whenever lj and l m are physcally adjacent.e. they have a common node or ether (1) or (2) hold, they cannot be panted by the same color. Usng (1) and (2), ln adjacency matrx whch s used to desgn colorng algorthm s defned. B. Conventonal Schedulng In general conflct free schedulng, for a gven nterference margn, s based on conventonal graph colorng technques. A smple and low complexty algorthm for mnmal colorng.e. pantng all lns wth mnmum requred number of colors s as follows, Step 1. Frst ln s assgned to the frst (=1) partal topology (color),.e. T 1 ={l 1 }. Step 2. Next unassgned ln l j s chosen. Step. 3. Usng (1), f for any value of, l j can be smultaneously actvated wth all members of T, t s added to T as T T U l j. Step 4. If for all, where max, last step cannot be run, l j s consdered as frst element of T max+ 1 and max max +1. Step 5. If all lns are allocated to T, we have a complete set of partal topologes and f not, go to Step 2 and run the last two steps for all remaned lns. T s computed by a colorng scheme le ths, form a complete set of partal topologes where each ln s member of only one of partal topologes. Followng smple example shows neffcency of colorng when lns must be fred wth dfferent usage rates. Assume 3 lns l 1, l 2, l 3 wth rates r 1 =3, r 2 =1, r 3 =2, where ln 1 can be actvated wth other two smultaneously and two others cannot be actvated together. Mnmal colorng schemes for these 3 lns wth ther assgned rates are as follows ) T 1 ={l 1, l 2 } and T 2 ={l 3 } requred number of tme slots s 5. ) T 1 ={l 1, l 3 } and T 2 ={l 2 } requred number of tme slots s 4. But obvously none of these colorng schemes are optmum and optmum schedulng for these lns s, ) T 1 ={l 1, l 2 } durng the frst tme slot and T 2 ={l 1, l 3 } durng the next two slots; requred number of tme slots s 3. Therefore n optmum soluton T have non-empty ntersecton and we call them components of the topology. A component s a set of lns that can be actvated at the same tme. Any sngle ln s also a component and we call them as frst generaton components and τth generaton of components means set of all components each wth τ members. Components of each generaton are parents of one n the next generaton. For our 3-lns example, we have 5 components as follows, T 1 ={l 1 }, T 2 ={l 2 }, T 3 ={l 3 }, T 4 ={l 1, l 2 } and T 5 ={l 1, l 3 }, where frst 3 components are parents of the last two ones. Consequently n general case to optmze the schedulng approprate components and ther optmal rates must be found.
III. MATRIX GAME FORMULATION Gven ln actvaton rate vector r wth I elements and component set C wth J elements, we defne followng payoff matrx H as follows, where s actvaton rate for th ln and s jth component. Assume H as payoff matrx of the mn-max zero sum game between two players where ther strategy sets are lns and components sets respectvely. In the sequel we prove that the mxed strategy vector of the second player gves optmum component rate. Theorem- mxed equlbrum of the zero sum game defned by payoff matrx (3) gves optmum schedulng for rate vector r. Proof: Assume x and y as mxed strateges of the players at equlbrum. Snce y s normalzed to 1 actvaton of components n average needs one tme slot. r as actvaton rate of th ln supported by y components s computed as follows, r~ = [ H]y (4) r where [ ] H s th row of the H. Therefore by component rate vector y, the ln wth mnmum supported rate whch s bottlenec of the game s, mnmum (3) = arg mn ( [ H] (5) Consequently optmum component rate vector must maxmze mn ( [ H] a y = arg max mn ( [ H] y (6) And theoretcally (6) s equvalent to x ( x T H y = arg max y mn, (7) where (7) defnes equlbrum for game defned by payoff matrx H and value mn ( x T H max y. Formulaton of the problem as a game gves us the chance to smplfy the model usng especal propertes of the games, le domnancy theory. For nstance n general case whle solvng ths game, because of domnancy, components tagged x as parent can be gnored because apparently any parent s domnated by ts chld. Therefore we just need to consder last generaton generated born from any ln. On the other hand, calculaton of mx strategy for lnear games s easy and can be performed by fcttous playng method (FP). FP property of the games s a feature of statonary games where players can update ther belef on other player s strateges based on hstory of ther decsons. Ths technque whch s used to solve matrx games, was proposed by Brown [13] and ts convergence for dfferent condtons was proved n [14-19]. FP algorthm for a mn-max zero-sum game has followng steps, Step 1. Intalzaton of xˆ as, ˆ [ ] T x =, (8) 0 H where [ ] H s and arbtrary row of H ( xˆ actually wll be Hy ). Then set teraton number =1. Step 2. Fndng best strategy for the frst player at th teraton as, = arg mn (9) m xˆ where 1,m xˆ, m s mth element of the ˆx. Step 3. Updatng ŷ as, ˆy = ˆy + [ H]. Step 4. Fndng best strategy for the second player at th teraton as, j n ŷ,n where = arg max (10) ŷ, n s nth element of the ŷ. Step 5. If We algorthm has not converged to equlbrum set + 1 then update ˆx as, xˆ = xˆ + [ H], (11) 1 j and then return to Step 2. Algorthm s δ converged f, ŷ,j xˆ, δ, (12) When the algorthm approaches equlbrum mxed strateges for both players are calculated by averagng over best strateges calculated at each teraton. 1
IV. SIMULATION RESULTS The proposed algorthm s smulated over randomly generated graphs wth nodes unformly dstrbuted over unt square. Sources and sns are also selected randomly and for each source-sn par shortest path (physcally shortest part that requres mnmum power) s calculated usng Djstra algorthm [20]. Attenuaton factor s assumed to be 4 and number of pacets to delver from each source to ts correspondng sn s randomly chosen by Posson dstrbuton. The proposed optmal algorthm s compared to the conventonal schedulng based on networ graph colorng and no schedulng case. The average number of requred tme slots to carry each pacet from source to ts correspondng sn s consdered as performance crteron and results are averaged over 1000 ndependent runs. Results are summarzed n Fgs. 1-4. Fg. 1, presents average number of requred tme slots versus nterference margn (β) for dfferent numbers of source-sn sessons when N =10 nodes. We can see from the fgure that the proposed schedulng outperformance conventonal scheme. In the case where we have 10 parallel source-sn sessons, optmum schedulng offers more schedulng gan compared to 5 source-sn sessons because when hgher number of paths must be scheduled, more paths s partcpatng n the schedulng process. Therefore every ln has more partners for smultaneous transmsson. In other word, number of last generaton of components s hgher and consequently schedulng gan s also hgher. Fg. 2, presents the same results for N=20. In ths case, when we have 10 parallel source-sn pars, both optmum and conventonal schedulng offer more schedulng gan compared to the 5 parallel pars but gan of the optmum schedulng s stll much hgher. Fgs. 3 and 4 present average number of requred tme slots versus number of parallel source-sn pars. We can see that the schedulng gan advantage of optmum schedulng compared to the conventonal schedulng ncreases wth the number of source-sn pars. For nstance, n the case of 10 source-sn pars, optmum schedulng needs between 11 and 25 percent less tme slots n average. V. CONCLUSION In ths paper a novel technque based on matrx games s proposed to solve optmal schedulng problem s wreless adhoc networ. In the proposed method, frst wreless topology s dvded to components whch actually are overlapped partal topologes. Then by assgnng approprate usage rates to the selected components optmum schedulng s acheved. The optmal rates for graph components are obtaned by calculatng mx equlbrum of a matrx game between lns and components. Usng matrx games for ths nd of problem helps us to smplfy the problem through specal features of the games, and usng fcttous playng method whch wors teratvely, provdes quc convergence. The proposed game model has been smulated for multpleuncast sessons and the results are averaged over 1000 ndependent runs. Smulaton results prove the effcency of the proposed method compared wth the conventonal schedulng. REFERENCES [1] D. D. Vergados, D. J. Vergados, C. Doulgers, and S. L. Tombros, QoS-aware TDMA for end-to-end traffc schedulng n ad hoc networs, IEEE Transactons on Wreless Communcatons, vol. 13, no. 5, pp. 68-74, Oct. 2006. [2] K. Oonomou and Stavraas, Analyss of a probablstc topology-unaware TDMA MAC polcy for ad hoc networs, IEEE Journal on Selected Areas n Communcatons, vol. 22, no. 7, pp. 1286-1300, Sep. 2004. [3] T. C. Hou, C. M. Wu, and M. C. Chan, Dynamc channel assgnment n clustered multhop CDMA/TDMA ad hoc networs, n proceedngs of 15 th nternatonal Conference on Personal, Indoor and Moble Rado Communcatons, PIMRC2004, pp. 145-149, Sep. 2004. [4] H. S. Ahn and S. W. Ra, A study on synchronzaton of hybrd TDMA/bnary CDMA, n Proceedngs of 57st IEEE Vehcular Technology Conference, VTC Sprng 2003, Jeju, S. Korea, Aprl 2003, pp. 1681-1684. [5] W. We, L. Xn, and D. Krshnaswamy, Robust Routng and Schedulng n Wreless Mesh Networs, proceedngs of SECON 2007, pp. 471-480, June 2007. [6] T. El-Batt and A. Ephremdes, Jont Schedulng And Power Control For Wreless Ad-Hoc Networs, n proceedngs of INFOCOM 2002, vol. 2, pp. 976-948, June 2002. [7] Y. E. Sagduyu, and A. Ephremdes, On Jont MAC and Networ Codng n Wreless Ad Hoc Networs, IEEE Transactons on Informaton Theory, vol. 53, no. 10, pp. 3697-3712, Oct. 2007. [8] B. Lorenzo, S. Glsc, Optmzaton of Relayng Topology n Cellular Multhop Wreless Networs, IEEE Mltary Communcatons Conference MILCOM 2008, pp. 1-8. [9] A. Ghonem, H. Abbass, and M. Barlow, Characterzng Game Dynamcs n Two-Player Strategy Games Usng Networ Motfs, IEEE Transactons on System, Man, and Cybernetcs, Part B, vol. 38, no. 3, pp. 682-690, June 2008. [10] D. Nyato, and E. Hossan, A Noncooperatve Game- Theoretc Framewor for Rado Resource Management n 4G Heterogeneous Wreless Access Networs, IEEE Transactons on Moble Computng, vol. 7, no. 3, pp. 332-345, Mar. 2008. [11] R. Banner and A. Orda, Bottlenec Routng Games n Communcaton Networs, IEEE Transactons on Selected Areas n Communcatons, vol. 25, no. 6, pp. 1173-1179, Aug. 2007.
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