Joint Routing and Link Scheduling for Wireless Mesh Networks through Genetic Algorithms

Transcription

1 Jont Routng and Lnk Schedulng for Wreless Mesh Networks through Genetc Algorthms Leonardo Bada, Alesso Botta IMT Lucca Insttute for Advanced Studes pazza S. Ponzano 6, 55100, Lucca, Italy Lucano Lenzn Dept. of Informaton Engneerng, Unversty of Psa va Dotsalv 2, Psa, Italy Abstract Wreless Mesh Networks (WMN) are emergng as an attractve technology for provdng broadband connectvty to moble clents who are just on the edge of wred networks, and also for buldng self-organzed networks n places where wred nfrastructures are not avalable or not deemed to be worth deployng. Ths paper nvestgates the jont lnk schedulng and routng ssues nvolved n the delvery of a gven backlog from any node of a WMN towards a specfc node (whch acts as a gateway), wthn a gven deadlne. As n a real WMN, schedulng and routng are assumed to be aware of the physcal nterference among nodes, whch s modeled n the paper by means of a Sgnal-to-Interference Rato (SIR). Frstly, usng a theoretcal model of a WMN we formulate the problem as an Integer Lnear Programmng (ILP) problem. Secondly, snce the problem cannot be dealt wth usng exact methods, we propose and use a technque based on Genetc Algorthms (GAs). To the best of our knowledge, GAs have never been used before for workng out these knds of optmzaton problems n a WMN envronment. We show that our technque s sutable for ths purpose as t provdes a good trade-off between fast computaton and the overall goodness of the soluton found. Our experence has n fact shown that GAs would seem to be qute promsng for solvng more complex WMN models than the one dealt wth n ths paper, such as those ncludng multple flows and mult-rado mult-channels. Keywords- Wreless Mesh Networks, Routng, Lnk Schedulng, Integer Lnear Programmng, Genetc Algorthms. I. INTRODUCTION WMNs are an emergng class of networks, usually bult on fxed nodes that are nter-connected va wreless lnks to form a mult-hop network [1]. Ther man goal s to provde broadband access to moble clents who are just on the edge of wred networks. WMNs can be used where cable deployment s not feasble or s too expensve, such as n remote valleys or rural areas, but also n offces and home envronments. End-users are served by nodes called mesh routers, whch are generally assumed to be statonary. Mesh routers are n turn wrelessly nterconnected so as to form a network backhaul, where rado resource management challenges come nto play. Moreover, some mesh routers are generally provded wth access (e.g. through wres) to the Internet and therefore can act as gateways for the entre WMN. Communcaton between any two mesh routers as well as from any router to gateways s mult-hop. Many of the WMN ssues are thus common to those of mult-hop wreless networks, such as determnng lnk schedulng n order to obtan hgh throughput effcency [2], [3] or selectng approprate routes between source and destnaton [4], [5]. However, the fact that mesh routers are fxed makes the backhaul of a WMN nherently dfferent from dstrbuted wreless networks (e.g. ad hoc networks), where the nodes may be portable devces. For example, problems such as energy consumpton are no longer an ssue. Also, the uncertanty about postonng termnals due to moblty or dffculty to communcate, as well as ther computatonal capablty, are mtgated. Ths makes t sensble to opt for a centralzed network management, as opposed to the dstrbuted approaches used for ad hoc wreless networks. In ths case, nodes act n a coordnated fashon under the supervson of a network entty whch determnes the management based on global knowledge of the network topology and addtonal condtons. A cross-layer approach where the routng and lnk schedulng functonaltes are jontly addressed has been extensvely studed n mult-hop wreless networks [4]-[7]. However, the problem of determnng, for example, the shortest deadlne wthn whch a specfed backlog vector can be jontly routed and scheduled between WMN nodes and a gateway, does not seem to have been analyzed n the lterature. Ths s the prmary objectve of the paper, whch focuses on two major nnovatons. Frstly, we formulate our problem through an ILP framework [8] by capturng the characterstcs both of the WMN topology and of the rado channel, whch allows us to determne the feasblty condtons for our problem. In the desgn of our framework we gve partcular emphass to nterference related aspects. In partcular, we employ the so-called physcal nterference model, whch computes the Sgnal-to-Interference Rato (SIR) at each actve node and compares t wth an approprate threshold [9]. ILP formulatons generally use another approach, named protocol nterference model, whch s smpler to apply but n our case may actually lead to oversmplfcatons. To the best of our knowledge, our ILP formulaton s the only one avalable that explctly addresses the physcal nterference model n ts orgnal verson wth lnear constrants and bnary varables. However, we beleve that one more mert of our ILP framework s to leave room for possble extensons to specfc cases of nterest, n whch a gven objectve functon s proposed. Secondly, we use Genetc Algorthms (GAs) to solve the cross-layer problem, and ths technque copes reasonably well wth our framework. It s known from the lterature [2] that fndng lnk actvaton patterns that satsfy traffc requrements and keepng nterference under control typcally causes NPcomplete problems. Ths means that the soluton to the problem

2 cannot be guaranteed to be found n polynomal tme. Exact approaches fal to solve the problem n a reasonable tme, even wth not very large topologes, e.g. wth 8-10 nodes. GAs are an optmzaton technque whch mtates evolutonary processes exstng n nature [10]. They do not guarantee to fnd the best possble soluton wthn a gven tme: even f they are customzed approprately, they only solve the problem optmally wth unlmted tme at ther dsposal. However, GAs often work n practcal cases as they are able to provde a good enough soluton n a reasonable tme. Moreover, they appear to be deal for handlng dscrete values, multple constrants and also multple objectves, as happens n problems of network plannng [11], [12], as well as wth the problem dscussed n ths paper. We would lke to stress that although GAs are often seen as a standard technque that can be used wthn any optmzaton framework, our problem requres an orgnal soluton to mplement n the GA, whch wll be examned n detal n the paper. The rest of ths document s organzed as follows: n Secton II we dscuss the lterature. In Secton III we outlne the basc assumptons of the model by descrbng the varables and the notaton utlzed. These are employed n Secton IV to formulate the ILP model. In Secton V we descrbe GAs and dscuss ther applcaton to our case study. Fnally, n Secton VI we present numercal results, and we draw conclusons n Secton VII. II. RELATED WORK There are many papers n the lterature [2]-[7], [13], [14] that can be related to the present work, as they nvestgate routng, schedulng, or both, through what can be seen as a Lnear Programmng framework. For example, [4] dscusses routng optmzaton for wreless networks, but the man focus s on sensor networks, and, as s common for such systems, energy effcency s consdered as the objectve. Also, there s no consderaton about mutual nterference of the nodes, whch s mportant n WMNs. The analyss of [13] s, on the other hand, more applcable to our scenaro, snce t deals wth throughput maxmzaton and focuses on nterference relatonshps. As far as the nterference model s concerned, usng the physcal nterference model s very rare n the lterature. A notable excepton to ths s [2]. In ths very recent paper, the evaluaton of the SINR relatonshps s used to fnd feasble schedules n a WMN, and computatonally effcent solutons are proposed to ths end. Unlke our nvestgaton, a lnk actvaton pattern s sought n order to meet pre-determned lnk weghts whch can correspond to the routes, whereas n our analyss we solve both routng and schedulng jontly. Ths places our work n the feld of cross-layer solutons, whose nvestgaton ncludes also channel assgnment, as n [14], where a jont channel assgnment and routng problem s approached, but s solved through heurstcs. Exact solutons are dscussed n [5], where a jont channel assgnment and routng s proposed through an ILP framework. The schedulng ssue s consdered as the soluton to a prelmnary optmzaton, where only lnks that can be scheduled together are used. However, ths paper only accounts for the protocol nterference model, and n the routng phase non-nteger lnk actvaton values are utlzed. Smlar consderatons hold for [3], where a two-phase algorthm s ntroduced. Frst, a routng LP s solved that takes the protocol nterference model nto account but does not do any schedulng. Ths soluton s then scheduled over tme usng a dfferent algorthm. In [6], an optmzaton approach s proposed to jontly solve lnk schedulng and routng, as we do n the present paper. However, the meanng of lnk schedulng s dfferent, as n [6] the feasblty of a vector of rates s smply sought, whereas our am s to determne a lnk actvaton pattern, whch delvers the backlog of every node to the gateways wthn an assgned deadlne. Ths framework s further extended n [7] to nclude channel assgnment as well. However, agan the fact that the bnary varables of lnk actvaton are relaxed to ratonal values s shown n [7] to be a lmtng assumpton, whch may lead to naccuraces n the soluton. Also, all prevous contrbutons presentng a cross-layer approach take nto account the protocol nterference model, whereas we use the more sutable physcal nterference model. Fnally, our paper also presents an orgnal contrbuton n terms of the technque used to solve the optmzaton. GAs are n fact very commonly used for a prelmnary plannng of wreless networks n general [11], and ths s true for WMN as well. For example, n [12] there s an overvew on how to use GAs to help the deployment of nter-urban mesh networks. Another very recent paper [16] employs GAs for sensor networks. Even though the knd of network s dfferent, the authors present some physcal layer consderatons about channel actvaton and the mutual nterference of nodes. However, n all these nvestgatons dealng wth network deployment, the usage of GAs s mostly motvated by the hgh complexty of the problem, whch prevents t from beng solved wth exact technques. On the other hand, the speed of GA n gvng good solutons to the problem s not exploted, snce the tme for the optmzaton process to converge s not as relevant as n shorter tme-scale problems such as routng or schedulng. Indeed, we beleve that n our problem we are able to show ths addtonal advantage offered by the computatonal effcency of GAs. To our knowledge, ths fact s not very frequently explored n the lterature, thus makng our work nnovatve. III. BASIC ASSUMPTIONS OF THE MODEL We represent the backhaul of a WMN as a drected graph G = ( N, E ), whch conssts of N = N nodes representng the mesh routers of the WMN, connected by drected edges of the set E representng potental lnks between termnals. Notaton e = (, j) E means that N s the transmtter node of lnk e and j N s the recever. We denote wth Y N the set of the gateways, whch are any-cast end destnatons for the mesh routers. In general, not all pars of nodes are connected through an edge. We denote the one-hop nput and output neghbors set of a node as S and R. In other words, S and R are the set of nodes for whch an edge exsts n E to and

3 from node, respectvely,.e. S = { j N ( j, ) E } and R = { N (, ) E }. j j Note that n a WMN t s reasonable to consder every par of nodes as beng connected through an edge n E, whch means E = {(, j) N N j} and R = S = N \{ }. Ths s due to the absence, n real WMNs, of transmttng power lmtatons (mesh routers can be easly attached to a power outlet), and, through approprate power control, t s therefore vrtually possble to reach any other node. Also, the edges n E only descrbe a vrtual lnk between nodes of the WMN, whch can even be unused f the routng algorthm detects that they are not worth actvatng. In general, other technques often consder an a pror network prunng, but ths may lead to approxmatons when formulatng the problem, whch we want to avod. In fact, our methodology apples wthout any restrctons to every scenaro, even strongly connected ones, as we leave open the possblty for any route through the set of nodes. Of course, ths also leads to a heaver problem n terms of computatonal complexty. Hereafter for the sake of smplcty we also assume that nodes can use a sngle power level. Ths s not a lmtng assumpton, as multple power levels can be taken nto account by consderng multple edges for the same par of nodes, wthout changng the ratonale of the analyss. Smlarly, we assume that all nodes are homogeneous n terms of the number and knds of rado nterfaces they own, as well as the frequency bands they are enabled to transmt on. Indeed, the extenson of WMN management to the multple channel case looks promsng and several standards are envsoned to explctly nclude support for such a case. All these extensons (multple channels, multple power levels, etc) can be seen as extensons of the basc framework dscussed here and are left for future research on ths topc. We assume that the WMN system operates n synchronous tme slotted mode where tmeslots are labeled va nteger numbers 0,1,..., t,.... Every edge (, j) E s also assocated wth a transmsson rate r j and a path gan g j. The former descrbes the number of packets, assumed to be constant, that can be sent durng a tmeslot over the edge (, j ), whereas the latter s the nverse of the channel attenuaton (transmtted power over receved power) between nodes and j and wll be used n the followng when modelng nterference between transmsson lnks. Both r j and g j varables can be collected nto matrces R = ( r j ) and G = ( g j ). Another assumpton made for analytcal tractablty s that t s not possble to underutlze an edge below the avalable rate r j, unless the transmtter does not have enough packets to send. Ths generally prevents the sender from splttng the data nto parts smaller than the whole rate of an edge However, ths would be really benefcal n a neglgble number of cases; thus, ths assumpton s not restrctve at all n practce. To solve the jont lnk schedulng and routng problem, we defne a 0-1 schedulng varable ( ) j x t for every (, j) E, as 1 f j s actve on tme slot t xj ( t) = 0 otherwse In other words, xj ( t ) denotes whether or not there s a data transmsson (.e. the lnk s actvated) on tme slot t. These varables are bound to be nteger, varyng over a dscrete (slotted) tme, so as to determne a tme-dvson schedulng pattern for the WMN backhaul [2]. Smlarly to the analyss presented by [6], we remark that the dervaton of a schedulng pattern of lnks mplctly determnes the routng as well. However, rather than approachng the routes on a per-flow bass, we derve the routes by lookng at the dynamcs of the lnk actvaton over tme. Unlke other papers [5], [7], n ths work we mpose the xj ( t ) varables to be strctly bnary and varyng over dscrete tme t. In other words, we explctly avod relaxng constrants about varables to be nteger, whch s an approxmaton that can lead to strongly sub-optmal results n the ILP. For the sake of analytcal tractablty, we wll focus on perodc schedulng, where a frame of duraton T slots s assumed to set the cycle of lnk actvatons. Ths means that lnks are actvated accordng to the soluton found for t between 0 and T 1, and ths pattern can be repeated dentcally every T slots. We assume that each node supports a sngle flow towards a gateway. The amount of traffc per node s known n advance and s already avalable at the begnnng of the frame at the non-gateway nodes. We leave for future work any extenson about packet arrvals delayed throughout the whole frame. The goal wthn a sngle frame s to delver the traffc to the gateways. Ths can be done by sendng t drectly to a gateway or by relayng to one or more nodes before reachng the destnaton gateway, dependng on the status of WMN backhaul lnks. In the latter case there s flow traffc aggregaton at some ntermedate node towards a gateway. The progress status of the transmsson to the gateways s modeled through varables q ( t ), whch descrbe at every tme slot t the queue length at each node. In realty, these are more lke auxlary varables, snce, as shown n the followng, they can be put n relatonshp through flow constrants wth the bnary varables xj ( t ). We assume that, for every t n 0, 1,, T 1, q ( t ) represents the amount of traffc n queue at node, that needs to be delvered to one of the gateways before the end of the frame. The connecton between q ( t ) and xj ( t ) s such that q ( t ) represents the amount of data before the applcaton of the transmssons dentfed by xj ( t ), whereas q ( t + 1) descrbes the outcome of these transmssons. For ths reason, q ( t ) vares over tme, so that at the begnnng of the frame q (0) represents the overall amount of data (.e. the aggregated demand from ts assocated users) to delver for node, and q ( T ) descrbes the resdual backlog at node after the applcaton of the jont routng and lnk schedulng pattern. IV. PROBLEM FORMULATION AND MAIN CONSTRAINTS The problem of assgnng meanngful 0-1 values to xj ( t ) can be seen as a flow optmzaton problem subject to three knds of constrants. The frst one s related to the flow conservaton and delvery of all traffc to the gateways. Also, two other types of condtons are needed to check the feasblty of the lnk actvaton pattern. Both of them are related to the feasblty of

4 smultaneous actvatons of lnks, whch s generally benefcal as t mproves the transmsson parallelsm. Only compatble transmssons can be scheduled n the same tme slot, where compatblty means possblty to be used smultaneously. Modelng ths property among wreless lnk transmsson s challengng, and several models have been proposed [9]. To check whether two transmssons can coexst, two condtons must be met: the rado equpment of a sngle node cannot be used for too many tasks (.e., transmsson/recepton). Accordng to whether the channel s full duplex or half duplex [7], t s ether possble to have at most one transmsson and one recepton at the same node per each slot, or one sngle task comprsng both recepton and transmsson. nterference ssues also need to be checked. Several models can be used, and we wll refer to the physcal nterference model [9]. We classfy the three knds of constrants: flow constrants, drect compatblty constrants, and nterference constrants. These are dscussed n ther respectve subsectons. A. Flow constrants The flow constrants nclude flow conservaton for every tme slot t at each node: ( ) q ( t + 1) = max 0, q ( t) x ( t) r + j j j R (1) + mn( q ( t), x ( t) r ) N, t = 0,, T 1 ( j j j ) S j The formulaton of ths constrant n a lnear verson accounts for the possblty of havng transmsson and recepton smultaneously,.e. a full duplex case s consdered. In fact, the rght-hand term sum accounts for both ncomng and extng packets. However, the fact that the actve ncomng lnks (n the frst term) and the actve outgong lnks (n the second term) can be at most one s mplctly accounted for. If the channel s half duplex, no modfcaton s needed, snce the condton s even more restrctve: at most one among all ncomng and outgong lnks can be actve. Addtonally, at tme T everythng has to be delvered to the gateways: q (0) = q ( T) (2) N Y We also assume that the gateways do not generate traffc. The formulaton of a related constrant s not strctly necessary, but t s useful to smplfy the resultng algorthm. j R q (0) = 0, Y, (3) x ( t) 0, Y, t 0,1,, T 1. (4) j B. Drect compatblty constrants The constrants that we call drect compatblty constrants relate to the mpossblty of utlzng a transcever equpment of a node for more purposes than s desgned for. For full duplex lnks, the drect compatblty constrants can be wrtten as j R S j x ( t) 1 N, t = 0,, T 1 (5) j x ( t) 1 N, t = 0,, T 1 (6) j However, wreless lnks are ntrnscally half-duplex, unless specal technques are employed, whch mplement full duplexng, such as drectonal antennas [17] or multple channels [7]. If there s no frequency or spatal separaton between transmtter and recever, f a node s transmttng, any smultaneous recepton wll be destroyed by the self-nterferng transmtted power. Thus, the rght constrant for a WMN s a half duplex one. To account for a half duplex channel, the constrants above are smply merged so as to form j R x ( t) + x ( t) 1 N, t = 0,.., T 1 (7) j S j C. Interference compatblty constrants j The physcal nterference model evaluates the Sgnal-to- Interference Rato (SIR) of every transmsson and assumes that, n order to be successful, the receved power at every actve recever has to overcome a SIR threshold called γ. Even though γ can be a dfferent value for every node, f the traffc flows are homogeneous and the modulaton technques are the same, t s sensble to use the same threshold for all the nodes. Also, for the sake of smplcty and wthout loss of generalty, we omt ambent nose terms, whch could be ncluded by consderng the SINR (Sgnal-to-Interference-plus-Nose Rato) nstead of the SIR. Ths does not lead to any sgnfcant changes n the mathematcal formulaton. The nterference compatblty constrant can be wrtten as γ x ( t) j g x ( t) g x ( t) (, j) E, t = 0,, T 1, j kj k k S j\ { } R k \{ j} j whch s n accordance wth the most commonly used defnton of SIR [9]. The physcal meanng of ths expresson s as follows. Assumng all lnks use the same power, the actvaton of lnk from to j at tme t, correspondng to havng xj ( t) equal to 1, s subject to havng an SIR on ths lnk greater than or equal to γ, whch s obtaned by checkng whether the rato between the useful power (numerator term) over the nterferng power plus nose (denomnator) s greater than or equal to γ. Note that, to be meanngful, the nterference constrant must be appled to actve lnks only. Ths s the reason behnd the mathematcal formulaton of (8), where f xj ( t ) = 1, the above nequalty holds and the term xj ( t) can be removed from both left-hand and rght-hand terms, whereas f xj ( t ) = 0, the above nequalty s trvally always verfed. In order to have a lnear constrant rather than a quadratc one, the followng artfce s employed. Rearrange (8) as: g x ( t) γ x ( t) g x ( t) j j j kj k k S \{ } R \{ j} (, j) E, t = 0,, T 1. j k (8) (9)

5 Ths s stll a quadratc constrant. However, after some manpulatons we can derve the followng lnear relatonshp as the nterference constrant: g γ g (( x ( t) ) + x ( t) 1 ) j j kj k j k S \{ } R \{ j} j (, j) E, t = 0,, T 1 k (10) Ths formulaton can be shown to be equvalent by consderng each possblty of xj ( t ) beng ether 0 or 1 and relyng on constrant (7), whch bounds the nner-most summaton to be always less than or equal to 1. Note that ths would also hold true n the full duplex case, snce the role of constrant (7) could be played by (5) and (6) together. Even though the problem can be entrely formulated wthn an ILP framework, the soluton s hard to fnd wth exact methods. Ths happens snce the problem can be shown to be NPcomplete [2]. As t wll be shown n secton VI, the problem becomes untreatable even wth a lmted number of nodes,.e. more than 5 mesh routers ncludng a gateway. The computatonal complexty s also strongly dependent on T. Heurstc solutons [14] mght work n certan cases, but they fal to adapt to dfferent network scenaros. For these reasons, we propose n ths paper a selfconfgurable and effcent methodology based on Genetc Algorthms, whch wll be explaned n detal n the next secton. V. A GENETIC APPROACH Genetc Algorthms (GAs) are a meta-heurstc technque employed to solve optmzaton problems, whch mtate Natural Selecton,.e. the process of adaptaton to the envronment performed by lvng bengs [10], [18]. GAs are an appealng approach to solve the complex problem stated n the prevous sectons. Among ther most nterestng features, GAs are able to fnd good solutons to an unconstraned problem n a reasonable tme, and they always fnd at least one good suboptmal soluton, does not requre a dfferentable objectve functons and can be talored to handle any sort of constrant, can easly handle dscrete problems by choosng a dscrete alphabet of symbols (e.g., nteger numbers) for the chromosome, can scale wth the problem by changng the setup of some parameters (e.g., number of ndvduals n the populaton), can be customzed to nclude some heurstcs and experts knowledge n the generaton of the ntal populaton and n the desgn of the genetc operators. For these reasons, we chose GAs as the heurstc approach to solve the problem formulated n Secton IV but wthout relaxng any of the constrants, ncludng the nteger constrant of varables xj ( t ). A. Genetc Algorthms: Background A GA determnes, rather than a sngle soluton, a whole populaton consstng of ndvduals, whch are all canddate solutons to the problem. The dstnctve features of each ndvdual are coded nto a structure called chromosome. The chromosome s a strng of genes, whose values can be chosen n a set of symbols. An applcaton-dependant process generates the ndvdual by decodng ts chromosome. The symbols used as values of the genes are usually bnary, nteger or real numbers, dependng on the nature of the problem. Once an ndvdual s generated, a ftness functon s used to evaluate ts goodness as a soluton to the problem. Usually, low values of ftness functon are gven to the best ndvduals (mnmzaton problem). For the sake of smplcty, n the followng we wll blur the defntons of ndvdual and chromosome. A GA starts wth an ntal populaton generated ether randomly, or wth some heurstc approach that explots the knowledge of an expert n the problem doman. The algorthm then proceeds n steps called generatons. At each generaton t, a new populaton P(t+1) s evolved from P(t). As generatons pass, the populaton should mprove globally thanks to the applcaton of genetc operators that mmc the natural evoluton mechansms. To ths am, the best ndvduals are chosen from P(t) (selecton) to be mated (crossover) and slghtly modfed (mutaton), so as to create the new populaton P(t+1). The selecton operator s used to decde whch ndvduals n P(t) should be chosen to generate P(t + 1). Optonally, an elte of the selected ndvduals (.e. a small number of the best performng ndvduals) survves and s moved from P(t) to P(t+1) wthout any change. The crossover operator conssts n choosng some of the ndvduals and matng them, that s, substtutng them wth ther chldren,.e., ndvduals generated by mxng the genetc materal n the parents chromosomes. The actual mplementaton of a crossover operaton very much depends on the codng schema of the chromosome. Fnally, the mutaton operator ntroduces some new genetc materal n the populaton by randomly modfyng the values of some genes. Agan, dfferent knds of mutaton operatons can be defned to handle dfferent sets of symbols. The populaton contnues to evolve untl a stoppng crteron s fulflled, the smplest beng a maxmum number of generatons. The overall basc GA algorthm s shown n pseudo-code n Fgure 1. If crossover and mutaton are general enough, GAs can be shown to allow the exploraton of the whole soluton space. If an optmzaton goal s set, they are bound to fnd the optmal soluton, even though t cannot be guaranteed that t wll be the optmal one, nor can the tme to fnd t be predcted. However, snce the executon tme s generally rapd, GAs are also nterestng for practcal purposes as they can be seen as fast procedures to fnd a good enough soluton to the problem. Ths gves them an advantage wth respect to exact technques such as Branch and Cut used n commercal solvers, snce any soluton produced by a GA s drectly applcable. Therefore, GA could be used to operate onlne WMN management, where the soluton may be teratvely updated. Ths could also be an nterestng development of the present analyss for future work. B. A GA-based approach for the ILP problem Whle some classes of problems can be solved by drectly applyng a basc verson of a GA, more often the development of such an algorthm for a specfc problem s an engneerng process that nvolves a good amount of desgn and talorng.

6 Indeed, the desgn of a GA ncludes fndng sutable representaton schemas, codng strateges, genetc operators, values of parameters, etc. Furthermore, f the problem s constraned, lke the ILP formulated n Secton IV, we are forced to select and adapt approprate constrant-handlng methods from the ones avalable n the lterature [19], [20]. The frst step when desgnng the GA s to dentfy how to mathematcally represent a soluton as an ndvdual, n order to create a populaton for the algorthm. In our case, gven the natural bnary formulaton of the problem, we consder genes to be bnary dgts. Thus, the chromosomes are coded as sequences of bts representng the varables xj ( t ), sorted frst nternally to each frame by any orderng of the edges, then frame-by-frame n an ncreasng order. Formally, the genetc map of any ndvdual s: ( x1 (0), x2 (0),, x E (0), xe ( t), x1 ( T 1), x2 ( T 1),, xe ( T 1)), where ndces 1,2,, E refers to a sutable orderng of the set E. To generate an ntal populaton, composed of 200 ndvduals, we nvestgated the use of several heurstcs for routng and schedulng problems, but currently none of them seems to sgnfcantly mprove the solutons wth respect to a completely random ntal populaton. Nevertheless, we observed that good canddate solutons show overall a count of actve lnk that s much less than the number of nactve lnks. Thus, to speed the convergence of the GA, we non-unformly generate the random ntal populaton wth a rato of actve lnks equal to 0.2 over all lnks. However, ths pont may deserve further nvestgaton n future research. The GA proceeds by teratvely modfyng the populaton, that s, by cyclcally applyng the selecton, the crossover, and the mutaton operators as descrbed n Secton V.A. As the selecton operator, we use the robust and well-known stochastc unversal samplng [18]. As regards the other operators (crossover and mutaton), we developed our customzed versons. Our codng schema has two granularty levels: the lnk level, represented by a sngle gene, and the frame level, coded by the overall confguraton of the network for one tme slot. We desgned our operators so as to work on both levels of granularty. The crossover operator s the 0.5-unform crossover [18]. The standard verson of ths operator chooses the value of each gene n the chromosome of a chld between the two values of the parents, wth a unform probablty. Ths s a lnk-level granularty. Nevertheless, t can be useful to apply the same approach on the frame level, so as to explot the knowledge already dscovered about whole tme slots. Thus, our modfed unform crossover may act, wth a unform probablty, on one of the two dfferent granulartes, mxng sngle bts or whole tme slots from the two parents to generate the chld. A smlar approach was used to develop the mutaton operator. We recall that the am of ths operator s to ntroduce some new prevously unexplored solutons n the populaton by slghtly modfyng the current ones. Thus, our mutaton operator can perform, wth unform probablty, one of the followng operatons: ntalze P(0 ) repeat evaluate P(t ) va ftness_functon; apply selecton to choose parents; apply crossover to generate offsprng apply mutaton to offsprng generate P(t+1) ncrease t by 1 untl a termnaton condton s verfed Fg. 1. The pseudo-code of a basc GA. mutate the chromosome on a lnk-level granularty by the unform random mutaton [18], scramble some of the tme slots of the chromosome (e.g. swtch slot number 1 and number 5), replace some tme slots wth a duplcate of other slots of the same chromosome (e.g., replace slot number 5 wth a copy of slot number 1), replace some slots wth empty slots. To satsfy the constrants, we used two dfferent technques. We dvded constrants nto two classes: constrants that have to be satsfed by each ndvdual generated durng the algorthm, and constrants that can be unsatsfed by some ndvduals. The frst class ncludes the drect compatblty constrants of (7). The second class ncludes the flow constrants of (1) and (2), and the nterference compatblty constrants of (10). Constrants n the frst class are always satsfed by means of a repar process, whch s performed after the applcaton of any genetc operator that mght produce an nfeasble ndvdual. For nstance, suppose that the mutaton operator generated an ndvdual n whch, at some tme, a node actvates two output lnks n the same tme slot. In ths case, the repar randomly deactvates one of the lnks, and fxes the ndvdual. Another case handled by repar s the actvaton of an output lnk by a node that has no more packets to send. Snce repar s performed each tme a genetc operator s appled, t must be desgned to be an extremely fast and effcent routne. Thus, we decded to deactvate conflctng lnks n a random fashon, and to repar only easy constrants that are the bass to derve all the ILP problems n Secton IV. Nevertheless, further research could lead to a more effectve repar process based on a preevaluaton of all the possble fxed ndvduals generated by an nfeasble one. Constrants n the second class are handled by allowng nfeasble ndvduals to survve n the populaton. Those ndvduals are gven hgher values n the ftness functon by means of penalty functons. The penalty s computed n the followng way: T 1 j ρ = q (0) q ( T ) + p ( t), N Y t = 0 (, j) E (11)

7 Fg. 2. Grd topology wth 5 or 9 nodes, ncludng a gateway. where pj ( t) descrbes the nterferences volatons at tme slot t, that s p ( t) j j j kj( ( k ) j ) = k S j \{ } R k \{ j} 1 f g γ g x ( t) x ( t) + 1 < 0 0 otherwse (12) The ftness functon can also ncorporate, by a lnear combnaton, some metrcs of the network that we want to optmze. Interestngly, the best results, both n terms of convergence speed and goodness of the soluton found, were gven by also ncludng a (small) penalty proportonal to the number of actvated lnks per slot,.e. to the sum of xj ( t ) over all edges and the whole frame. Ths s because actvatng too many lnks not only causes more nterference, but also prevents the GA from tryng alternatve routes by means of crossover or random mutaton, whch s due to the reparaton. Ths vanlla approach can be easly modfed to ncorporate any other network measures, such as global nterference, throughput, mnmum number of tme slots, etc. As stated above, the typcal termnaton condton of a GA s a fxed maxmum number of generatons. We used a hybrd stoppng condton whch stll stops the GA after a maxmum of 200 generatons and tres to perform an early stop n two cases: a good feasble soluton s found quckly, or the problem seems to be nfeasble. The dea s that, f we already are n the feasble regon, we are not nterested n optmzng the network metrc much more, and that f the problem seems nfeasble, we should gve up early wth the best soluton found. Thus, n the former case, we perform an early stop f, after a frst feasble soluton s found (that s, a soluton wth > 0), we do not fnd any other better soluton n 5 generatons. In the latter case, we perform an early stop f we have not found any feasble soluton and we have notced no mprovements n the last 50 generatons. VI. NUMERICAL EVALUATIONS To evaluate the performance of our GA, we consder a grd consstng of 35 m 35 m squares, as reported n Fg. 2. Each square can be occuped by at most one node, accordng to a pre-determned pattern ndcated below. A specfc square occupancy dentfes a dfferent scenaro nstance. The node s randomly placed (wth unform dstrbuton on both coordnates) wthn the square. The ratonal behnd ths approach s to model channel varatons through changes n the network topology, whle at the same tme keepng constant some parameters, e.g., the number of nodes. For ths reason, any nstance of the same scenaro can also be vewed as a dfferent topology whch can be created on the same physcal network, where nodes have fxed placements but the channel s tme-varyng. We consder two scenaros, both wth a gateway placed n the black square. In the frst one, whch conssts of 5 nodes, beyond the gateway node, 4 mesh routers are placed n each of the dark grey squares. Four addtonal routers are placed n the lght grey squares so as to form the second scenaro, wth a 9- node topology. For each scenaro we generated 20 dfferent topology nstances by varyng the node poston wthn the square t belongs to. The graph resultng from node placement s also determned by consderng the path gan of each edge of length d as 3.5 d proportonal to and ts rate as a dscrete value functon of the dstance, determned as constantly equal to 8 packets/slot for d below 50 meters, 4 packets/slot between 50 and 75 meters, 2 packets/slot between 75 and 100 meters, and 1 packet/slot for larger dstances. We assumed half duplex channels and SIR target γ equal to 3.0 (n lnear scale) for all the recevers. We mplemented the GA algorthm as dscussed n Secton V, usng the procedures contaned n the genetc package of MATLAB Release 2006a [21] as a bass. The GA was run fve tmes for each topology nstance, n order to avod partcularly unfortunate cases where the GA termnates n a dead end of the state space. The plotted results refer to the best soluton out of the fve trals. Whenever feasble, to test the goodness of the solutons found by the GA, we also mplemented an exact ILP soluton technque usng the LPSOLVE model solver [22]. Both algorthms were tested for the 20 dfferent topology nstances mentoned above. As performance metrcs, we consdered both the fracton of cases (.e. topology nstances) n whch the GA fnds a feasble soluton (.e., a soluton whch allows the delvery of the backlog from any node to the gateway wthn the frame duraton) wthn the above termnaton condtons, and the delvery rato (.e., the rato of delvered packets over the total traffc of each node) of the best soluton found. We show detaled results consderng GA performance, and also comparng t wth exact optmzaton technques for the 5- node topology. It s hard to make a detaled comparson of both genetc and exact algorthms on topology wth a hgher number of nodes due to the computatonal complexty of exact technques. In a sense, not only s the Genetc Algorthm more computatonally effcent, but t also has the consderable advantage of beng more scalable. In Fg. 3, we show the performance of the GA n the 5-node network, for the case where the load to delver to the gateway s fxed for each node to 10 packets, and we vary the frame length T. For comparson, exact results are also plotted. As expected, the fracton of feasble solutons found s an ncreasng functon of the frame length, for both GA and the

8 Fg node scenaro, GA performance as a functon of the frame length. Fg node scenaro, GA performance as a functon of the number of packets per node. exact technque, snce a larger T offers a hgher degree of freedom n accommodatng the packets over the schedule. It s possble to see that for low and hgh values of T, the performance of the GA matches the exact results perfectly. For ntermedate values, the GA slghtly underestmates the solvablty of some nstances, snce t may fal to fnd an exstng feasble soluton. However, n the worst case the rato n fndng a feasble soluton whenever t exsts s 61.3% for T = 9, whereas t s consderably hgher for any other case. However, even when the GA fals to fnd an exact soluton, ether because the optmzaton stops to a suboptmal value or snce t does not actually exst, the delvery rato acheved by GA s stll farly hgh. For small values of T, the GA s always able to delver 80% of the traffc or more, whereas for T 8 ths rato s above 96%. Ths represents a very mportant advantage of GA n practcal mplementaton, as t gves a soluton n any case, and when ths s not the optmal one t s stll very close to t. Fg. 4 shows the result of another smlar nvestgaton, where T s kept constantly equal to 10 and nstead the load per node s changed. The trend s reverted, snce the hgher the load the more dffcult t s to have a soluton and also to fnd t through the GA. However, n ths case too the worst performance of the GA s an exact soluton of 60% of the cases, n relaton to a delvery rato whch s very hgh n any case. For example, f the load per node equals 14 packets, only 20% of the topologes admt a soluton, 75% of whch are found by the GA. However, the delvery rato s larger than 95%. From an nformaton theory pont of vew [9], these curves may also be used to dscuss network capacty. In ths case, the "crtcal" load of the network,.e., the value around whch the fracton of feasble allocatons drops sgnfcantly, s 12 packets per node, whch corresponds to a maxmum capacty of around 4.8 packets/slot. In Fg. 5 the same results as Fg. 3 are reported for the 9- node scenaro. The trend s more or less qualtatvely smlar, though due to the larger number of nodes the tme to delver all packets becomes larger. Here, t was mpossble to nclude exact results as well, snce the network s already too large to keep the executon tme of any exact algorthm wthn reasonable bounds. Lkewse, Fg. 6 reports the same analyss as Fg. 4 for the 9-node scenaro. Due to larger network sze, n ths case the tme frame was fxed to T = 20. Agan, the range between 8 and 12 packets per node s crtcal for the network. The solutons found by the GA ndcate that the 9-node scenaro s able to accommodate 10 packets per node n 50% of the cases, that s, the capacty s approxmately 4.0 packets/slot. The slght decrease wth respect to the 5-node topology s perfectly n lne wth the larger network sze and also wth the fact that the nodes added n ths scenaro are further from the gateway (so they have both a lower rate for ther connectons and a hgher number of hops). These results seem to suggest that the GA scales suffcently well for larger topologes. To understand n more detal the complexty of the algorthm, and n order to have comparson results wth the exact technques, we can refer to Fg. 7, where a complexty analyss s performed. Here, we focus on the only nstances of the 5-node scenaro where a feasble soluton was found by both algorthms, and we measure the complexty through the followng performance ndces: a) number of evaluatons of the ftness functon made by GA; b) smplex teratons performed by LPSOLVE. Ths gves a rough dea of how the algorthms scale when the sze of the problem ncreases. Moreover, we vary the frame sze T, snce the complexty of the problem strongly depends on t. As shown n the fgure, whereas the exact technque explodes already when T s changed from 7 to 10, the GA complexty stays more or less constant. Indeed, t even slghtly decreases when the frame length s very hgh, snce n these cases a soluton s found very rapdly, as s reasonable to expect. Ths proves how good the GA s n fndng a quck vald soluton to easy problems. In practcal cases, t s possble that the network resources are not fully utlzed, e.g., the traffc per node s sgnfcantly lower than what can be allocated over an entre frame. However, exact technques can fal to quckly solve the problem, due to ts large sze In ths case, GAs can be seen as a

9 Fg node scenaro, GA performance as a functon of the frame length. Fg node scenaro, GA performance as a functon of the number of packets per node. very good alternatve to heurstcs, snce by modfyng ther meta-parameters they are able to adapt themselves to dfferent problem nstances. VII. CONCLUSIONS In ths paper we have nvestgated jont lnk schedulng and routng strateges for Wreless Mesh Networks. We have proposed an optmzaton framework makng use of an entrely ILP formulaton, where we partcularly amed at keepng the nteger constrant of lnk actvaton varables and adoptng the more realstc physcal nterference model. Ths led us to the formulaton of an ILP problem whose soluton captures both levels of lnk schedulng and routng n a cross-layer fashon, by supplyng a perodc lnk actvaton pattern whch s able to delver a gven amount of traffc to the network gateways. The man fndngs are that the physcal nterference model s stll treatable wthn the ILP framework. The hard part of the problem s due to the nteger constrant, whch causes the computatonal complexty to grow exponentally, both n the number of nodes and n the length of the tme frame. Due to the nherent complexty of solvng such a problem, we also proposed a fast and effcent soluton technque, namely Genetc Algorthms. After havng dscussed theoretcal prncples of GAs, we ntroduced several orgnal mplementaton parts n order to obtan effcent GAs for the problem under nvestgaton. Fnally, the proposed GA has been tested n two wreless mesh network scenaros. The numercal evaluatons show that the GA s able to solve both scenaros reasonably well, and also scales well, whereas exact optmzaton technques are unable to solve the larger topologes. The soluton found by GA s not always optmal. However, t s always very close to the optmum. Moreover, the GA s a very good approach for realstc cases where feasble solutons are easy to fnd, snce n these cases they converge very rapdly, compared to other technques, to a soluton whch s good n practce. For these reasons, we beleve that GAs could be very useful tools for a Fg. 7. Computatonal complexty benchmark. centralzed management of WMNs due to ther good level of effcency n a reasonable computatonal tme. Future research could be devoted to further optmzng the proposed GA, for example to enable t to deal wth non-bnary structure n order to better manage larger networks and/or decrease the computatonal complexty even more. Also, we envson that GAs could be used n more complex problems characterzed by multple flows and mult-rado mult-channels, due to ther ablty to cope wth mult-dmensonal constrants and objectves. REFERENCES [1] R. Bruno, M. Cont, E. Gregor, Mesh networks: commodty multhop ad hoc networks, IEEE Communcatons Magazne, vol. 43, no. 3, pp , March [2] G. Brar, D. Blough, P. Sant, Computatonally effcent schedulng wth the physcal nterference model for throughput mprovement n wreless mesh networks, Proc. ACM Mobcom 2006, pp. 2 13, [3] V. S. Anl Kumar, M. V. Marathe, S. Parthasarathy, and A. Srnvasan, Algorthmc aspects of capacty n wreless networks, Proc. ACM SIGMETRICS, 2005, [4] J.-H. Chang and L. Tassulas, Maxmum Lfetme Routng n Wreless Sensor Networks, IEEE/ACM Transactons on Networkng, vol. 12, no. 4, pp , 2004.

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