Game-theoretic Particle Swarm Optimization forWMNs

Liqiang Zhao and Hailin ZhangState Key Laboratory of Integrated Services Networks

Xidian UniversityXi'an, Shaanxi, China

E-mail: lqzhao@mail.xidian.edu.cn

Abstract-Game theory is a useful and powerful tool forperformance analysis and optimization of wireless meshnetworks (WMNs). Based on incompletely cooperative gametheory, a mesh router can estimate the game state (e.g., thenumber of competing nodes), and broadcast this information toits clients. Then all the clients playa cooperative game based onthe estimated game state, and achieve the optimal equilibriumstrategy. For each client to implement the game independently asit cannot get the game state accurately or timely sometimes,particle swarm optimization is introduced into the game, and agame-theoretic particle swarm optimization scheme (G-PSO) forWMNs is presented in this paper. Simulation results show that .G-PSO can increase system throughput and decrease delay, jitterand packet-loss-rate.

Keywords-Mesh, MAC, Game Theory, Particle SwarmOptimization

I. INTRODUCTION

A wireless Mesh network (WMN) is a wireless networkthat consists of routers and clients connected through wirelessmedium using a mesh architecture. It is a cooperative networkand represents an important networking technology that willshape the next generation wireless systems. WMNs find manyapplications in broadband home networking, community andneighborhood networking, metropolitan area networks; theycan also be used to set up temporary networks at battlefields~

disaster and emergency zone [1]. Due to their importance,WMNs have attracted a lot of interests from both academia andindustry. Performance analysis and optimization of WMNs,especially its Medium Access Control (MAC) protocols, is animportant research topic for WMNs. So far, most WMNs areimplemented using Wireless LANs (WLANs). IEEE 802.11x isone of the most influential WLAN standards, and its basicMAC protocol is called Distributed Coordination Function(DCF). DCF is based on Carrier Sense Multiple Access withCollision Avoidance (CSMA/CA), one of typical contention­based MAC protocols. Currently, CSMA/CA has been the defacto contention-based MAC standard ofWMNs, and is widelyused in almost all of the testbeds and simulations for WMNresearch.

In CSMA/CA, all nodes must compete to transmit theirpackets and the channel access of each node has a directinfluence on its neighboring nodes. The interactions between

This work is supported by the 111 Project (B08038), National NaturalScience Foundation of China (No. 60772137), Natural Science BasicResearch Plan in Shaanxi Province of China (Program No. 2006F30), and theED FP6 "GAWIND" project on broadband wireless access network designunder the grant number MTKD-CT-2006-042783.

978-1-4244-2644-7/08/$25.00 ©2008 IEEE

Wei Ding and Jie ZhangCentre for Wireless Network Design

University of BedfordshireLuton, Bedfordshire, UK

the nodes give us an intuition that game theory would be a verygood tool for WMN performance analysis and optimization [2].When using game theory in WMNs rather than mathematics oreconomics, much attention should be paid to the relevantcontext of WMNs. [3] presents a simple Nash equilibriumbackoff strategy to resolve the unfairness problem. However, todo so each node has to broadcast its local signal-to-noise ratioperiodically to its neighbors. This is clearly difficult to beimplemented in current WMNs. [4] presents a novel concept ofincompletely cooperative game theory to improve theperformance of CSMA/CA without any explicit cooperationamong nodes. For mesh routers to implement the game, [5]introduces cross-layer design and a V-CSMA/CA estimationmechanism. For clients to implement the game, [5] provides asimplified G-CSMA/CA protocol. G-CSMA/CA is easy to beimplemented in nodes, but it performs worse than the game.

In this paper, to take full advantage of the mesh topology,the hierarchy architecture is presented for WMNs. Toimplement the incompletely cooperative game effectively inthe hierarchy WMNs, particle swarm optimization (PSO) [6] isintroduced into the game, and a game-theoretic PSO scheme(G-PSO) is provided in this paper.

The rest of this paper is organized as follows. CSMA/CA,game theory and PSO are introduced respectively in section II.In section III, after presenting the hierarchy architecture, PSOis introduced into the game, and G-PSO is presented forWMNs. In section IV, simulation studies are carried out toevaluate the performance of G-PSO. The concluding remarksare given in Section V.

II. PRELIMlNARY

A. Description ofCSMA/CA

CSMA/CA uses an acknowledgment (ACK) mechanism forverifying successful transmissions and optionally, an RTS/CTShandshaking mechanism for decreasing collisions overhead. Inboth cases an exponential backoff mechanism is used. Beforetransmitting, a node generates a random slotted backoff interval,and the number of the backoff slots is uniformly chosen in therange [0, CW-l]. At the first transmission attempt, thecontention window, CW, is set equal to a value CWmin calledthe minimum contention window. After each unsuccessfultransmission, CW is doubled up to the maximum value Cw,nax.

Once CW reaches CWmax, it will remain at the value until thepacket is transmitted successfully or the retransmission timereaches retry limit. While the limit is reached, retransmissionattempts will cease and the packet will be discarded.

B. Description ofGame Theory

Game theory is a powerful tool to study situations ofconflict and cooperation, which is concerned with finding thebest actions for individual decision makers (Le., players) inthese situations and recognizing stable outcomes [7]. Gamesmay generally be categorized as non-cooperative andcooperative games. Non-cooperative game theory is concernedwith the analysis of strategic choices and explicitly models thedecision making process of a player out of its own interests.Unlike in non-cooperative games, in cooperative games, theplayers can make binding commitments.

C. Description ofParticle Swarm Optimization

The PSO algorithm is based on the evolutionarycomputation technique [6], and is often used to solve complexengineering problems such as structural and biomechanicaloptimizations [8]. PSO [9-11] optimizes an objective functionby conducting population-based search. The populationconsists of potential solutions, called particles, similar to birdsin a flock. The particles are randomly initialized and then freelyfly across the multi-dimensional search space. While flying,every particle updates its velocity and position based on itsown best experience and that of the entire population. Theupdating policy will cause the particle swarm to move toward aregion with a higher object value. Eventually, all the particleswill gather around the point with the highest object value.

III. GAME-THEORETIC PARTICLE SWARM OPTIMIZATION

A. Hierarchy Architecture for WMNs

WMNs were originally used in defence applications and arelikely to find further applications in emergency serviceoperations where planned infrastructure is unavailable.However, current works have concluded that for a pure meshWLAN, subscribers cannot self-generate capacity at a ratesufficient to maintain a target level of per-user throughputregardless of network size and population. Thus this type ofmesh topology is unlikely to find widespread commercialapplications. Current works also concluded that another viableway scalability can be achieved is by providing additionalcapacity in the form ofa secondary backbone mesh network, sothat a hierarchy architecture for WMNs is presented in thispaper.

In the hierarchy WMNs, the full mesh topology is reservedfor the secondary (fixed) backbone network, with an interfaceto the public telephone network and the Internet, and the partialmesh topology is reserved for peripheral (mobile) networksconnected to the full mesh backbone, as shown in Fig.l. In thetraditional WMNs, the full mesh topology occurs when everynode has a circuit connecting it to every other node. This isvery expensive to implement although it yields the greatestamount of redundancy. In Fig. 1, the multi-beam smartantennas can be used in the routers, so one router can

communicate simultaneously with several (the maximumnumber is the number of its beams) neighbouring routers. Withthe partial mesh topology, a client is connected to one or twoneighbouring clients. The partial mesh topology is lessexpensive and easy to implement but it yields less redundancythan the full mesh topology. In the hierarchy architecture, it isassumed that each router has unlimited resources, and theresources (e.g., CPU, memory, and power) in each client arelimited.

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B. Incompletely Cooperative Game Theory

The incompletely cooperative game is a stochastic game,which starts when a new packet arrives at the node'stransmission buffer and ends when the packet is moved out ofthe buffer (Le., transmitted successfully or discarded). Eachgame process includes many time slots and each time slotcorresponds to one game state. In each time slot, each player(Le., node) estimates the current game state (e.g., the number ofcompeting nodes) based on what happened in the past timeslots. After estimating the game state, the player adjusts its ownequilibrium strategy by tuning its local contention parameters(e.g., CWm;n). Then all the nodes take actions simultaneously,Le., transmitting or backoff. Although the player does notknow which action the other nodes (i.e., its opponents) aretaking now, it can predict its opponents' actions according toits history.

The game state includes not only the player's local states,such as bit rate, but also its opponents' states, such as numberof its opponents (n) [4]. The former can be retrieved fromCSMA/CA operation and the latter cannot. [12] presents across-layer-based framework for the game, as seen in Fig. 2,which consists of three major components, a detector, anestimator and an adjustor. The first component, the detector,detects and records the experienced bit-specific information,such as bit rates, and frame-specific information, such as frametransmission and collision probability. The second component,the estimator, uses the above measurements to estimate thecurrent game state. The term cross-layer design in the gamemeans that the knowledge of the PHY and MAC layers issimultaneously used by the estimator to implement efficientmethods for calculating the game state. The third component,

the adjustor, makes decisions according to which strategy theplayer transmits its packets.

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Obviously, the key problem in the game is how to estimatethe game state accurately and timely. [4] shows that thenumber of a player's opponents n is a function of its framecollision probability (P) and transmission probability (t), asfollows:

(1)

However, [13] shows that (1) is accurate only undersaturated conditions (i.e., each node always has a packetwaiting for transmission), and far from accurate underunsaturated conditions if not filtered. [14] provides two run­time estimation mechanisms, Le., auto regressive movingaverage (ARMA) and Kalman Filters. The two mechanisms arevery accurate even in unsaturated conditions. However, theyare too complex to implement in clients.

As each mesh router has unconstrained resources, it can runa complex estimation mechanism, e.g., V-CSMA/CA andcross-layer-based frame\vork [5] [12], to estimate the gamestate. After estimating the game state, the mesh routerbroadcasts its estimated information to all the clients in its cell.However, this may involve some delay and in a fast-changingWMN, more frequent information updates are needed. Thisresults in a significant overhead. Hence, each mesh client hasto take a distributed approach of detecting the channel,estimating the game state and adjusting its local parameters tothe estimated game state. However, a complex estimationalgorithm cannot be implemented easily in current mesh clients.Actually, it is very difficult, or sometimes impossible, for eachclient to estimate the number of its opponents accurately andtimely. [5] introduces game theory and cross-layer design intoWMNs, but does not consider how to implement them in realscenarios. Hence, PSO is introduced into the game to solve thisproblem.

C. Game-theoretic Particle Swarm Optimization

Assume there are M particles (Le., nodes) in a swarm (Le., aWMN cell), along with N geometrical parameters (Le.,performance parameters) to be optimized. The particles'positions and velocities (Le., strategies) are stored in two MxNmatrices, X and ~ respectively. At each iteration, the velocitiesof each particle is determined by the distances from its currentposition to two important locations, which are denoted by the"personal best" (Pbest) and the "global best" (gbest). The Pbest isthe location where each particle attains its best fitness value upto the present iteration, which represents the cognitive

contribution to the velocity v. On the other hand, the gbest, is theglobal best location among all the particles in the swarm,which represents the social contribution to the velocity v. Atthe n-th iteration, the particles' velocities are calculatedaccording to

where the variable w is the particle inertia, which is reduceddynamically to decrease the search area in a gradual fashion,while Cl = Cz = 2.0 are the cognitive and social scalingparameters, respectively. Two random variables, 1'/1 and 112, areboth uniformly distributed within [0, 1] to inject theunpredictability in the swarm behavior. P and G are MxNmatrices where the Pbest and the gbest are refreshed and stored ateach iteration. All the M rows of G are identical since all theparticles have the same gbest.

And the particle's positions are updated by

(3)

Please note that in the game, each player estimates the gbest,

while in the traditional PSO algorithm, normally, it is assumedthat each particle knows the gbest.

In this paper, a simplified binary PSO algorithm is used inG-PSO, as follows:

{Tit = 0 172 = 1 if a node obtains the game state

.(4)171 =1 17'2 =0 if a node cannot obtain the game state

There are two kinds of particles in the swarm: nucleus (Le.,router) and electrons (i.e., clients). The router can obtain theoptimal path which the swarm passes through to the optimaldestination. The clients can update their paths according to theinformation from the router. So in the moving process of theswarm, when the router finds the path and the destination, theclients will trail and reach the destination quickly; otherwise,the clients seek the destination independently. Of course, everyclient's behaviour will gradually be the same within someiterations.

In WMNs, each mesh router can run a complex estimationmechanism, e.g., ARMA or Kalman Filters, to estimate thegame state, and then broadcast it to all the nodes. Based on theestimated game state, the mesh router and nodes playcooperative game to achieve the gbest.

After getting the number of competing nodes n, each nodetunes its local CWmin to the optimal value, as follows [12]:

[n x rand (6,7)J its bit rate is 11 Mb/s[n x rand(9,10)J its bit rate is 5.5 Mb/s

CWmin = [n x rand(17,18)] its bit rate is 2 Mb/s ' (5)

[n x rand(25,26)] its bit rate is 1Mb/s

where rand (x, y) returns a random value between x and y, and[z] is the largest integer that is not more than z. The value ofCWmin in the above equation is the gbest for the group of n

competing nodes [12], for simplicity, which is called CW~in in

the following.

As each node only has constrained resource, it cannotobtain the game state by running the above estimationalgorithms. On the other hand, sometimes the node cannot getthe accurate game state from the router timely. However, if therequirements on timeliness are relaxed so that each nodeadjusts its strategy after a game process (i.e., after transmittingor discarding a packet) rather than in each time slot, asimplified estimate algorithm G-CSMAICA is obtained [5].

After transmitting a packet, to maintain the actualcontention level, each node adjusts its CWmin as follows:

IV. SIMULATION RESULTS

To evaluate the proposed protocol, we performedsimulations in an ideal channel with none hidden terminals.The values of the parameters used to obtain numerical resultsfor simulations are specified in IEEE 802.11b protocol. Forsimplicity, the channel rate is fixed at 11 Mb/s. The packetswill be discarded only due to the re-transmission time reachesthe retry limit, and do not consider the delay limit. Supposethere are 50 nodes and each node generates new packets undera Poisson process. The packet arrival rate is initially set to belower than the saturation case, and it is subsequently increasedso that, at the end of the simulation time, all nodes are almostin saturation conditions [15]. For comparison, the idealperformance of the game is simulated too, where it is assumedthat each node always knows the game state.

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where max(x, y) returns the larger value of x and y. Theparameter CWmin' CWmax, and CW at the right hand are thevalues of the nominal CWmim CWmax and the final contentionwindow used in the previous game process respectively. Theparameter CWmin at the left hand is used in the current gameprocess to transmit a new packet.

In CSMAlCA, a node starts a contention process alwayswith the nominal CWmin. So, CSMAICA has one maindrawback: in a heavily loaded network, the increase of CW isobtained at the cost of continuous collisions. In (6), aftertransmitting a packet, the node does not start the nextcontention process with the nominal CWmin. The value of CWmin

is chosen in the following manner. If the previous packet istransmitted successfully, the final value of CW is the optimalone. The best strategy for the node is to set CWmin = CWI2, tomake use of the channel effectively. If the previous packet isdiscarded, the best strategy for the node is to set CWmin = CW~to decrease collision.

Obviously, the obtained value of CWmin in (6) is the Pbest for

the node, for simplicity, which is called CW~in in the

following.

Moreover, it is simple to implement (6) in each node as noestimation algorithm is needed.

Based on (5) and (6), each node tunes its CWmin as follows:

CW . = {CW~in a node knows the game state . (7)mm CW~in a node does not know the game state

Obviously, on one hand, this is a cooperative game. Whenall the nodes can obtain the game state, they make a bindingcommitment and thus play cooperative games to achieve thegbest. On the other hand, this is a Nash equilibrium strategy. Ifthe player cannot obtain the game state, it takes the best actionsfor its own interests, i.e., Pbest.

Fig. 3 shows that the three protocols, CSMA/CA, theincompletely cooperative game and G-PSO, have almost thesame unsaturated throughput, and then CSMAICA getssaturated and its throughput keeps almost constant, while thethroughput of G-PSO and the game continue increasing as theyare still unsaturated. G-PSO and the game almost get saturated

at the same time. And the two protocols have almost the samesaturation throughput.

In general, the performance of G-PSO is very close to thatof the incompletely cooperative game, while G-PSO is muchbetter than CSMA/CA.

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v. CONCLUSION

In this paper, firstly, the incompletely cooperative game isused to improve the MAC protocols for WMNs. Secondly,after considering the context of WMNs, the problem ofestimation algorithm in the game is discussed in detail. Finally~

particle swarm optimization is introduced into the game tosolve the above problem. And a game-theoretic particle swarmoptimization scheme (G-PSO) is proposed, which can be easilyimplemented in current WMNs. Based on G-PSO, each nodescan achieve the optimal performance no matter it can obtain thegame state or not. The simulation results show that G-PSO isan appropriate tool to improve the performance ofWMNs.

Fig. 4-6 show that before saturation, the three protocolshave almost the same access delay, jitter and packet-loss-rate.For simplicity, we assume that the packets are discarded onlybecause re-transmission time reaches retry limit, and do notconsider delay and buffer limit. Before saturation (about 60s),almost all packets can be transmitted and packet-loss-ratekeeps constant at zero in DCF. After saturation, delay, jitterand packet-loss-rate increase sharply, and those in DCFincrease most sharply, and those in G-PSO are a little largerthan those in the game.

Delay, jitter and packet-loss-rate in G-PSO mayoccasionally be large as the players cannot estimate the gamestate and have to take actions to achieve Pbest instead.

Figure 5. Jitter

Figure 6. Packet-loss-rate

50 100

Simulation time (sec) REFERENCES

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