Using Game Theory and Bayesian Networks to Optimize Cooperation in Ad Hoc Wireless Networks

Transcription

1 Using Game Theory and Bayesian Networks to Optimize Cooperation in Ad Ho Wireless Networks Giorgio Quer, Federio Librino, Lua Canzian, Leonardo Badia, Mihele Zorzi, University of California San Diego La Jolla, CA 9293, USA DEI, University of Padova, via Gradenigo 6/B 35131, Padova, Italy Abstrat Infrastruture sharing has been reently investigated as a viable solution to inrease the performane of oexisting wireless networks. In this paper, we analyze a senario where two wireless networks are willing to share some of their nodes to gain benefits in terms of lower paket delivery delay and redued loss probability. Bayesian Network analysis is exploited to ompute the orrelation between loal parameters and overall performane, whereas the seletion of the nodes to share is made by means of a game theoreti approah. Our results are then validated through use of a system level simulator, whih shows that an aurate seletion of the shared nodes an signifiantly inrease the performane gain with respet to a random seletion sheme. I. INTRODUCTION AND RELATED WORK Cooperation is one of the most promising enabling tehnologies to meet the inreasing rate demands and quality of servie requirements in wireless networks, espeially sine nowadays many tehniques to share the spetrum resoures among different networks are envisioned. Beyond spetrum sharing, also infrastruture sharing is possible, namely, when a network deides to ooperate, it shares some or all of its nodes, that beome relays for another network. In suh a senario, ooperation an leverage the benefits of diversity, obtaining a onsiderable gain in the effiieny of shared resoures. Usually, sharing the whole set of nodes an grant the biggest advantage to both networks. However, this learly omes at the ost of additional traffi that should be handled by some of the shared nodes. In addition, in a realisti environment, an operator may not be willing to share too many nodes to improve the traffi of another operator, e.g., for seurity or privay reasons. Therefore, the operator may deide to share only a limited number of nodes, reeiving the same treatment from the operator of the other network. If this is the ase, an optimal hoie of the shared nodes, aording to ertain riteria, is needed. Indeed, some nodes may be deployed in ruial positions, and they may be partiularly suited for helping the other network; on the ontrary, nodes plaed lose to the network border are likely to be less useful or even useless. Furthermore, sharing a node implies that a higher amount of traffi will be routed through it, whih may result in a higher lateny for the traffi of its own network. In this paper we onsider two wireless networks deployed in the same region but operated by different entities. In the first senario, the two oexisting networks perform their operations separately: eah network only uses its own resoures to deliver the data pakets generated by its nodes. Clearly, sine they are assumed to share the same spetrum resoures, ross network interferene may limit the overall performane. For suh a senario, we selet a set of loal parameters: some of them are diretly observable and depend only on the topology of the network, like the number of neighbors at a given node, and some others are not observable and depend on the link harateristis and on the traffi load. We exploit Bayesian Network (BN [1] analysis to estimate the joint probability distribution of this set of parameters, and we use BN also to predit, given the evaluation of the observable parameters, the values of the other parameters that will be used to alulate a performane metri. The use of this probabilisti tool is very promising for wireless network optimization, and it has been reently exploited, e.g., for prediting the ourrene of ongestion in a multi-hop wireless network [2]. Suh an approah an also be used to improve the performane of both networks in our senario. The key idea is that a higher node density may help both networks to augment the available diversity, and thus to find shorter routes for multihop ommuniations. It is straightforward that this may be obtained if eah network an take advantage of some of the nodes of the other one. We model the interation between the two networks through Game Theory. In spite of the onsiderable theoretial gain that a ooperative transmission allows, modeling the involved agents as smart selfish deisionmakers usually leads to ineffiient non-ooperative equilibria. For example, [3] shows that the IEEE distributed Medium Aess Control (MAC protool leads to an ineffiient Nash Equilibrium (NE and in [4] a situation similar to the Prisoner s Dilemma ours in a slotted Aloha MAC protool. To improve the performane of the network, ooperation among the players is often desirable. In the present paper, we ahieve this goal formulating the problem as a repeated game, whih onsists in a number of repetitions of a base game. It aptures the idea that a player has to take into aount the onsequenes of his urrent ation on the future ations of other players. Cooperation is in fat obtained by punishing deviating users in subsequent stages. Similar approahes have been used for example in [5] [7]. In brief, the main ontributions of this paper are: the definition of the ooperation problem of two networks sharing the same spetrum resoures as a strategi game; the use of BN theory to learn the probabilisti relationships among a set of parameters of interest in the network, in order to infer the performane metri parameters from some observable topologial parameters;

2 the proposal of a game theoreti algorithm to hoose the best nodes to share; the implementation of the strategi game and the BN preditor in an atual wireless network simulator, that simulates the network behavior at the Physial, the MAC and the network layers of the protool stak; to numerially show the effetiveness of our algorithm in improving the performane of the wireless networks by aurately seleting the nodes to be shared. The rest of the paper is divided as follows. In Setion II we briefly review the Bayesian Network and Game Theory approahes, then in Setion III we introdue our network senario. In Setion IV we desribe the ooperation strategy adopted, while in Setion V we detail the simulation setup and show the main results. Setion VI onludes the paper. II. MATHEMATICAL PRELIMINARIES In this setion we briefly desribe the mathematial tools, i.e., Bayesian Networks and Game Theory, that we adopt to identify tehniques for the seletion of the best ooperating nodes in the network. The former is a method to learn an approximate joint probability distribution among a set of random variables from a set of instanes of suh variables. The latter is a branh of mathematis studying interations between deision makers. A. Bayesian Networks (BNs A Bayesian Network (BN is a probabilisti graphial model [1] desribing onditional independene relations among a set of M random variables through a Direted Ayli Graph (DAG. This graph is used to effiiently ompute the marginal and onditional probabilities of the M variables. A node i in the DAG represents a random variable x i, while an arrow that onnets two nodes i and j represents a diret probabilisti relation between the orresponding variables x i and x j. The absene of a diret arrow between two variables implies that the variables are independent, given ertain onditions on the other variables of the graph. The orientation of the arrow is also relevant to desribe the relationship between the two variables. If the arrow is direted from node i to node j, we say that i is a parent of j, and we write x i pa(x j. To larify this onept, onsider the following example. If nodes h, i, and j are represented in a BN suh that h is a parent of i and i is a parent of j, the joint probability of the orresponding variables is P(x h,x i,x j = P(x h P(x i x h P(x j x i, (1 that is simpler than a general joint probability among three variables. See [1] for further details on the BN properties. a Learning the struture: The tehnique to learn the approximate joint probability distribution through a BN is divided into two phases, struture learning and parameter learning. The former is a proedure to define the DAG that represents the qualitative relationships between the random variables, i.e., the presene of a diret onnetion between a ouple of variables, not onditioned by other variables. Aording to the sore based method, e.g., see [8], we do not assume any a priori knowledge on the data, but we just analyze the realizations of the variables and we sore eah possible DAG with the Bayesian Information Criterion (BIC [9], that we have hosen as a sore funtion. The BIC is easy to ompute and is based on the Maximum Likelihood (ML riterion, i.e., how well the data suits a given struture, and penalizes DAGs with a higher number of edges. If eah variable is distributed aording to a multinomial distribution, i.e., it has a finite number of possible outomes, then the BIC beomes very simple to ompute, involving only summations for all possible outomes of the variables and all possible outomes of the parents of eah variable, see [8]. b Learning the parameters: The parameter learning phase onsists in estimating the parameters of the simplified joint distribution aording to the probability struture defined by the DAG hosen in the struture learning phase. In order to have the joint distribution, it suffies to estimate the probability of eah variable onditioned by the variables that orrespond to its parent nodes in the graph. Coherently with the hoie of the BIC as a soring funtion, we use the ML estimation tehnique also to determine all the onditioned probabilities for eah variable onsidered. B. Game Theory Game Theory [1] is a branh of mathematis that studies games, i.e., strategi situations where many players interat together. The aim of game theory is to provide interation models and predit the outomes of a game. In game theory, utility funtions are used to represent the players appreiation of the outomes. A utility funtion is assoiated to eah player and its value depends on the ations taken by all the players. It is ommon to assume that the players are selfish and rational. The former implies that eah of them only wants to maximize its own utility, independently on the utilities of the others. The latter means that they hoose their strategies on the basis of all the information they have about the game. We desribe a game in the normal form as a triplet: G = (P,S,U, (2 where P = {1,...,N} is the set of the N players, S = (S 1,...,S N is the N-tuple of strategy sets, where S i is the set of all ations that i an take, and U = (u 1,...,u N is the N-tuple of utility funtions, where u i : S R is the utility funtion of player i. A vetor s = (s 1,...,s N, where s j S j for all j = 1,...,N, is alled a strategy profile. An important onept in Game theory is the Nash Equilibrium (NE, defined as the strategy profile s NE = (s NE 1,...,s NE N where eah player obtains its maximum utility given the strategies of the other players, i.e., suh that ( u j s NE ( u j s NE 1,...,s NE j 1,s j,s NE j+1,...,s NE n, (3 for all j = 1,...,N, for all s j S j. In other words, it is an equilibrium against unilateral deviations. However, the existene and uniqueness of suh an equilibrium point in S are not guaranteed. Further, NEs an be ineffiient from a soial point of view, leading to the so alled tragedy of

3 the ommons [11], an ineffiient situation ourring when individuals share a ommon resoure in a selfish manner. A possible way to inrease the effiieny of an NE is to formulate the problem as a repeated game, i.e., a base game repeated in time. In this ase a rational player is fored to take into aount how his urrent ation an influene the future ations of other players. A ooperative behavior is indued by punishing deviating users in subsequent stages. III. SYSTEM MODEL In this setion, we desribe the network senario under investigation from the physial up to the routing layer. In our senario, two ad ho wireless networks oexist and share the ommon spetrum resoure. Eah network onsists of N terminals randomly deployed, and eah node is a soure of traffi, whih generates pakets aording to a Poisson proess with intensity λ. The end destination of eah paket is another node in the network, hosen at random. Furthermore, time is divided in slots and slot synhronization is assumed aross the whole network. A. Physial Layer At the physial layer, CDMA with fixed spreading fator is employed to separate simultaneous transmissions, sine both networks share the same spetrum resoures, and a training sequene is transmitted at the beginning of eah transmission to help hannel estimation. The reeiving node, D (, uses a simple iterative interferene anellation sheme to retrieve the desired paket when M simultaneous ommuniations, namely T (1,...,T (M, are heard. In order to desribe this sheme, we need to define the Signal to Interferene plus Noise Ratio (SINR at D ( for the inoming transmission T (i from node D (i, i.e., Γ (i N s P (i = N +, (4 j ip(j where N is the noise power and N s is the spreading fator. P (j indiates the inoming power due to T (j, i.e., for all j = 1,...,M: P (j = P T h D (j,d ( 2 d α j, (5 A where P T is the transmission power, whih is onsidered to be the same for all the nodes in the network, A is a fixed path-loss term, d j is the distane between the reeiving node and the soure of T (j, α is the path loss exponent, and h D (j,d( is a omplex zero mean and unit variane Gaussian random variable, whih represents the effet of multi-path fading. More preisely, in our senario, we onsider a time orrelated blok fading. Therefore, for the hannel between nodes D (j and D (, the multi-path fading oeffiient in time slot t is h D (j,d ((t = ρ h D (j,d ((t 1+ 1 ρ 2 ξ, (6 where ρ is the time-orrelation fator and ξ is an independent omplex Gaussian random variable with zero mean and unit variane. Now we an desribe the iterative interferene anellation sheme as follows: the destination node D ( sorts the M inoming transmissions aording to the reeived SINR, in dereasing order (for simpliity, we assume Γ (1 Γ (M ; starting from transmission T (1, D ( tries to deode the orresponding paket, with a deoding probability that is a funtion of Γ (1 ; if the paket is orretly reeived, its ontribution is subtrated from the total inoming signal; D ( attempts to deode the transmission with the next highest SINR, T (2, and goes on until it an try to deode the paket of interest. B. MAC Layer At the MAC layer, we implement a simple transmission protool based on a Request-To-Send/Clear-To-Send (RTS/CTS handshake. Every time node D (i wants to send a paket to node D (j, it heks the destination availability by sending a RTS paket; if D (j is not busy, it replies with a CTS so that D (i an start transmitting the paket. Corret reeption is aknowledged by means of an ACK paket. In the ase of deoding failure, after a random bakoff time, node D (i shedules a new transmission attempt, or disards the paket, if the maximum number of retransmissions has been reahed. The signaling pakets are very short, i.e., they are transmitted within a single time slot, and are proteted by a simple repetition ode of rate 1/2. Instead, data pakets may span several time slots, so error detetion oding is used to verify their orret reeption, i.e., redundany bits are added at the end of eah paket. C. Network Layer The soure node and the destination node are not neessarily within overage range of eah other, so we onsider multi-hop transmissions. Two nodes are neighbors, i.e., they an ommuniate diretly, if their distane is lower than a threshold value l. In order to transmit to a node that is not within overage, the nodes use a stati routing table, whih is built using Optimized Link State Routing (OLSR [12], a traditional routing protool, and is available at every node of the network. Eah time a node generates a new paket, or reeives a paket to be forwarded, the paket is put in the node queue, with FIFO poliy. The maximum queue length is fixed and equal for all nodes. If a new paket arrives when the queue is full, it is simply disarded. IV. COOPERATION STRATEGY In this setion we desribe how the two networks that oexist in our senario an share effiiently the spetrum resoures by means of ooperation. A. Performane metri Given the path from D (i to D (j, we define the delivery delay ζ (i,j as the average end-to-end delay of a paket sent along the path, given that the paket is reeived; and the paket loss probability p (i,j as the probability that a paket is lost along the path. The former depends on the hannel

4 and interferene onditions, whih may require one or more retransmissions, and on the overall traffi level. Indeed, for multi-hop routes, a paket has to wait at eah relay node until all the pakets it finds in the FIFO queue have been sent. Regarding the latter, the paket loss, there are two main events to be aounted for. One is a high interferene level, that may lead to a paket drop due to an exessive number of retransmissions; the other is buffer overflow, i.e., the paket is disarded if the next relay has no room for it in its queue. We onsider a metri to measure the gain offered by the various ooperation strategies, whih takes into aount the average end-to-end delay of a paket sent along the path from D (i to D (j. Sine no end-to-end paket retransmission mehanism is implemented in our network, the effet of lost pakets must also be onsidered. Ignoring lost pakets (i.e., omputing the delay statistis only on orretly delivered pakets may lead to an optimisti evaluation of the network performane under heavy traffi, where few pakets atually reah the destination. In this ase, a high-loss path might end up being onsidered better than a more reliable path with a slightly higher delivery delay. The other extreme, i.e., defining the delay ontribution of a lost paket as infinite, makes the delay evaluation meaningless sine the average delay would also be infinite for any positive loss probability. Clearly, neither option is desirable in our ase. Therefore, we propose another definition that gives a finite bias to the average delay in ase of a paket loss. In partiular, when a paket is lost when going from D (i to D (j, we inrease the delay of the following paket in the same path by the interarrival time between pakets routed on that path. 1 This additional delay is given by (N 1/λ, i.e., the inverse of the per-path average traffi intensity (reall that eah paket generated at D (i has a randomly hosen destination among the remaining nodes of the network, so that the per-node traffi λ needs to be divided by the number of possible destinations, N 1. Aording to this reasoning, we reursively define the weighted delivery delay of a data paket sent via multi-hop transmission by node D (i to node D (j as: ( ( N 1 W (i,j = 1 p (i,j ζ (i,j +p (i,j +W (i,j. λ (7 In this alulation, the hannel and interferene onditions, and thus the loss probability, are assumed to be independent for different pakets. This is due to the fat that the destination for eah paket is hosen at random, and the time between two subsequent paket transmissions over the same path is deemed to be long enough. From Eq. (7 we obtain: W (i,j = N 1 λ p (i,j 1 p (i,j +ζ (i,j. (8 The delivery delay ζ (i,j and the loss probability p (i,j depend on the nodes that the routing protool selets as 1 Equivalently, we assign to lost pakets a delay ontribution equal to the interarrival time, to reeived pakets the atual delay inurred, and then divide the sum of all ontributions by the number of orretly reeived pakets only. relays. In a stati network, it is possible to estimate these values during a training period. Instead, if the network is dynami (mobile nodes or time-varying traffi statistis, this is not possible. We propose a different way of estimating the delay and the loss probability, based only on instantaneous geographi and routing information. Sine a paket sent over a multi-hop path has to traverse a number of nodes before reahing the destination, we make the assumption that both the overall path delivery delay and the overall path loss probability an be deomposed into ontributions given by the various traversed nodes. More preisely, the overall delivery delay is given by the sum of the average delays required to traverse every single node (time in queue plus transmission time, whereas the overall loss probability is obtained from the loss probabilities at every node (probability of transmission failure and probability of buffer overflow. If R (i,j is the set of nodes belonging to the path between D (i and D (j (exluding D (i and D (j, we have: ζ (i,j = ζ q (i + ζ q (h, (9 h R (i,j where ζ q (h is the average time between the arrival of a paket at node D (h and its reeption at the next hop. This delay depends on the next relay; indeed, while the time needed for traversing the queue is the same for all pakets, the time required for a suessful transmission depends on the hannel ondition, and hene on the next hop hosen. We estimate ζ q (h averaging over all the possible next-hop relays, thus over all the neighbors of node D (h. The paket loss in the multi-hop path is alulated in a similar way, i.e., p (i,j = 1 (1 p (i t (1 p (j q (1 p (h t (1 p (h q, h R (i,j (1 where p (h t is the probability that a transmission from node h to the next hop fails beause the maximum number of retransmissions is reahed, and p (h q is the probability that a paket orretly reeived at node D (h is disarded due to buffer overflow. Furthermore, we notie that p (h q depends on the queue of the reeiving node D (h, while p (h t depends also on whih node is used as next hop. For this reason, similarly to what we have done for ζ q (h, we onsider a value averaged over all the neighbors of D (h. With (9 and (1 we an alulate the weighted delivery delay W (i,j, defined in (8. This parameter should be estimated for eah ouple of nodes, with a suffiiently long training period. From (8, we define W as the average over all the ouples of nodes belonging to the network. This will be used in the following as the performane metri of the whole network. B. Stohasti estimation of loal parameters In a real network, the values of the parameters ζ q (i, p (i t, and p (i q should be estimated based on loal information. Our idea is to use some parameters that an be easily alulated at eah node D (i. We onsider in partiular the number of

5 Fig. 1. Bayesian Network showing the probabilisti relationships among the 5 parameters of interest: ζ q, p t, p q F, and N. flows F (i, that an be easily alulated from the routing table, and the number of neighbors, N (i. We have estimated the probabilisti relationships among ζ q, p t, p q, F, and N. Notie that we removed the dependene on the speifi node. In fat, the Bayesian Network approah exploits the olleted data, whih are speifi for eah node, to find out the orrelation between the loal parameters and the values of N and F. The result is a set of general onditional distributions (one per eah loal parameter whih an be therefore applied to any node of the network. It follows that one the number of flows or neighbors of a given node is known, the distributions of ζ q (i, p (i t, and p q (i for that node are also known. We first olleted the measures of these parameters in our senario as a funtion of the traffi load λ, for different topologies. Then we alulated the struture of the Bayesian Network (BN. We should notie that this proedure is different from using a training period to diretly derive the loal parameters. In fat, in this ase a training period would be needed every time the topology hanges, so as to evaluate their value for eah speifi node or path. On the ontrary, with our proedure we an estimate the general joint probability among these parameters, that does not depend on the speifi topology. The struture of the BN is reported in Fig. 1. The struture of this BN is the same for all the values of λ, while quantitatively the probabilisti relationships hange with λ. We notie that N does not influene, to a first approximation, the values of the three performane parameters, one the value of F is observed. In other words, one we alulate from the routing table the value of F, we an have an estimate of the probability distribution of the three performane parameters. From these estimated parameters, we an alulate also the overall network performane W. C. Cooperation When ooperation is exploited, some nodes are shared between the two networks, and the routing tables alulated via OLSR hange aordingly. By using the framework introdued above, we an estimate the overall performane of the two networks with and without ooperation. We denote with W k (D 1,D 2 the weighted delivery delay of network k, with k = 1,2, when the two networks share the set of nodes D 1 and D 2, respetively. In partiular, W k (, is the performane metri of network k when no nodes are shared. Thus, for any hoie of the nodes shared we an alulate TABLE I SIMULATION PARAMETERS Number of nodes per network 1 Transmission power [dbm] 24 Noise floor [dbm] -13 Modulation used BPSK Time slot duration [ms] 1 Paket length [bit] 496 λ [pkt/s/node] 1 to 5 Spreading fator 16 Fading orrelation fator ρ.9 the variation in W k for the two networks. Then, we an model the ooperation strategy by means of Game Theory, by onsidering eah network as a selfish agent whose utility funtion an be any dereasing funtion of W k. To sum up, the following steps are followed in our framework: we learn the network behavior by measuring the parameters of interest over several random topologies with fixed setup; we use the BN method to infer the joint distribution among ζ q, p t, p q, F, and N ; we evaluate the utility funtions W k (D 1,D 2, for the two networks k {1,2}, for all the possible hoies of the sets D 1 and D 2. we selet the two subsets D 1 and D 2 to be shared, based on the game theoreti approah desribed in Setion IV-D. D. Game theoreti approah The problem is formulated as a repeated 2-player game, where the players are the two networks. We name the nodes of the networks from 1 to 2N, where the nodes in the sets Q 1 = {1,...,N} and Q 2 = {N+1,...,2N} belong to network 1 and 2, respetively. The strategy of eah network is represented by the set of nodes D 1 and D 2 they deide to share, therefore in the most general formulation the strategy sets are the power sets S 1 = 2 Q1 and S 2 = 2 Q2. The utility funtion of eah network, u k : 2 Q1 2 Q2 R, k = 1,2, is the reiproal of the average weighted delay per path for that network, that is, W 1 k (D 1,D 2. Eah of these metris jointly depends on the strategies of both players: if a network deides to share a given node, that node is loaded by the traffi of the other network that passes through it. On the other hand, an additional shared node dereases the overall amount of traffi that passes through the other nodes. In this paper, we assume for simpliity that the networks do not have the freedom to hoose the number of nodes to share. They an share either no nodes or exatly 2 nodes, therefore the ardinality of eah strategy spae is ( N Although our approah an be extended to a larger number of ooperating nodes, our numerial results show that a large fration of the available ooperation gain is already ahieved with this simple hoie. If we onsider a single stage of this game it is immediate to see that the unique NE is the strategy profile s = (,, i.e., no network ooperates. In fat, given the strategy of the

6 λ = 1 λ = 2 λ = 3 λ = 4 λ = λ = 1 λ = 2 λ = 3 λ = 4 λ = λ = 1 λ = 2 λ = 3 λ = 4 λ = ζ q p q.25 p t F F F Fig. 2. BN estimation of the average delivery delay ζ q as a funtion of the number of flows F passing through the node. Fig. 3. BN estimation of the probability of buffer overflow p q as a funtion of the number of flows F passing through the node. Fig. 4. BN estimation of the probability of transmission failure p t as a funtion of the number of flows F passing through the node. other, eah network prefers to share no nodes in order not to inrease the total traffi through its nodes. However, in the repeated formulation it an be shown that eah strategy profile that allows to reah a better utility for both players is a NE. A player deviating from that strategy profile an be punished by the other player during subsequent stages. The duration of this punishment an be set so that the gain obtained during the deviating stage does not ompensate the loss during the subsequent stages. Punishment strategies in repeated games allow multiple equilibria with varying utilities for eah player. Inspired by the Nash bargaining solution [1], we deide ( u2 u NC 2, where ( u1 u NC 1 2 are the status quo utilities, i.e., the utilities to maximize the produt u NC 1 and u NC W 1 1 (, and W 1 2 (, obtained when networks do not ooperate. We additionally impose the mathematial onstraint u k u NC k, k = 1,2, to avoid the situation where the maximum orresponds to a derease in the utilities of both networks. The solution found results in inreased utilities for both networks ompared to the non ooperative ase, therefore it is a NE for the repeated game formulation. V. RESULTS In this setion we present the simulation setup and the main results of our approah for ooperation. A. Simulation Setup In order to prove the effetiveness of our ooperation strategy, we developed a network simulator whih enompasses the layers from physial to routing, as desribed in Setion III. The system parameters are reported in Table I. Eah simulation run is performed with randomly generated onneted networks, and lasts for 1 time slots, inluding an initial transient phase. Different values of the traffi generation intensity λ were onsidered, from 1 paket/s, orresponding to a lightly loaded network, up to 5 paket/s, whih is instead the ase of an overloaded network. In eah senario, 5 simulation runs were performed to ollet the data required for the BN inferene. Based on this information, the empirial distributions and the average values of ζ q, p t and p q, onditioned on F, were derived. In the subsequent steps, a new set of 5 simulation runs was performed for eah value of λ. In eah run, two networks are again randomly deployed; the overall system performane is theoretially evaluated by omputing the values of W k, based on the routing tables, and the values of W k (D 1,D 2, with k {1,2}, where D 1 and D 2 are the optimal sets of nodes to be shared, aording to the game theoreti framework proposed in Setion IV. The aim is to verify how muh gain is ahievable with our approah with respet to a random seletion of the nodes shared and a fully ooperative strategy. Therefore, the network performane obtained by using our Game Theoreti node seletion strategy is ompared to those ahieved by the following strategies: 1 no ooperation, 2 two nodes shared, randomly hosen by eah network, and 3 all nodes shared. B. Bayesian Network estimation Exploiting the stohasti estimation of loal parameters through the BN approah proposed in Setion II-A, we an evaluate the expeted value of the three parameters of interest, namely the average delivery delay ζ q, the probability of buffer overflow p q and the probability of transmission failure p t, as a funtion of the number of flows F passing through the node and of the traffi intensity λ. The expeted values of ζ q, p q, and p t are shown in Figs. 2, 3 and 4, respetively. We notie that the highest number of flows through a single node is reahed when that node beomes the only onnetion among three separate lusters of nodes. If these groups have similar ardinalities, and the number of nodes in eah network is N, we an rise up to a maximum of about 4(N 1 2 /3 flows through a single node, that is lose to the maximum value of F represented in the figures. We also observe in Fig. 2 that for very high values of F and λ, the average delivery delay dereases. We onjeture that this happens for two reasons: (1 the queue of these nodes are always almost full, so that the time to traverse them annot grow muh further, whereas (2 a node traversed by a high number of flows is often hosen as reeiver by most of his neighbors. For these reasons, when it transmits, a lower number of ommuniations an interfere, thus leading to a lower time needed to deliver a paket to the next hop.

7 Weighted delay [slot] No Coop 2 Rand 2 GT Full Coop λ nodes whih an guarantee the highest benefit was made. Even when a small fration of nodes is shared, we obtained a signifiant gain. In partiular, both for lightly and heavily loaded senarios, the seletion sheme based on Game Theory an ahieve almost the same performane as a full ooperation sheme. ACKNOWLEDGMENTS This work was partially supported by the U.S. Army Researh Offie, Grant. No. W911NF , and by the European Commission under the SAPHYRE projet, Grant No Fig. 5. Weighted delay as a funtion of the paket generation intensity λ, for the four ompared senarios: with no nodes shared (No Coop; with two nodes shared, randomly hosen (2 Rand; with two nodes shared, hosen via Game Theory (2 GT; and with all the nodes shared (Full Coop. C. Cooperation performane In Fig. 5, we present the atual gain, in terms of delay redution, offered by the onsidered ooperation strategy. The urves are obtained by averaging over 5 random topologies, eah onsisting of two networks of N = 1 nodes. The other system parameters are reported in Tab. I. We plot the average weighted delay of eah network (due to the symmetry of the senario, it is not neessary to distinguish between the networks in four different ases, that is: (1 when no nodes are shared, namely No Coop; (2 when 2 nodes randomly hosen are shared, namely 2 Rand; (3 when 2 nodes, seleted through the proposed Game-theoreti approah, are shared, namely 2 GT; (4 when all nodes are shared, namely Full Coop. It an be observed that, as intuition suggests, full ooperation grants the highest benefits, due to the higher diversity. Hene, this is the maximum ahievable gain for the senario investigated. This gain is more pronouned when the networks are heavily loaded, sine ongested paths are more frequent, and spatial diversity beomes more advantageous. When only two nodes an be shared, the hoie of the shared nodes makes the differene. In fat, Fig. 5 shows that a areful seletion of the resoures to be shared an signifiantly inrease the ahievable gain when ompared to a blind random seletion. A random seletion an not offer a signifiant gain for lightly loaded networks, while, for heavily loaded networks, it an offer only one third of the gain granted by full ooperation. On the ontrary, if the same number of nodes are shared, but hosen by means of our game-theoreti approah, the maximum ahievable gain is fully obtained for lightly loaded networks and losely approahed for heavily loaded networks. REFERENCES [1] D. Koller and N. Friedman, Probabilisti Graphial Models: Priniples and Tehniques. The MIT Press, 29. [2] G. Quer, H. Meenakshisundaram, B. Tamma, B. S. Manoj, R. Rao, and M. Zorzi, Using Bayesian Networks for Cognitive Control of Multihop Wireless Networks, in Proeedings of IEEE MILCOM, San Jose, CA, US, Nov. 21. [3] G. Tan and J. Guttag, The MAC protool leads to ineffiient equilibria, Proeedings of the 24th Annual Joint Conferene of the IEEE Computer and Communiations Soieties (INFOCOM 5, vol. 1, pp. 1 11, Mar. 25. [4] R. Ma, V. Misra, and D. Rubenstein, Modeling and Analysis of Generalized Slotted-Aloha MAC Protools in Cooperative, Competitive and Adversarial Environments, Proeedings of the 24th IEEE International Conferene on Distributed Computing Systems (ICDCS 6, p. 62, July 26. [5] L. Lifeng and H. El Gamal, The Water-Filling Game in Fading Multiple-Aess Channels, IEEE Trans. on Information Theory, vol. 54, no. 5, pp , May 28. [6] M. Cagalj, S. Ganeriwal, I. Aad, and J. P. Hubaux, On Selfish Behavior in CSMA/CA Networks, in Proeedings of the 24th Annual Joint Conferene of the IEEE Computer and Communiations Soieties (INFOCOM 5, vol. 4, 25, pp [7] V. Srivastava, J. Neel, A. B. MaKenzie, R. Menon, L. A. DaSilva, J. E. Hiks, J. H. Reed, and R. P. Gilles, Using Game Theory to Analyze Wireless Ad Ho Networks, IEEE Communiations Surveys and Tutorials, vol. 7, no. 4, pp , 25. [8] F. V. Jensen and T. D. Nielsen, Bayesian Networks and Deision Graphs. Springer, 27. [9] G. Shwarz, Estimating the Dimension of a Model, The Annals of Statistis, vol. 6, no. 2, pp , [1] G. Owen, Game Theory, 3rd ed. New York: Aademi, 21. [11] G. Hardin, The Tragedy of the Commons, Siene, vol. 162, no. 3859, pp , De [12] P. Jaquet, P. Muhlethaler, T. Clausen, A. Laouiti, A. Qayyum, and L. Viennot, Optimized Link State Routing protool for ad ho networks, in Proeedingsof the IEEE International Multi Topi Conferene IEEE INMIC., 21. VI. CONCLUSIONS In this paper, we developed a framework whih an be used to selet the best ooperation strategy between two oexisting wireless networks sharing some of their nodes. Bayesian Network theory was used to derive the statistial orrelation between loal parameters and global system performane. Based on this information, a game theoreti seletion of the