Channel Allocation in Non-Cooperative Multi-Radio Multi-Channel Wireless Networks

Transcription

1 Cannel Allocation in Non-Cooperative Multi-Radio Multi-Cannel Wireless Networks Dejun Yang, Xi Fang, Guoliang Xue Arizona State University Abstract Wile tremendous efforts ave been made on cannel allocation problems in wireless networks, most of tem are on cooperative networks wit few exceptions [6, 8, 31, 32]. Among tose works on non-cooperative networks, none of tem considers te network wit multiple collision domains. Instead, tey all assume te single collision domain, were all transmissions interfere wit eac oter if tey are on te same cannel. In tis paper, we fill tis void and generalize te cannel allocation problem to non-cooperative multi-radio multi-cannel wireless networks wit multiple collision domains. We formulate te problem as a strategic game, called CAlloc. We sow tat te CAlloc game may result in an oscillation wen tere are no exogenous factors to influence players strategies. To avoid tis possible oscillation, we design a carging sceme to induce players to converge to a Nas Equilibrium (NE). We bound te convergence speed and prove tat te system performance in an NE is at least (1 r ) of te system performance in an optimal solution, were r is te maximum number of radios equipped on wireless devices and is te number of available cannels. In addition, we develop a localized algoritm for players to find an NE strategy. Finally, we evaluate our design troug extensive experiments. Te results validate our analysis of te possible oscillation in te CAlloc game lacking te carging sceme, confirm te convergence of te CAlloc game wit te carging sceme, and verify our proof on te system performance compared to te upper bounds returned by an LP-based algoritm. I. INTRODUCTION Te development of te IEEE 82.11a/b/g standards as spurred te emergence of broadband wireless networks. Due to te common transmission media sared by communication devices, interference arises if communication devices are operating on te same frequency. It as been sown tat interference severely limits te network capacity [1]. Frequency Division Multiple Access (FDMA) is a widely used tecnique to enable multiple devices to sare a communication medium. In FDMA, te available bandwidt is divided into multiple sub-bands, named cannels. Using multiple cannels in multi-radio wireless networks can greatly alleviate te interference and improve te network trougput [26]. Ideally, if tere are a sufficient number of cannels and eac device assigns different cannels to its radios, tere would be no interference in te network at all. However, since te spectrum is a scarce resource, we are only allowed to divide te available bandwidt into a limited number of cannels. For example, tere are 3 and 12 non-overlapping cannels for te IEEE 82.11b/g standards in 2.4 GHz and te IEEE 82.11a standard in 5 GHz, respectively. A fundamental problem in Te autors are affiliated wit Arizona State University, Tempe, AZ {dejun.yang, xi.fang, xue}@asu.edu. Tis researc was supported in part by NSF grants 9563 and Te information reported ere does not reflect te position or te policy of te federal government. multi-radio multi-cannel (MR-MC) wireless networks is ow to allocate cannels to radios, wic is commonly referred to as te cannel allocation problem (also known as te cannel assignment problem). Wile tremendous efforts ave been made on te cannel allocation problem, most of tem are on cooperative networks were devices are assumed to be cooperative and unselfis; owever tis assumption may not old in practice. Usually, a wireless device is owned by an independent individual, wo is only interested in selfisly maximizing its own profit witout respecting te system performance. Tere are a few works considering non-cooperative networks [6, 8, 31, 32]. However, all of tese works only consider te problem in a single collision domain, wic means all te transmissions will interfere wit eac oter if tey are on te same cannel. In tis paper, we study te cannel allocation problem in non-cooperative MR-MC networks wit multiple collision domains. To caracterize te network wit multiple collision domains, we introduce interference models into te network. Te results in tis paper are independent of te interference model adopted as long as te model is defined on pairs of communications, for example, te protocol interference model is used in tis paper. We model te cannel assignment problem in non-cooperative MR-MC wireless networks as a strategic game. We sow tat te game may oscillate indefinitely wen tere are no exogenous factors to influence players beavior. Tis possible oscillation can result in significant communication overead and te degradation of te system performance. To avoid tis undesirable outcome, we develop a carging sceme to induce players beavior. Te design of te carging sceme ensures te convergence to a Nas Equilibrium (NE). Players are in an NE if no player can improve its utility by canging its strategy unilaterally. Altoug NE is usually not social optimal, we can prove tat te system performance in an NE is guaranteed to be at least a factor of te system performance in te optimal solution. We summarize our main contributions as follows: To te best of our knowledge, we are te first to study te cannel allocation problem in non-cooperative MR-MC wireless networks wit multiple collision domains. We model te problem as a strategic game, called CAlloc. We sow tat te CAlloc game can result in an oscillation, were players keep canging teir strategies back and force trying to improve teir utilities. To avoid te possible oscillation, we design a carging sceme to influence players beavior. We prove tat, under te carging sceme, te CAlloc game converges

2 to an NE. We also prove tat te system performance in an NE is guaranteed to be at least (1 r ) of te system performance in te optimal solution, were r is te maximum number of radios equipped on wireless devices and is te number of available cannels. We design a localized algoritm for players to find an NE and prove tat it takes O( rn 3 (n + log )) time for te CAlloc game to converge to an NE, were n is te number of players. In order to verify our proof of te system performance in an NE, we give an LP-based algoritm to derive efficiently computable upper bounds on te optimal solution. Troug extensive experiments, we validate our analysis of te possible oscillation in te CAlloc game lacking te carging sceme and confirm te proof of te convergence of te CAlloc game wit te carging sceme. Te results also sow tat te system performance in an NE is very close to te optimal solution and tus verify our proof of te system performance. Te remainder of tis paper is organized as follows. In Section II, we review te current literature on te cannel allocation problem. In Section III, we present te system model considered in our paper and formulate te cannel allocation problem as a game, called CAlloc. In Section IV, we use an example to sow tat it is possible for te CAlloc game to oscillate endlessly wen tere are no exogenous factors to influence players beavior. In Section V, we design a carging sceme to induce players to converge to an NE, compute te price of anarcy of te CAlloc game, and develop a localized algoritm for players. In Section VI, we give an LP-based algoritm to find an upper bound on te optimal solution. In Section VII, we evaluate te performance of te CAlloc game troug extensive experiments. Finally, we form our conclusion in Section VIII. II. RELATED WORK Most previous works on cannel allocation can be categorized into two categories, cannel allocation in cooperative networks [5, 15, 21, 24, 25, 27 29] and cannel allocation in non-cooperative networks [6, 8, 31, 32]. We summarize te related works in Table I. A. Cannel Allocation in Cooperative Networks Tere is a considerable amount of works on te cannel allocation problem in Wireless Mes Networks (WMNs). In [5], Das et al. presented two mixed integer linear programming (ILP) models to solve te cannel allocation problem in WMNs wit te objective to maximize te number of simultaneously transmitting links. However, it is known tat solving an ILP is NP-ard. In [24], Ramacandran et al. proposed a centralized cannel allocation algoritm utilizing a novel interference estimation tecnique in conjunction wit an extension to te conflict grap model, called te multiradio conflict grap. In [28], Sridar et al. proposed a localized cannel allocation algoritm called LOCA, wic is a euristic algoritm. Subramanian et al. [29] and Marina et al. [21] studied te cannel allocation problem were eac link is TABLE I RELATED WORK Single Collision Domain Multiple Collision Domains Cooperative none [5, 15, 21, 24, 25, 27 29] Non-cooperative [6, 8, 31, 32] our work assigned a cannel wit te constraint tat te number of different cannels assigned to te links incident on any node is at most te number of radios on tat node. Subramanian et al. [29] developed a centralized algoritm based on Tabu searc and a distributed algoritm based on te Max-K-cut problem. Marina et al. [21] proposed a greedy euristic cannel allocation algoritm, termed CLICA. In [15], Ko et al. studied te cannel allocation problem wit a different objective function and proposed a distributed algoritm witout any performance guarantee. In [27], Sin et al. considered te cannel allocation problem to maximize te trougput or minimize te delay, and presented te an allocation sceme, called SAFE, wic is a distributed euristic. All te above related works are based on te assumption tat wireless devices in te network cooperate to acieve a ig system performance. However, tis assumption migt not old in practice. Usually, a wireless device is owned by an independent individual, wo is only interested in selfisly maximizing its own profit witout respecting te system performance or considering oters profits. B. Cannel Allocation in Non-cooperative Networks Game teory as been widely used to solve problems in non-cooperative wireless networks, for instance, Aloa networks [19] and CSMA/CA networks [3, 16]. Based on a grap coloring game model, Halldórsson et al. [12] provided bounds on te price of anarcy of te cannel allocation game. However, teir model does not apply to multi-radio networks. In an earlier work, Félegyázi et al. [6] formulated te cannel allocation problem in non-cooperative MR-MC wireless networks as a game, analyzed te existence of Nas Equilibria and presented two algoritms to acieve an NE. Along tis line, Wu et al. [32] introduced a payment formula to ensure te existence of a strongly dominant strategy equilibrium (SDSE). Furtermore, wen te system converges to an SDSE, it also acieves global optimality in terms of system trougput. In [8], Gao et al. extended te problem to multi-op networks and also addressed coalition issues. Most recently, Wu et al. [31] studied te problem of adaptive-widt cannel allocation in non-cooperative MR-MC wireless networks, were contiguous cannels may be combined to provide a better utilization of te available cannels. However, all te above results can only be applied to a single collision domain, witout considering multiple collision domains. In tis paper, we fill tis void and study te cannel allocation problem in non-cooperative MR- MC networks wit multiple collision domains. III. SYSTEM MODEL AND GAME FORMULATION A. Network Model Te network model in tis paper closely follows te models in [6, 8, 31, 32]. We consider a static wireless network consist-

3 ing of a set L = {,,..., L n } of n communication links. Eac link L i is modeled as an undirected link between two nodes v i and u i, were v i and u i denote two wireless devices communicating wit eac oter. Te use of te undirected link model reflects te fact tat te IEEE DCF requires te sender to be able to receive te acknowledgement message from te receiver for every transmitted packet. Since links are undirected, two nodes are able to coordinate to select te same cannels for communication. As in [6, 8, 31, 32], we assume tat te links are backlogged and always ave packets to transmit. Eac wireless device is equipped wit multiple radio interfaces. We furter assume tat eac transmission must be between two radios, of wic one functions as a transmitter and te oter as a receiver. Tus, it is reasonable to assume tat bot nodes of L i ave te same number of radios, denoted by r i. We assume te wireless devices ave te same maximum transmission power, but eac of te devices adjusts its actual data transmission power according to te lengt of te transmission link, denoted by l i. Let R denote te transmission range under te maximum transmission power. Furtermore, tere are > 1 ortogonal cannels available in te network, e.g. 12 ortogonal cannels in te IEEE 82.11a protocol. We denote te set of cannels by C = {c 1, c 2,..., c }. To communicate, two nodes of a link ave to tune at least one of teir radios to te same cannel(s). Parallel communications are allowed between two nodes if tey sare multiple cannels on radios. In order to avoid te co-radios interference in a device [8], we assume tat different radios on a node sould be tuned to different cannels. Terefore, it is reasonable to assume tat r i < for all L i L as it would be straigtforward to allocate cannels oterwise. Allocating eac cannel to at most one radio as also been proved to be a necessary condition to maximize te device s transmission data rate [6, 8, 31]. B. Interference Model Due to te common transmission medium, wireless transmission along a communication link may interfere wit te transmissions along oter communication links, especially tose witin its vicinity. Wile existing works [6, 8, 31, 32] ave studied te cannel allocation problem in non-cooperative networks for bot single-op and multi-op models, all te results can only be applied to a single collision domain. In oter words, tey assume tat all transmissions interfere wit eac oter if tey sare at least one cannel. However, te strengt of a wireless transmission signal decays exponentially wit respect to te distance it travels from te transmitter. Terefore te signal from a distant transmission is, if not negligible, not destructive enoug to prevent anoter transmission from succeeding. In tis paper, we generalize te cannel allocation problem to networks wit multiple collision domains. To caracterize networks wit multiple collision domains, an appropriate interference model is necessary. Various interference models ave been proposed in te literature, for example, te primary interference model [11], te protocol interference model [1, 14], and te pysical interference model (a.k.a SINR interference model) [1, 14]. Te results in {1,2,4} {2,3} {2,3,4} L 5 {3} L 4 {2,4} Fig. 1. A 5-link network, were L = {,,, L 4, L 5 }, r 1 = 3, r 2 = 3, r 3 = 2, r 4 = 2, r 5 = 1 and C = {1, 2, 3, 4}. L 5 L 4 L 5 L 4 (a) PING (b) MPING (c) MING Fig. 2. Te PING, MPING and MING for te example in Fig. 1 L 5 L 4 tis paper are independent of te specific interference model used as long as te interference model is defined on pairs of communication links. For te sake of presentation, te protocol interference model is adopted trougout tis paper. Tis model as been used by most of te works on cannel allocation problems [1, 2, 21, 26, 29, 3]. In tis model, eac node as an interference range γl i, wic is at least as large as te transmission range (equal to l i ), i.e., γ 1. We assume tat l i R γ, for any L i. Any node u will be interfered by node v if u is witin v s interference range. We can imagine tat, associated wit eac L i, tere is an interference disk D ui centered at u i, and an interference disk D vi centered at v i. Te union of D ui and D vi, denoted by D ui D vi, constitutes te interference area of L i. Link L i interferes wit link L j if and only if eiter of v j and u j is in D ui D vi, and two links sare at least one common cannel. Before te cannels on te radios are known, we can only say tat L i potentially interferes wit L j. As an illustrating example, Fig. 1 sows a 5-link network, were dased peanut-saped curves represent te boundaries of interference areas (numbers in parenteses will be explained in Section III-D). In tis example, potentially interferes wit wile cannot interfere wit. Conflict graps are widely used to facilitate te design of cannel allocation algoritms [14, 21, 22, 24, 29]. We use a similar concept, called potential interference grap (PING), to caracterize te interfering relationsips among te links. Different from te conflict grap, te edges in te PING are directed due to te eterogeneity of te interference range. In a PING, G p = (V p, A p ), nodes correspond to communication links. Hereafter, we also use L i to denote te corresponding node in G p. Tere is an arc from L i to L j if L i potentially interferes wit L j. Fig. 2(a) sows a PING of te example in Fig. 1. Unfortunately, te above defined PING does not accurately model te devices wit multiple radios. For example, if L i potentially interferes wit L j and bot links ave two radio

4 pairs, tere sould be two interference arcs. Terefore, we extend te PING to model multi-radio networks and call te new model multi-radio potential interference grap (MPING). An MPING is a directed multigrap, G m = (V m, A m ), were nodes still represent transmission links, arcs represent potential interference between links, and parallel directed arcs may exist between two vertices. Tere are min{r i, r j } arcs from L i to L j if (L i, L j ) A p. Let A m(l i ) and A + m(l i ) be te set of in-arcs and te set of out-arcs, respectively. Te in-arc set A m(l i ) of L i is te set of arcs going into L i and te out-arc set A + m(l i ) of L i is te set of arcs going from L i. Te in-arcs in A m(l i ) are called te potential interference arcs of L i. Let Nm(L i ) be te set of in-neigbors and N m(l + i ) be te set of out-neigbors in te MPING G m. Te in-neigbor set Nm(L i ) of L i is te set of vertices, wic are te tails of in-arcs and te out-neigbor set N m(l + i ) of L i is te set of vertices, wic are te eads of out-arcs. Te MPING corresponding to Fig. 1 is sown in Fig. 2(b). C. Game Teory Concepts in a Nutsell Game teory [7] is a discipline aimed at modeling scenarios were individual decision-makers ave to coose specific actions tat ave mutual or possibly conflicting consequences. A game consists of a set P = {P 1, P 2,..., P n } of players. Eac player P i P as a non-empty strategy set Π i. Let s i denote te selected strategy by P i. A strategy profile s consists of all te players strategies, i.e., s = (s 1, s 2,..., s n ). Obviously, we ave s Π = Pi PΠ i. Let s i denote te strategy profile excluding s i. As a notational convention, we ten ave s = (s i, s i ). Te utility (or payoff) function u i (s) of P i measures P i s valuation on strategy profile s. We say tat P i prefers s i to s i if u i(s i, s i ) > u i (s i, s i). Wen oter players strategies are fixed, P i can select a strategy, denoted by b i (s i ), wic maximizes its utility function. Suc a strategy is called a best response [7] of P i, wic is formally defined as follows. Definition 1: [Best Response] Given oter player s strategies s i, a best response strategy of P i is a strategy s i Π i suc tat b i (s i ) = arg max si Π i u i (s i, s i ), were Π i is te strategy space of P i. In order to study te interaction of players, we adopt te concept of Nas Equilibrium (NE) [7]. Definition 2: [Nas Equilibrium] A strategy profile s ne = (s ne 1, s ne 2,..., s ne n ) constitutes a Nas Equilibrium if, for eac P i, we ave u i (s ne i, s ne i ) u i(s i, s ne i ) for all s i Π i. In oter words, none of te players can improve its utility by unilaterally deviating from its current strategy in an NE. Matematically, it means b i (s ne i ) = sne i for all P i P. To caracterize and quantify te inefficiency of te system performance due to te lack of cooperation among te players, we use te concept of price of anarcy (POA) [17]. Definition 3: [Price of Anarcy] Te price of anarcy of te game is te ratio of te system performance in te worst Nas Equilibrium to te system performance in te social optimal solution, tat is P OA = min s ne Π ne U(sne ), max s Π U(s) were Π ne Π is te set of all NEs and U is a function of te strategy profile measuring te system performance. Note tat U(s) is not necessarily te sum of te utilities of all players as player s utility may not always reflect te system performance. Te POA in game teory is an analogue of te approximation ratio in combinatorial optimization. If a game as a POA lower bounded by α 1, it means tat for any instance of te game, te system performance in any NE is at least α times te system performance in te optimal solution. D. CAlloc Game Formulation We formulate te cannel allocation problem in noncooperative MR-MC wireless networks as a game, called CAlloc. In tis game, eac transmission link is a player, wose strategy space is te set of all te possible cannel allocations on its radios. We assume tat players are selfis, rational and onest. We leave te case were players can ceat for our future work. Trougout te rest of tis paper, we will use link and player intercangeably. Te cannel allocation of L i is defined to be a vector s i = (s i1, s i2,..., s i ), were s ik = 1 if L i assigns cannel c k to one radio pair and s ik = oterwise. To sufficiently utilize te cannel resource, we require tat k=1 s ik = r i, wic is also proved to be optimal for eac player for te single-collision domain case [6]. Te strategy profile s is ten an n matrix defined by all te players strategies, s = (s 1, s 2,..., s n ) T. Altoug previous works in te literature [6, 8, 31, 32] ave used acievable data rate as te utility function, tey assume tat all te links are in a single collision domain. Because of te idden terminal problem, it is unlikely to ave a closedform expression to calculate te acievable data rate for eac player in te network wit multiple collision domains. Tis difficulty as also been discussed in [6, 32]. An alternative is to use te interference as a performance metric. As sown in [34, Eq.(4)], te data rate is approximately a linear function of te interference tat te link can overear. Te use of te interference as a performance metric can also be found in [24, 29, 3]. Given a strategy profile s, we say a communication radio pair interferes wit L i if tis radio pair belongs to a link interfering wit L i and as been tuned to a cannel tat is also allocated by L i. We define te interference number of L i, denoted by I i (s), to be te number of communication radio pairs interfering wit L i. Matematically, we ave I i (s) = s i s j, were te symbol is te dot product between two vectors. Note tat I i (s) A m(l i ) for all L i L. Wen s is given, we can construct te multi-radio interference grap (MING), G m (s) = (V m (s), A m (s)), from te MPING by removing corresponding potential interference arcs. Te number of arcs from L i to L j is equal to s i s j. In tis paper, we define te utility function of a player to be a function of its interference number. More specifically, te utility function u i (s) of player L i is defined as u i (s) = A m(l i ) I i (s). (1)

5 In oter words, te objective of L i is to remove as many of te potential interference arcs as possible from N m(l i ) by allocating cannels to its radios. Wen te network is given, A m(l i ) is a constant. Hence maximizing (1) can acieve te goal of minimizing I i (s), wic is te interference suffered by L i under te strategy profile s. Intuitively, te system performance function is defined as U(s) = A m I i(s), (2) wic is te total potential interference removed from te MP- ING under allocation profile s. Likewise, A m is a constant, ence maximizing (2) can acieve te goal of minimizing I i(s), wic is te overall network interference. Use te example in Fig. 1 for illustration. Te numbers in te parenteses associated to eac link represent te allocated cannels. Te corresponding cannel allocation vectors are s 1 = (1, 1,, 1), s 2 = (, 1, 1, 1), s 3 = (, 1, 1, ), s 4 = (, 1,, 1), and s 5 = (,, 1, ). Te interference grap under s is sown in Fig. 2(c). Under tis strategy profile, we ave u 1 (s) = 2, u 2 (s) =, u 3 (s) =, u 4 (s) = 1 and u 5 (s) = 1. Te system performance is U(s) = 4. IV. OSCILLATION IN THE CHALLOC GAME In tis section, we sow tat players migt not converge to any stable status, i.e. NE, according to te current defined utility function. Consider te network illustrated in Fig.3(a). Obviously, we ave te MPING as sown in Fig.3(b). Assume tat eac link is equipped only one radio pair and tere are two cannels {c 1, c 2 } available. Due to te special topology and te dependency relation, we ave te following conclusions. If L 4 uses c 1, bot and will use c 2. If bot and use c 2, will use c 1. If uses c 1, L 4 will use c 2. If L 4 uses c 2, bot and will use c Tis process turns into an infinite loop. L 4 (a) A four-link example (b) MPING Fig. 3. An example were players oscillate forever Tis possible oscillation is definitely undesirable for two reasons: 1) Te cannel switcing delays can be in te order of milliseconds [4], an order of magnitude iger tan typical packet transmission time (in microseconds). 2) It can introduce a significant amount of communication overead, as two devices need to coordinate to switc cannels. V. NASH EQUILIBRIA As we ave discussed in Section IV, te CAlloc game can run into an oscillation problem, wic is undesirable from te system s perspective. In order to induce players to converge to L 4 an NE, we design a carging sceme to influence te players in tis section. We ten prove tat, based on te newly defined utility function considering te carge, te CAlloc game must converge to an NE. In addition, we prove tat even at te worst NE, te system performance is at least (1 r ) times te system performance in te optimal cannel allocation, were r is te maximum number of radios equipped on te nodes and is te number of cannels available in te network. Finally, we present a localized algoritm for players to converge to an NE. A. Carging Sceme Design Similar approaces ave also been used in [31, 32]. Teir carging functions are designed based on te globally optimal cannel allocation. Essentially, players wo deviate from te optimal cannel allocation will be punised according to te carging function. Unfortunately, it as been proved tat te optimization problem of maximizing te system performance function (2) is NP-ard [29]. Different variations of te cannel allocation problem ave also been sown to be NP-ard [1, 21, 24, 26]. Terefore, we focus on designing a carging sceme, wic can make te CAlloc game converge to an NE and acieve guaranteed system performance. As in [31 33], we assume tat tere exists a virtual currency in te system. Eac player needs to pay certain amount of virtual money to te system administrator based on te strategy profile s. We define te carge p i of player L i as p i (s) = s i s j, (3) wic is te total interference player L i imposes on te oters. Te carge can be considered to be te fee for accessing te cannels. We ten redefine te utility function for eac player L i L as u i (s) = A m(l i ) I i (s) p i (s), (4) wic is equal to te original utility minus its payment to te system administrator. B. Existence of Nas Equilibria Having defined a new utility function for te player, we next prove te existence of Nas Equilibria wit te elp of te concept of potential game [23]. Definition 4: [Potential Game] A function Φ : Π Z is an exact potential function for a game if te cange of any player s utility can be exactly expressed in te function. Formally, Φ sould satisfy Φ(s i, s i ) Φ(s i, s i ) = u i (s i, s i ) u i (s i, s i ), for all s i and s i, s i Π i. A game is called a potential game if it admits an exact potential function. A nice property of being a potential game is tat if Φ is bounded, we can prove tat te game possesses an NE and any improvement pat leads to an NE. An improvement pat is a sequence of strategy profiles, eac of wic (except te first one) is formed from te previous one by canging a unique player s strategy to improve te player s utility. Terefore we first prove tat te CAlloc game is a potential game and ten prove tat its corresponding potential function is bounded.

6 Lemma 1: Te CAlloc game is a potential game. Proof: We prove tis lemma by constructing an exact potential function Φ. Define Φ as Φ(s) = 1 u i (s). 2 We next prove tat Φ(s i, s i ) Φ(s i, s i) = u i (s i, s i ) u i (s i, s i) for all s i and s i, s i Π i. First, we ave Φ(s) = 1 A 2 m (L i ) s i s j s i s j L j Nm(L i) L j N m(l + i) = A m 1 s i s j + s i s j, (5) 2 2 were te second equality follows from te fact tat A m = A m(l i ). We ten ave Φ(s i, s i ) Φ(s i, s i ) = 1 s j s i = s j s i s i s j + s i s j + s i s j + s i s j + =u i (s i, s i ) u i (s i, s i ), s j s i s j s i s i s j s i s j s i s j s i s j were te first equality follows from te fact tat te cange of player L i s strategy only affects players in Nm(L i ) and N m(l + i ), and te last equality follows from u i (s i, s i ) u i (s i, s i) = L j Nm(L i) s i s j + L j N m(l + i) s i s j ( Lj Nm(L s i) i s j + ) L j N m(l + s i) i s j. We ave proved tat Φ(s) is an exact potential function (Definition 4) of te CAlloc game. Hence te CAlloc game is a potential game. Te bound of Φ(s) is given in te following lemma. Lemma 2: For any s Π, Φ(s) is bounded by O( rn 2 ). Proof: By (5), we ave Φ(s) A m 2 L r i L i(n 1) rn2. Tis completes te proof. Now we give te main teorem in tis section. Teorem 1: Te CAlloc game possesses an NE. Proof: Combining Lemma 1 and Lemma 2, tis teorem directly follows from Corollary 2.2 in [23], wic states tat every finite potential game possesses a Nas Equilibrium. C. Price of Anarcy Altoug we ave proved tat tere exist Nas Equilibria in te CAlloc game, we know tat NE is usually not socially efficient in te sense tat te system performance in an NE is not optimized. Neverteless, we prove in tis section tat te POA of te CAlloc game is independent of te number of players involved in te game and is lower bounded by a constant wen te number of cannels and te number of radios equipped on devices are fixed. Teorem 2: In te CAlloc game, P OA ( 1 ) r. Recall tat r is te maximum number of radios equipped on wireless devices and is te number of available cannels. Proof: Before proving te POA of te CAlloc game, we first find a lower bound of te utility of any player in an NE. Let s ne = (s ne 1, s ne 2,..., s ne n ) T be any NE of te CAlloc game. Let s opt = (s opt 1, sopt 2,..., sopt n ) T be a social optimum. We ave u i (s ne ) = A m(l i ) A m(l i ) s i Π i s ne i s ne j s i s ne j s ne i s ne j s i s ne / Π i (6) = A m(l i ) r i ( A m(l i ) + A + m(l i ) ) (7) A m(l i ) r ( A m(l i ) + A + m(l i ) ), (8) were (6) follows from te definition of NE, (7) follows from te fact tat eac arc is counted ( ) 1 r i 1 times and Πi = ( ) r i, next (8) follows from r = max Li L r i. Ten te system performance is U(s ne ) = u i (s ne ) + s ne i s ne j (9) L j N m(l + i) ( A m(l i ) r ( A m(l i ) + A + m(l i ) )) + s ne i s ne j (1) = A m 2 r A m + j s ne i s ne j (11) = A m 2 r A m + A m U(s ne ), (12) were (9) follows from (2) and (4), (1) follows from (8), and (12) follows from te fact tat A m = U(s ne ) + L j N +m(l i) sne i s ne j. Considering te obvious fact tat U(s opt ) A m, we ave ( U(s ne ) 1 r ) U(s opt ). (13)

7 Since (13) olds for any NE of te CAlloc game, it is straigtforward to prove tat P OA ( 1 r ). Note tat we were very conservative wen we derived (1), making te bound of POA very loose. We will leave te derivation of a tigter bound for te future study. D. A Localized Algoritm for te CAlloc Game Since te CAlloc game is a potential game (Lemma 1), any improvement pat leads to an NE [23]. To form an improvement pat, we need to require te sequential action of te players. Eac player takes its best response strategy upon its turn. Note tat b i (s i ) can be computed just based on te strategies of te players in N m(l i ) and N + m(l i ). Terefore we can design a localized algoritm for players to find an NE. A localized algoritm needs no information to propagate troug te wole network. Tus it is scalable to te network size and robust to te topology cange. Algoritm 1: A Localized Algoritm for L i 1 Randomly allocate r i cannels as s i ; 2 W i i, ctr ; 3 wile true do 4 if W i = ten 5 Get te current cannel allocation; 6 s i b i(s i ); 7 if s i = s i ten 8 if ctr = n ten break; else ctr ctr + 1; 9 else s i s i, ctr ; 1 W i n; 11 else W i W i 1; 12 end Te localized algoritm is illustrated in Algoritm 1. Te implementation issue will be discussed later. To avoid te simultaneous cange in cannel allocations of different players, we let eac player L i ave a counter W i, wic is initially set to i. At te beginning of te algoritm, eac player L i randomly picks r i cannels as its initial strategy. In every iteration, L i cecks te value of W i. If W i is equal, L i gets te current cannel allocations and calculates its best response strategy b i (s i ). If te best response strategy is te same wit its current strategy, it increases anoter counter ctr by 1. Oterwise, it updates its strategy and resets ctr to. Te value of ctr indicates ow many times its current strategy as been te best response strategy consecutively. Te use of ctr is to avoid early termination before te CAlloc game converges to an NE. If te counter W i as not reaced, L i decreases its value by 1. Lemma 3: Te best response strategy b i (s i ) can be computed in O((n + log )) time. Proof: We prove tis lemma by giving an algoritm to compute b i (s i ). For eac cannel c k C, we compute te value of L j Nm(L e i) k s j + L j N m(l + e i) k s j, were e k denotes te vector wit a 1 in te kt coordinate and s elsewere. Tis can be finised in O(n) time. Sort te cannels in a nondecreasing order, wic can be finised in O( log ) time. Since cannels are independent, L i selects te first r i cannels as b i (s i ). Hence te above algoritm can be finised in time bounded by O((n + log )). Teorem 3: For any instance of te CAlloc game, if all te players follow Algoritm 1, it takes O( rn 3 (n + log )) time to converge to an NE. Proof: According to Lemma 1 and Lemma 2, every time a player canges its strategy (to one introducing better utility), te potential function Φ(s) will be increased accordingly and te value of Φ(s) is bounded by O( rn 2 ). Terefore te number of strategy updates is bounded by O( rn 2 ). Since tere will be at least one strategy update in eac round, te number of rounds is also bounded by O( rn 2 ). Using Lemma 3 and te fact tat n players take actions sequentially in eac round, we can prove tat it takes O( rn 3 (n + log )) time for te CAlloc game to converge to an NE. Implementation Issue: Existing works [13, 15, 27, 28] on distributed or localized cannel allocation algoritms all assume tat te interference sets are given, but do not discuss ow to find tem in a distributed manner. We assume tat during te cannel allocation stage, all te players are using te same cannel on one of teir radios, wic is called te control cannel, and using te maximum transmission power to send packets. Because of te assumption tat l i R γ, it is guaranteed tat all te links in te interference range of link L i can overear te packets. Te packet to be excanged during te cannel allocation stage is of form (i, s i ). For eac player L i, upon its turn, it sends out te packet. During oter time periods, it listens to te control cannel and receives packets from oters. For te packet received from L j, L i computes its distance from L j according to te received signal strengt 1, puts L j in N m(l + i ) if L j is witin its interference range, and puts L j in Nm(L i ) if it is witin L j s interference range. VI. UPPER BOUNDS ON OPTIMAL CHANNEL ALLOCATION In tis section, we derive efficiently computable upper bounds on te cannel allocation problem, wic will be used in Section VII to evaluate te system performance of te CAlloc game. We first formulate te cannel allocation problem as an integer linear program (ILP) and ten relax te constraints to acieve an upper bound on te optimal solution. Let s ik {, 1} denote L i s allocation on c k, were s ik = 1 if L i allocates c k to one of its radios and s ik = oterwise. Let x ijk {, 1} denote te interference from L i to L j via c k. Our ILP can be formulated as follows, max s.t. A m n n i=1 j=1,j i k=1 x ijk s ik = r i (L i L) (14) k=1 x ijk s ik + s jk 1 ((L i, L j ) A m, c k C) (15) s ik {, 1} (L i L, c k C) x ijk {, 1} (L i, L j L, L i L j, c k C), 1 Oter distance measurement tecniques can also be used [2].

8 were Constraints (14) follow our system model, and Constraints (15) guarantee tat x ijk = 1 if and only if bot L i and L j ave one radio tuned to cannel c k. Unfortunately, solving an ILP is in general NP-ard [9], wic means it may take exponential time to find te optimal solution. Hence we relax te above ILP to an LP by allowing s ik and x ijk to be real values between and 1. Te LP as been sown to be solvable in polynomial time [18]. As te constraints are relaxed, te LP only gives an upper bound on te ILP s optimal solution. A. Experiment Setup VII. EVALUATIONS In te simulations, links were randomly distributed in a 1m 1m square. Te lengt of eac link was uniformly distributed over [1, 3]. Te interference range of te node was set to 2 times of te link lengt. Te number of available cannels was varied from 5 to 12 wit increment of 1. Te number of links was varied from 1 to 1 wit increment of 1. Te number of radio pairs on eac link was uniformly selected over [1, r], were r {2, 3, 4, 5}. Note tat r = r in most cases. For every setting, we randomly generated 1 instances and averaged te results. 1) Cannel Allocation Algoritms: To evaluate te system performance of te CAlloc game, we compare te CAlloc game wit oter two algoritms listed as below. LP-based Algoritm (LP): Tis algoritm is based on te LP formulation in Section VI. Random Allocation Algoritm (Rand): In tis algoritm, eac link L i randomly select r i cannels out of te cannels. To certain extent, Rand serves as a lower bound of te system performance in any cannel allocation. 2) Performance Metric: Te performance metrics include te system performance defined by (2), te player s removed interference defined by (1), and te convergence speed defined as te number of rounds before an NE is reaced. B. Convergence of te CAlloc Game We first verify tat te CAlloc game wit te carging sceme, denoted by CAlloc, does converge wile te one witout te carging sceme, denoted by No-Carge, may oscillate. In tis set of simulations, we set n to 5, r to 3, and to 8. Te x-axis represents te number of runs, eac of wic is an iteration of te wile-loop in Algoritm 1. Using runs can sow te results in at a more granular level tan using rounds. Altoug te algoritm will terminate wen te game reaces an NE, we let it keep running for te sake of comparison. We ave te results for 1 runs, but only sow te first 1 runs due to te space limitation. Fig. 4(a) sows te system performance of tese two difference game settings. As expected, CAlloc converges to an NE after 222 runs, wile No-Carge still oscillates even after 1 runs. Te factors affecting te convergence speed will be investigated in te next section. Fig. 4(b) sows te removed interference for a random player (player 44). We observe tat te removed interference of te player stays te same after about 2 runs in CAlloc, U(s) CAlloc No Carge Number of runs 28 CAlloc No Carge Number of runs (a) System performance (b) Player s performance Fig. 4. Convergence of te CAlloc game but oscillates even after 1 runs in No-Carge. Note tat oter players ave te similar results. C. Convergence Speed We next verify our analysis of te convergence speed, measured by te number of rounds. According to Teorem 3, te teoretical value is O( rn 2 ). We set to 8 and r to 3 in Fig. 5(a). We set n to 5 and r to 3 in Fig. 5(b). We set n to 5 and to 8 in Fig. 5(c). Fig. 5(a) and Fig. 5(b) sow te impact of n and te impact of r on te convergence speed, respectively. We observe tat te convergence speed is muc faster tan te teoretical speed, and tat all te instances can converge witin 1 rounds on average. Fig. 5(c) sows tat te convergence speed is almost independent of, wit te varying range of te average being less tan 1. D. System Performance We now compare te system performance of te CAlloc game wit tose of oter algoritms in Section VII-A. Fig. 6 sows all te results. As expected, LP as te best performance wile Rand as te worst. Te first observation is tat all te results are consistent wit our performance analysis in Teorem 2. In particular, Fig. 6(a), Fig. 6(b) and Fig. 6(c) sow te impact of n, r and on te system performance, respectively. Te results confirm tat te system performance of CAlloc compared to LP is independent of n. Te less radios or te more cannels tere are, te closer te performance of CAlloc is to te performance of LP. Anoter observation is tat te gap between te performance of Rand and te performance of CAlloc gets narrower wen te number of cannels increases or te number of r radios decreases. Te reason is tat wen te value of decreases, te probability tat two interfering links sare te same cannels decreases as well. Removed Interference VIII. CONCLUSION In tis paper, we ave studied te cannel allocation problem in non-cooperative MR-MC networks. Compared wit existing works, we removed te single collision domain assumption and considered networks wit multiple collision domains. We modeled te problem as a strategic game, called CAlloc. Via an example, we sowed tat CAlloc may result in an oscillation wen no exogenous factors exist. To avoid tis possible oscillation, we design a carging sceme to influence players beavior. We ten proved tat CAlloc will converge to an NE in polynomial number of steps. We furter proved tat te system performance in any NE is guaranteed to be at least (1 r ) of tat in te optimal solution, were r is te maximum number of radios equipped on wireless devices

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