A Game Theoretic Analysis of Distributed Power Control for Spread Spectrum Ad Hoc Networks

Transcription

1 A Game Theoretic Analysis of Distributed ower Control for Sread Sectrum Ad Hoc Networs Jianwei Huang, Randall A. Berry, Michael L. Honig Deartment of Electrical & Comuter Engineering, Northwestern University, 145 Sheridan Road, Evanston, IL 6008, USA {jianweih, rberry, Abstract We consider a distributed ower control scheme in a Sread Sectrum SS) wireless ad hoc networ, in which each user announces a rice that reflects his current interference level. Given these rices, we resent an asynchronous distributed algorithm for udating ower levels, and rovide conditions under which this algorithm converges to an otimal ower allocation. We relate this algorithm to myoic best resonse udates of a fictitious game, and characterize the algorithm s convergence using suermodular game theory. I. INTRODUCTION ower control is a basic technique for mitigating interference in wireless networs that enables efficient use of available sectrum. In ad hoc networs ower control is comlicated by the lac of centralized infrastructure, which necessitates the use of distributed aroaches. In this aer, we consider distributed ower control for SS ad hoc wireless networs, where each transmitter sreads its ower over the available bandwidth and all interference is treated as noise. We focus on a model where the users receive a variable transmission rate that deends on their received signal-to-interference lus noise ratio SINR). Our objective is to find distributed mechanisms for coordinating the users ower levels and otimizing overall erformance, measured in terms of total networ utility. We consider a ower control scheme in which the users exchange rice signals that indicate the cost of received interference. ricing mechanisms for allocating resources in networs has received considerable attention for both wire-line e.g. [1]) and wireless networs e.g. [] [4]). The roblem here differs from much of the revious wor because, due to interference, the users objective functions are couled with each other, and the overall networ objective is not concave in the allocated resource transmit ower). Also, in most revious wor, rices are Lagrange multiliers for some constrained resource such as ower or bandwidth; here the rices reflect the interference or externalities among the users instead of a resource constraint. Our model is similar to [5], which also considers combined ower and rate control for a SS ad hoc networ. The ower adatation in [5] solves a similar roblem to that considered here using gradient udates. Instead, we consider an aroach based on suermodular game theory [6], which allows for different utility functions and aears to have faster convergence. A variety of game-theoretic aroaches have been alied to networ resource allocation e.g., [7]). Suermodular game theory, in articular, has been used to study ower control in [8] [10]. Our aroach differs from those in that i) we focus on an ad hoc instead of a cellular networ; ii) we consider a different functional form for the utilities, and iii) we do not directly model the roblem as a non-cooerative game. Instead, the users voluntarily cooerate with each other by exchanging interference information. We introduce a fictitious game and aly a strategy sace transformation to view this algorithm as a suermodular game. Other wor on ower control in CDMA cellular and ad hoc networs includes [9] [13]. In much of this wor, a transmission is assumed to be successful only if a fixed imum SINR requirement is met. This is true for fixed-rate communications. However, this is not the case for elastic data alications, in which users can adat their transmission rate. In this aer, we focus on such rate-adative users, where the goal of ower control is to imize total networ erformance instead of guarantee interference margins for each user. II. SYSTEM MODEL We consider a sna-shot of an ad hoc networ with a set K = {1,..., K} of distinct transmission airs. Each air consists of one dedicated transmitter and one dedicated receiver. This can reresent a articular schedule of transmissions detered by an underlying routing and MAC layer rotocol. We will use the terms air and user interchangeably in the rest of the aer. Each user transmits an SS signal sread over the total bandwidth of B Hz. Over the time-eriod of interest we assume that the channel gains of the transmitterreceiver airs are fixed. The channel gain between user s transmitter and user j s receiver is denoted by h j. Note that in general h j =h j, since the latter reresents the channel gain between user j s transmitter and user s receiver. An examle of a networ with four users airs of nodes) is shown in Fig. 1. The value of the transmission by user Kis characterized by a utility function u γ ), which is an increasing and strictly concave function of the received SINR, h γ ) = n B j =, 1) jh j

2 T 1 T 3 h 14 h 11 h 31 h 13 h 34 h 33 R 1 h 1 h 3 h 4 R 4 h 1 T h Fig. 1. SS ad hoc networ of four users airs of nodes). T and R denote the transmitter and receiver of user, resectively. where n 0 is the bacground noise ower and = 1,, K ) is a vector of the users transmission owers. The users utility functions are couled due to mutual interference. An examle utility function is a logarithmic utility function u γ ) = θ log γ ), where θ is a user deendent riority arameter. For low SINR, a user s rate is aroximately linear in SINR; in this regime, the utility is therefore roortional to the logarithm of the rate. In the high SINR regime, the logarithmic utility aroximates the Shannon caacity log 1 + γ ) weighted by arameter θ. The ower control roblem is to secify the vector of transmission owers,, where each user must also satisfy an [ individual] transmission ower constraint, =,,where0. From a networ ersective, the objective is to imize the total utility summed over all users, i.e., 3 h 44 u tot ) = {,, K} h h 41 h 4 T 4 R 3 h 43 R K u γ )). =1 1) As a baseline, first consider the case where the users do not exchange any information and simly choose their transmission owers to imize their individual utilities. As in [9], this can be modeled as a non-cooerative ower control NCC) game, G NCC =[K, { }, {u }], where the layers corresond to the users in K; each layer ics [ a strategy transmission ] ower) from the strategy set =, and receives a ayoff u γ ). In this game, a ower rofile, = 1,..., K ), is a vector of the users transmission owers. The ower rofile of user s oonents is defined to be = 1,..., 1, +1,..., K ), so that = ; ).User s best resonse is B ) = arg u γ, )), i.e., the value that imizes u γ, )) given a fixed. A ower rofile is a Nash Equilibrium NE) of the NCC game if it is a fixed oint of the users best resonses, i.e. u γ ; )) u γ ; )) for any and any user. Since each user s ayoff u γ, )) is strictly increasing with for fixed, and there is no enalty for high transmission ower as long as, it is easy to verify that the unique NE of the NCC game is NCC = 1,..., K ), i.e., each transmitter uses its imum ower. This solution can be far from the socially otimal solution, which imizes the total utility. Although u γ ) is strictly concave in γ, the objective in roblem 1 may not be concave in. However, it is easy to verify that any local otimum, = 1,..., K ),ofthis roblem will be regular see age 309 of [14]), and so must satisfy the following KKT necessary conditions: Lemma 1 KKT necessary conditions:): For any local otimal solution of roblem 1, there exist unique Lagrange multilier vectors λ =λ 1,..., λ K) and µ =µ 1,..., µ K ) such that for all K, u γ, )) + u jγ j j, j)) = λ µ, j = = ) λ )=0,µ ) =0, and λ,µ 0. Let π j j, j )= u j γj j, j ) ), 3) I j j ) where I j j )= l =j lh lj is the total interference received by user j before bandwidth scaling). Here, π j j, j ) is always nonnegative and reresents user j s sensitivity to its current interference level, i.e., how much its utility would increase if the interference is decreased by one unit. This imlicitly incororates user j s utility information, i.e., the larger the utility e.g., larger θ ) the larger is π j j, j ). Using 3), condition ), forall K, can be written as u γ, )) ) = j = π j j, j) hj = λ µ. 4) Assume each user j announces π j = π j j, j )) as a rice that he charges other users for generating interference, and that users are rice taers i.e. each user regards other users rices as constants). Condition 4) is then a necessary and sufficient otimality condition for the roblem in which each user secifies a ower level to imize the surlus function, s, ; π )=u γ, )) π j h j, 5) j = assug fixed and π i.e., each user is a rice taer and ignores any influence it may have on these rices). The surlus in 5) can be viewed as user s utility us a ayment to the other users in the networ due to the interference it generates. The ayment is its transmit ower times a weighted sum of other users rices, with weights equal to the channel gains between user s transmitter and the other users receivers. This ayment can be interreted as a igovian tax [15]. III. ASYNCHRONOUS DISTRIBUTED RICING AD) ALGORITHM The ricing interretation of the KKT conditions motivates the following asynchronous distributed ricing AD) algo-

3 rithm, in which users iteratively udate their rice announcements and transmit ower decisions in an attemt to achieve a solution that satisfies Lemma 1. For each K, define a ower udate function 1 W,π ) = arg s ˆ, ; π ) ˆ ) = γ ) g j = π jh j γ ) and a rice udate function C ) = u γ )) I ) = u γ )) γ )), γ ) B h where /γ ) is indeendent of,and, 0 x u ), g x) = u ) 1 x), u ) <x<u 0), 0, u 0) x. The AD Algorithm is then secified as follows where t denotes the time immediately before t): 1.) Initialization: at t =0, each user Kchooses some initial ower 0) and rice π 0)..) There are K sets of infinite ositive time instances T, and T,π, K, at which each user udates ower t) and rice π t) as follows t) =W t ),π t ) ), if t T,, 6) π t) =C t ) ), if t T,π. 7) Note that a user need not udate his ower and rice at the same time. The asynchronous udate is imortant in ractice because it may be difficult to coordinate different users in the networ to udate their strategies in a synchronized manner. This algorithm is distributed in nature not only because of the distributed generation of owers and rices by users, but also because each user needs to acquire limited information. Secifically, to imlement the ower and rice udates, each user only needs to now: i) its own utility u,sinrγ and channel gain h, ii) the adjacent channel gains h j for j K and j =, and iii) the rice rofile π. By assumtion each user nows its own utility. The SINR γ and channel gain h can be measured at the receiver and fed bac to the user. Measuring the adjacent channel gains h j can be accomlished by having each receiver eriodically broadcast a beacon; assug recirocity, the transmitters can then measure these channel gains. Note that the adjacent channel gains account for only 1/K of the total channel gains in the networ; each user does not need to now the other gains. The rice information could also be eriodically broadcast through this beacon. Each user announces a single rice that is used by all interfering transmissions, so that the number of rices scales linearly with the size of the networ. Also, numerical results suggest that there is little effect on 1 Notation [x] b a denotes { {x, b},a}., erformance if users only convey their rices to nearby transmitters, i.e. those generating the strongest interference. Denote the set of fixed oints of the AD algorithm by F AD = {, π), π) =W, π), C ))}, 8) where W, π) = W 1 1,π 1 ),...,W K K,π K )) and C ) =C 1 ),...,C K )). Lemma : A ower rofile satisfies the KKT conditions of roblem 1 if and only if, C )) F AD. The roof relies on the strict concavity of u γ ). Furthermore, if the objective function of roblem 1 is strictly concave in or some monotonic transformation of ), then the KKT conditions become necessary and sufficient for the unique otimal solution to roblem 1, and the AD algorithm would reach that oint if it converges. In general, roblem 1 may not be a concave imization roblem, so that the AD algorithm may reach a locally otimal oint or saddle oint) of roblem 1. IV. CONVERGENCE ANALYSIS OF AD ALGORITHM In this section, we characterize the convergence of the AD algorithm by viewing it in a game theoretic context. We first consider a game similar to the NCC game, excet now each layer s strategy includes secifying both a ower and ariceπ to imize a ayoff given by the surlus in 5). Since there is no enalty for user announcing a high rice, it can be shown that each user s best resonse is to choose a large enough rice to force all other users to sto transmission. This is certainly not a desirable outcome and suggests that the rices should be detered externally by another rocedure. Instead of the receding game, we consider the following [ Fictitious { ower-rice } {F) control }] game: G F = FW FC, FW, FC, s FW,s FC, where the layers are from the union of the sets FW and FC. FW = {1,..., K} is the fictitious ower layer set, in which each layer chooses a ower from the strategy set FW = [ ],. FC = {1,..., K} is the fictitious rice layer set, inwhich each layer chooses a rice π from the strategy set FC = [0, π ]. Here π =su u γ )) / I )), which could be infinite for some utility functions. A layer FW receives a ayoff s FW, ; π )=u γ )) j = π j h j, 9) and a layer FC has a ayoff s FC π ; ) = π + u ) γ )). 10) I ) The ayoff s FC is defined so that the otimal strategy π = u / I. The layers best resonse functions are then B FW,π ) = W,π ), FW, and11) B FC ) = C ), FC. 1) Of course, no algorithm that relies only on first-order information can be guaranteed to converge to a global or even local otimum if the otimization roblem is non-concave.

4 In the F game, each user Kin the original ad hoc networ is slit into two fictitious layers, one in set FW who controls ower and the other one in set FC who controls rice π. Although users in the networ cooerate with each other by exchanging interference information instead of choosing rices to imize their surlus), each fictitious layer in the F game is selfish and imizes its own ayoff function. In the rest of the aer, a user refers to one of the K transmitter-receiver airs in set K, and a layer refers to one of the K fictitious layers in the set FW FC. It follows from 11) and 1) that the AD algorithm 6) and 7)) can be interreted as if the layers in the F game emloy an asynchronous myoic best resonse MBS) udates, i.e. the layers udate their strategies according their best resonses assug the other users strategies are fixed at the time of the udates. It is nown that the fixed oints of MBS udates are the same as the NEs of a game see Lemma 4..1 in [6]). Lemma 3:, π ) F AD if and only if, π ) is a NE of the F game. It follows that roving the convergence of MBS udates of the F game are sufficient to rove the convergence of the AD algorithm to a solution of KKT conditions. We next analyze this convergence using suermodular game theory [6]. We first introduce some definitions 3. A twice differentiable function f has increasing differences between variables x, t) if f/ x t 0 for any feasible x and t. 4 AgameG = [K, { }, {s }] is suermodular if for each layer K, the strategy sace is a nonemty and comact subset of R, the ayoff function s is continuous in all layers strategies, and s has increasing differences between layer s strategy and any other layer s strategy. Several imortant roerties of suermodular games are summarized in the following theorem. Theorem 1: In a suermodular game G =[K, { }, {s }], a) There exists a NE. In case of multile NEs, there is a smallest one and largest one comonent-wise). b) If the users best resonses are single-valued functions, and each user starts from the smallest largest) element of the strategy sace and uses MBS udates, then the strategies monotonically converge to the smallest largest) NE. c) If each user starts from any feasible strategy and uses MBS udates, then the strategies will eventually lie in the set bounded comonent-wise by the smallest and largest NE. If there exists a unique NE, then the users strategies globally converge to that NE from any choice of initial strategies. roerties a) follows from Lemma 4..1 and 4.. in [6]; b) follows from Theorem 1 of [10] and c) can be shown by Theorem 8 in [16]. Next we show that by an aroriate strategy sace transformation, certain instances of the F game are equivalent 3 More general definitions related to suermodular games are given in [6]. 4 If we choose x to imize a twice differentiable function f x, t), then the first order condition gives f x, t) / x x=x =0, and the otimal value x increases with t if f/ x t > 0. to suermodular games, and so Theorem 1 alies. We first study a simle two-user networ, then extend the results to a K-user networ. A. Two-user networs Consider a networ with only two users. There are four layers in the corresonding F game, two in FW and two in FC. First, we chec whether the F game is a suermodular game. It is obvious that each user FW has a nonemty and comact strategy set, and so does each user FCif π <. This bounded rice restriction is not satisfied with some utilities such as u γ )=θ γ α /α with α [ 1, 0), since π = θ γ α+1 / h B) is not bounded as 0. However, since no user would transmit with zero ower at the NE otherwise the ayoff is ), we can bound π byalarge number that would never be reached by the AD algorithm. This would not imact any of the users choices, and so not change any of these results. The remaining increasing difference condition for the layers ayoff functions does not hold with the original definition of strategies, π) in the F game. For examle, from 9), / π j = h j < 0 for any j =, e.g. a higher rice leads the other users to decrease their owers. However, if we define π j = π j and let each user j FCchoose a π j from the strategy set [ π j, 0], then s FW / π j = h j > 0, s FW i.e. s FW ) has increasing differences between the strategies,π j or equivalently j, π j )). If we can redefine users strategies so that each layer s ayoff function has increasing differences between his transformed strategy and any other layer s transformed strategy, then the transformed F game will be suermodular. Let γ ={γ ) : j j, j K}and γ = {γ ) : j j, j K} for all K. Also define G γ )= γ u γ )/u γ ). Then, an increasing and strictly concave utility function u γ ) is defined to be Tye I if G γ ) [1, ], γ [ ] γ,γ ; Tye II if G γ ) 0, 1], γ [ ] γ,γ. The term G γ ) is called the coefficient of relative ris aversion in economics [15] and measures the relative concaveness of u γ ). Many common utility functions are either Tye I or Tye II. Examles of Tye I utility functions include u γ )=θ log γ ) and θ γ α /α with α [ 1, 0). Examles of Tye II utility functions include θ log γ ), θ log 1 + γ ), 1 e θ γ with θ <n 0 / h ))and θ γ α /α with α 0, 1]. The logarithmic utility function is both Tye I and II. A Tye I utility function is more concave than a Tye II one. Namely, an increase in one user s transmission ower would induce the other users to increase their owers i.e., s FW / j 0 for j =); a Tye II utility would have the oosite effect i.e., s FW / j 0 for j =). The strategy saces must be redefined in different ways for these two tyes of utility functions to satisfy the requirements of a suermodular game.

5 roosition 1: The F game for a two-user networ is a suermodular game in the transformed strategies 1,, π 1, π ) if both users have Tye I utility functions. roosition : The F game for a two-user networ is a suermodular game in the transformed strategies 1,,π 1, π ) if both users have Tye II utility functions. The roof of these roositions consist of checing the increasing differences conditions for each layer s ayoff function. These results along with Theorem 1 enable us to characterize the convergence of the AD algorithm. For examle, if the two users have Tye II utility functions and π 1, π < ), then F AD is nonemty. In case of multile fixed oints, there exist two extreme ones L, π L) and R, π R), which are the smallest and largest fixed oints in terms of the strategies 1,,π 1, π ). That is, for any fixed oint, π ), 1 [ ] L 1, R 1 and [ ] R, L. Similar relations hold for the rices. If users initialize 0), π 0)) at ) 1,, 0, π or 1,, π 1, 0 ), the ower and rices converge monotonically to L, π L) or R, π R), resectively. If users start from arbitrary initial ower and rices, then the strategies will eventually lie in the sace bounded by L, π L) and R, π R). Similar arguments can be made with Tye I utility functions with the corresonding strategy transformation. Transforg the F game into a suermodular game shows that the AD algorithm is a stable feedbac mechanism in the sense that the changes in owers and rices reinforce each other until a fixed oint is reached. Convergence of the transmit owers under both tyes of utilities is illustrated in Fig.. AD trajectories R Fixed oints Set L Initializations AD trajectories L Fixed oints Set R Initializations Fig.. A two-user networ with Tye I left) or Tye II right) utility functions. AD trajectories illustrate the monotonic convergence of the transmit owers. Convergence of the rices is not shown here.) Here we assume 1 = =0. B. K-user Networs roosition 1 can be easily generalized to a networ with K> users: Corollary 1: For a K-user networ, if all users have Tye I utilities, then G F is suermodular in the transformed strategies, π). In contrast, the transformed strategy used in roosition cannot be alied with K>users while assuring that the increasing differences roerty holds for every air of users. With logarithmic utility functions, it is shown in [5] that roblem 1 is a strictly concave imization roblem over the transformed variables y =log. In this case roblem 1 has a unique otimal solution, which is the only oint satisfying the KKT conditions. Thus, G F has a unique NE corresonding to this otimal solution, and the AD algorithm will converge to this oint from any initial choice of owers and rices. With some or additional conditions, the next roosition states that these roerties generalize to other Tye I utility functions. roosition 3: If for all K, > 0, andg γ ) [a, b] for all γ [γ,γ ], where[a, b] is a strict subset of [1, ], then roblem 1 has a unique otimal solution to which the AD algorithm globally converges. V. CONCLUSIONS We have resented a distributed ower control algorithm for SS wireless ad hoc networs. Each user announces a rice, which is associated with the KKT conditions for the total utility imization roblem. This leads to an asynchronous ower and rice adation algorithm, which requires only local nowledge of channel gains by each user. By interreting the algorithm as myoic best resonse udates of a fictitious game, convergence can be roved with an aroriate strategy transformation using suermodular game theory. For users with logarithmic utilities, the algorithm globally converges to the unique socially otimal solution. REFERENCES [1] F.. Kelly, A. Maulloo, and D. 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