Game Theoretic Approach to Power Control in Cellular CDMA

Transcription

1 Game Theoretic Approach to Power Control in Cellular CDMA Sarma Gunturi Texas Instruments(India) Bangalore , INDIA gssarma@ticom Fernando Paganini Electrical Engineering Department University of California, Los Angeles paganini@eeuclaedu Abstract In this paper, we present the power control problem in CDMA wireless data networks in the analytical setting of non-cooperative game theory User satisfaction is represented as a net utility function, which is the difference of a strictly concave function, based on signal to interference ratio, and a cost term on user s power A detailed analysis for the existence and uniqueness of Nash equilibrium for the above non-cooperative game is presented Next, a decentralized power control algorithm is developed which converges to the Nash Equilibrium, as demonstrated by both analytical and simulation methods The framework is then extended to the multi-cell case, making user utilities depend on base-station assignment as well as powers We propose a generalized algorithm that can handle base station assignment and hand-off, as well as power control, and study by extensive simulations its performance in a dynamic environment I INTRODUCTION Traditionally much of the work on power control in wireless networks was limited to voice traffic [4], [5] Recently, an alternative approach to the power control problem in wireless networks has been proposed in [], [], based on concepts from non-cooperative game theory as each user selfishly tries to maximize its utility function One deficiency of this setup is that utility function is dependent on the system parameters like modulation, coding and packet/frame size The objective of our work and other recent references [3], is to formulate the game in terms of intrinsic properties of the channel ( and power), and thus decouple it from lower layer decisions such as modulation and coding In comparison to [3], our approach is able to handle maximum power constraints on the users, which requires a different proof technique and we also extend the approach based on utility functions to the multi-cell problem of power control/base station assignment In section II, we develop the concept of utility for data users In Section III we first formulate the power control problem as a non-cooperative game and prove the existence and uniqueness of the Nash equilibrium in such a game togther with a synchronous power control algorithm achieving the equilibrium In section IV, we extend the single cell analysis to the multi-cell case section V we present the simulation results for both single cell and multi-cell case Section VI provides the concluding remarks to the paper II UTILITY FUNCTIONS FOR DATA We consider a CDMA cellular system of total bandwidth W Hz and unspread bandwidth B Hz supporting n users The signal-to-interference ratio of the ith user at its intended receiver is given by γ i = W B h i p i j i h jp j + σ () where σ is the AWGN power at the receiver and {h i } is the set of path gains from the mobile to the base station In wireless voice communications, low delay is essential and transmission errors are tolerable upto a point In wireless data communications, signals can accept some delay but are intolerant to transmission errors The latter depends, of course, on the coding and modulation employed; however, rather than include these details in the utility function (as done in [], []), we believe it is more natural to use the channel capacity as figure of merit for the purposes of power control, and leave to lower-levels the decision as to whether this capacity will be fully exploited by the coding/modulation A similar philosophy is used in [8] For simplicity, and in order to have a decentralized implementation, we approximate the capacity by the classical Gaussian channel formula, which corresponds to treating all interference as noise This motivates the first term in the utility function below, u i = B log ( + γ i ) a i p i bits/sec () The second term is included to make the problem well-posed; namely, we include a linear cost on transmit power III POWER CONTROL IN SINGLE CELL AS A GAME A Formulation as Non-cooperative Game Let Γ be a n-person non-cooperative game where user i has the strategy set P i = [, p max ] and a utility function u i (p i, p i ) Let the power vector p = (p, p,, p n ) P denote the outcome of the game where P is the set of all power vectors Formally, the non-cooperative power control game can be written as max u i (p i, p i ), for i =,,, n (3) p i P i

2 As a result of the constraints on the power level of a mobile, the maximum utility of the user is attained at the power level given by projecting p i on the set P i, ie ˆp i = F i ( p i ) (4) where F ( ) is a diagonal mapping from R n to R n defined as F i (x) = p i if x i p i F i (x) = x i if x i p i F i (x) = if x i (5) Furthermore, (4) can be equivalently written as the vector equation ˆp = F ( p) = F (Mp + b) (6) where the matrix M R n n and if at an instant terminal i updates its power then the ith row of the matrix is given by m ii = and m ij = (B/W ) h j h i, for i j (7) and the ith component of the vector b R n is a constant given by b i = B a i ln Bσ (8) W h i B Existence and Uniqueness of the Equilibrium The concept of equilibrium used in game theoretic considerations is the Nash equilibrium and is defined as Definition : A power vector p = (p, p,, p n ) is a Nash equilibrium of the game Γ if for every i N, u i (p i, p i ) u i (p i, p i) for all p i P i Using () and () it is easy to show that u i is concave in p i Further, by our assumption P i is a compact, convex set Therefore, the existence of equilibrium in game Γ follows from the following theorem obtained from [6]-[7] Theorem : A Nash equilibrium exists in a game Γ = [N, P, P,, P n, u, u,, u n ] if for all i N ) P i is a nonempty, convex and compact subset of some Euclidean space R n ) u i (p) is continuous in p and quasi-concave in p i We will need the following results to analyze the uniqueness of the Nash equilibrium We also note that the algorithm is a synchronous algorithm therefore, all the terminals update their powers at the same instant Theorem : The spectral radius of M is less than for where B/W /n (9) Proof: The matrix, M can be written as M = B (I H) () W h ii = and h ij = h j h i () Noting that H is equivalent to H = h h h n h n () the only eigenvalues of the matrix H are, therefore, and n Therefore, from (), the only eigenvalues of M are B/W and (B/W )(n ) which for the suggested choice of B/W have the absolute value less than We also note that the mapping F has the following properties for any x, y R n : ) x y implies F (x) F (y), ) F (x) F (y) F (x y), 3) F (x + y) F (x) + F (y), 4) F (x) + F ( x) x, with equality if p max x k p max, k =,,, n Here x denotes the vector obtained by replacing each component of x by its absolute value Theorem 3: The Nash equilibrium for the non-cooperative game Γ is unique Proof: Suppose there exist two distinct equilibria p and p for the game Γ Therefore, from (6) we have p = F (Mp + b) (3) p = F ( Mp + b ) (4) Subtraction of these two equations yields : p p = F (Mp + b) F (Mp + b) F (M(p p )) (5) Therefore from (5) and Property of mapping F, we have, Similarly, F (p p ) F (M(p p )) (6) F (p p) F (M(p p)) (7) Adding (6) and (7) and using Property 4 results in F (p p )+F (p p) F (M(p p ))+F (M(p p)) Using Property 4 again in (8) M(p p ) (8) p p M(p p ) M p p Once again note that M denotes a matrix obtained by replacing each element of M by its absolute value Let M = G Since G is just M therefore, eigenvalues of G are Therefore, for some k B/W and (B/W )(n ) (9) p p G k p p ()

3 If we choose the value of B/W satisfying (9) and let k in (), we get lim p p = This implies that there is a unique equilibrium C Power Control Algorithm Suppose that terminals update their powers at time instants given by T = {τ, τ, τ 3, } with the update instances sorted in ascending order Algorithm : Consider the non-cooperative game Γ given in (3) Generate the sequence of powers as follows: ) Set the initial power vector p() = p where p is any vector in the strategy space P Set k = ) For all k such that τ k T and for all terminals, given p(τ k ), compute ˆp l (τ k ) = arg max u l (p l, p l (τ k )), where p l P l l =, N Theorem 4: Algorithm converges to the Nash equilibrium of the non-cooperative game Γ Proof: Let p(k) denote the power vector at any time instant τ k The implication of the step of the Algorithm is to calculate ˆp i which is the ith component of the vector ˆp given by (6) Therefore, we can write the following relations p(k + ) = F (Mp(k) + b) () p(k + ) = F (Mp(k + ) + b) () Proceeding in the same manner as in proof of Theorem 4, we get, p(k + ) p(k + ) G p(k + ) p(k) (3) Therefore, from (9) and (3) it follows that Now consider the following lim p(k + ) p(k) = p(k + N) p(k) which implies that, N+k j=k p(j + ) p(j) = ( G N + G N + + I) p(k + ) p(k) (I G) G k p() p() lim p(k + N) p(k) = If we denote the sequence of power vectors generated as {p k } then above equations show that the sequence {p k } has a subsequence {p ki } which converges to some limit, say p Now we show that the sequence {p k } converges to p First we note that sequence {p k } is bounded In analogy with (3) we have p(k + ) p) G p(k) p It follows from this relation that or, in other words, lim p(k) p = lim p(k) = p Now it remains to show that p is also an equilibrium of game Γ For every user i, at all update instances τ j T which gives u i (p i(τ j ), p i (τ j )) u i ( p i (τ j ), p i (τ j ) u i (p i, p i ) = lim j u i(p i(τ j ), p i (τ j )) lim i( p i (τ j ), p i (τ j ) j = u i ( p i, p i ) (4) Hence by definition of Nash equilibrium, p is an equilibrium of game Γ IV EXTENSION TO MULTI-CELL CASE A Formulation as a Game We consider a wireless CDMA data system with M cells and N users Let the signal to interference ratio of the ith user at jth base station be γ ij In this scenario, the base station also becomes a variable of optimization and the power control game, in multi-cell case, can therefore be written as max u ij(p i, p i ) (5) p i,j M Therefore, assuming no base station diversity, we can define a family of utility functions as u ij = B log ( + γ ij ) a ij p i + C ij (6) Here, C ij represents an incentive term to induce the sources to choose a reasonable base-station assignment We found that a reasonable option is to choose a ij h ij and C ij = a ij p max, which induced a localized assignment B Base Station Assignment and Hand-off Algorithm Initially the terminal uses very small powers to probe all the base stations Therefore, using the approximation of small power in (6), we have the utility for terminal i at base station j as B log ( + γ ij ) a ij p i + a ij p max Bγ ij / ln a ij p i + a ij p max (7) The terminal i chooses the base station j which provides the maximum value of the term on the right hand side of (7) After the initial base station assignment a terminal updates its power to maximize (6) Thus we have the following asynchronous algorithm for power control and handoff scheme for the non-cooperative game denoted by Γ m Suppose, that terminal i updates its power at time instants given by the set T i = {t i, t i, } where t ik < t i(k+) and t i = for all i Define T = {τ, τ, } as the set of update instances

4 TABLE I THE LIST OF PARAMETERS FOR A SINGLE-CELL CDMA SYSTEM 4 under various pricing factors n, number of users 9 W, spread spectrum bandwidth 6 Hz B, signal bandwidth 4 Hz standard deviation for shadowing 8 db dimensions of a cell m m σ, AWGN power at the receiver 5 5 Watts p, maximum power constraint Watts path gain (path gain) T T T N sorted in ascending order Similarly, let T hi be the set of instants when terminal i checks for hand-off Define T h = {τ h, τ h, } as the set of hand-off instances T h T h T hn sorted in ascending order Algorithm : Consider a multi-cell non-cooperative game Γ m For all i =,,, N initialize the power levels of all users to small value ) If terminal is assigned to a base station, then go to step Else compute u ij, the utility for terminal i at base station j, for all j Find j = arg max j u ij Assign terminal i to base station j ) For all k such that τ k T and for all terminals l {,,, N} such that τ k T l compute p l (τ k ) = max u lˆl(p l, p l (τ k )), where ˆl is the base station to which the lth terminal is assigned to at that instant 3) For all k such that τ hk T h and for all terminals l such that τ hk T hl, calculate u lj (τ hk ) for all j Let j = arg max j u lj (τ hk ) If j j, handoff terminal l to base station j C Mobility in Multi-cell Case We consider the scenario where the users are moving with constant speed and the path gains are varying The channel gain for every link is characterized by three components: distance loss, slow or shadow fading [9] and fast fading The distance loss model is usual inverse law with propagation exponent to be 4 The users are uniformly distributed in the area The mobility of the user i is modelled with a constant but random speed v i in the range -6 Km/hr Also, the initial direction for the user is picked from a uniform distribution in [ π, π] and later, after every seconds, the direction is selected from a triangular distribution with the mean being the old direction of the user V SIMULATION RESULTS AND DISCUSSION A Single Cell Case The system we examined for the single cell case had the parameters as listed in Table I The terminals are located at distance vector d = [, 3, 45, 55, 66, 74, 8, 94, ] meters from the base station Path gains are obtained using the simple path loss model with the exponent being 4, h i = K /d 4 i where K = 97 is a constant Intuitively, for the same power level p a user closer to base station should be penalized more than a user farther away from the base x Fig s of users under different pricing strategies under various pricing factors (path gain) / constant pricing Fig s of users under different pricing strategies station The simulations corroborated this idea and lead to fairer allocation of network resources among the users We simulated Algorithm for four different pricing strategies in which the pricing is a constant, is proportional to square root of path gains, is proportional to path gains and is proportional to square of path gains As seen from the Fig and Fig, when the pricing is proportional to the path gains the users receive a fairer allocation of s than under any other pricing strategy Fig 3 shows the power levels and s, under the linearly proportional pricing scheme, of the individual users each as a function of the distance B Multi-Cell Case The insights from single cell power allocation game in wireless data networks can be extended to the multi-cell case The system parameters used in the simulations are listed in Table I but for multi-cell case we consider 3 users The user locations are chosen at random with uniform distribution over the entire network area and the base stations are assumed to be located at the center of the cells Fig 4 shows the equilibrium scenario of a particular user under varying number of users in the network The simulations suggest that there exists a unique Nash equilibrium and the equilibrium powers of the users are always the same,

5 Power (Watts) User distribution in dynamic case before handoff BS BS BS 3 BS 4 handoff user y position(meters) x position(meters) Fig 3 Powers and s of users with pricing path gains Fig 5 Distribution of users before handoff User distribution in dynamic case after handoff BS BS BS 3 BS 4 handoff user 7 4 Power (Watts) y position(meters) 8 3 users 3 users 5 users time (sec) x position(meters) Fig 4 Power profile of a typical user under varying number of users in the network Fig 6 Distribution of users after handoff irrespective of the initial power levels of the users, provided the power levels are sufficiently small The scenario of base station and handoff of a user under mobility and dynamic channel conditions is depicted in Fig 5 and Fig 6 In Fig 5 the user marked is moving across from cell 3 to cell Therefore, the user gets hand over from base station 3 to base station and is shown as in Fig 6 VI CONCLUSIONS In this paper we applied game theoretic concepts to the power control problem in wireless networks We proposed power control algorithms for both the single cell and multicell case together with a base station assignment and hand-off scheme for the multi-cell scenario Under static scenario for single cell case, a synchronous power control algorithm has a unique Nash equilibrium and our simulations suggest that similar behavior is also exhibited by the multi-cell case In the single cell case our simulation studies show that there is a unique Nash equilibrium under an asynchronous power control algorithm as well but an analytical result was difficult to obtain An area of future work could be to prove this uniqueness analytically One other possible extension of this work might be to analytically prove the existence of Nash equilibrium in multi-cell case and also analyze the uniqueness of the equilibrium analogous to the single cell case REFERENCES [] DJ Goodman and NB Mandayam, Power control for wireless data, IEEE Personal Communications Magazine, 7(), pages 48 54, April [] CU Saraydar, NB Mandayam and DJ Goodman, Efficient power control via pricing in wireless data networks, IEEE Transactions in Communications, Vol5, No, pages 9 33, February [3] Tansu Alpcan, Tamar Basar, R Srikant and Eitan Altman, CDMA uplink power control as a noncooperative game, In Proceedings of the 4th IEEE Conference on Decision and Control, pages 97, December [4] R Yates, A framework for uplink power control in cellular radio systems, IEEE Journal on Selected Areas in Communications, Vol3, No7 995, pages [5] J Zander, Performance of optimum power control in cellular radio systems, IEEE Transactions in Vehicular Technology, Vol4, No, 99, pages 57 6 [6] G Debreu, A social equilibrium existence theorem, Proceedings of the National Academy of Sciences, 38, pages , 95 [7] IL Glicksberg, A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium points, Proceedings of the American Mathematical Society, 3, pages 7 74, 95 [8] L Xiao, M Johansson, and S Boyd, Simultaneous routing and resource allocation via dual decomposition, submitted to IEEE Transactions on Communications, boyd/srrahtml [9] MGudmundson, Correlation model for shadow fading in mobile radio systems, Electronics Letters,Vol7 No3, pages 45 46,7 Nov99