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IEEE COMMUNICATIONS LETTERS, VOL. 8, NO. 8, AUGUST 2004 Distributed Joint Rate and Power Control Game-Theoretic Algorithms for Wireless Data M. Hayajneh, Student Member, IEEE, and C. T. Abdallah, Senior Member, IEEE Abstract—In this letter, we consider two distributed game the-
the path gain from the th transmitter to the th receiver. This oretic algorithms to jointly solve the problem of optimizing the
gain may represent spreading gain and/or cross correlation be- transmission rates and transmit powers for future wireless data
tween codes in CDMA systems or any gain that captures the communication systems. We then establish the existence, unique-
effect of a fading channel.
ness and Pareto optimality of Nash equilibria of both games.
that optimally allocates the transmission rates Index Terms—Game theory, joint rate and power, Pareto effi-
for all users is given by TRANSMITTERS in multimedia wireless networks may where
is the pricing factor broad- require different quality of services (QoS) in order to es- casted by the base station (BS) to all users, and tablish a communication link with a receiver. Providing flex- sures the willingness of user to pay) is the utility factor of the ible transmission rates for each transmitter/receiver pair and ef- th user locally selected based on the desired transmission rate.
ficient use of the shared radio resources requires joint power and is a constant selected such that rate control optimization algorithms. Earlier work in this arena is the minimum required transmission rate. The first term of used centralized algorithms (c.f. Due to the difficulty of im- is chosen to maximize the transmission rate of user , while the plementing centralized algorithms, and to avoid control signals second term works as a barrier to prevent the th user's rate from that cause delays in the system operation, distributed algorithms , and to fairly allocate the transmission rates.
were proposed. Game theory was shown to be an appropriate is to prevent the greedy use of the available tool for finding power control algorithms in and for rate channel capacity.
flow control algorithms in and In particular, the authors below, allocates the transmit power levels that sup- in proposed a utility based joint power and rate optimiza- port the resulting Nash equilibrium rates tion algorithm, but the resulting Nash equilibrium (NE) point was Pareto inefficient and to guarantee the uniqueness of NE the
rates of all users were forced to be equal. In this letter, we use
game theory framework for finding a pure distributed algorithms
for the joint rate and power control optimization problem. To
solve the problem, we propose two, layered, but noncooperative
priced games as follows: The first game
allocates the optimal transmission rates for all users, then provides the second game with a vector of constants . The second game is the effective interference that user needs to overcome.
uate the optimal transmit power levels that support the resulting are the vectors of transmission rates and Nash equilibrium transmission rates of game transmit powers of all users except for the th user, and signal-to-interference ratio (SIR) defined by II. SYSTEM SETUP AND OUR APPROACH transmitter/receiver pairs (users) in a mobile cel- lular network. The th transmitter,transmits at a power level from its convex strategy space In applications where the spectrum and power are limited re- to the th receiver and sends data at a rate sources, it is recommended to use a spectrally and power effi- . The received power level at the th receiver cient modulation technique such as M-QAM. An empirical link from the th transmitter is given by -QAM of user is given by Manuscript received January 11, 2004. The associate editor coordinating the is the target BER review of this letter and approving it for publication was Prof. S. Pierre. The is a system constant. In this letter we use the work of C. T. Abdallah was supported in part by the National Science Founda- following approximation of (4) at high SIR tion under NSF-ITR 0312611.
The authors are with the Electrical and Computer Engineering Department, University of New Mexico, Albuquerque, NM 87131-0001 USA (e-mail:[email protected]).
is normalized by the channel bandwidth with units, Digital Object Identifier 10.1109/LCOMM.2004.833817 nats/s/Hz. A user can change the transmission rate by adapting 1089-7798/04$20.00 2004 IEEE IEEE COMMUNICATIONS LETTERS, VOL. 8, NO. 8, AUGUST 2004 different modulation formats (e.g., 2-QAM, 4-QAM, …).
B. Non-Cooperative Power Control Game With Pricing Therefore, the transmission rate of each user belongs to a To find the maximizing discrete set, but we assume in this letter that the transmissionrates are continuous for simplicity. In Section III we establishthe existence, uniqueness and optimality of the equilibrium of both games.
and by substituting for the value of , the maximizing transmit III. EXISTENCE OF NASH EQUILIBRIUM power level is thus given by A. Non-Cooperative Rate Control Game With Pricing (NRGP) The optimization problem of the th user defined in game The transmit power level represents the minimal power (i.e., is to find the transmission rate from the strategy space without waste) required to support the optimal transmission rate maximizes the utility function defined in (1). To do so, we set is a strictly concave function, and using the same ar- , there exists a Nash equilibrium point . In what follows we prove The maximizing transmission rate of user , the uniqueness of the Nash equilibrium point of game proposing the best response of user proposition 1.
Proposition 2: For game defined in (2), the best response of user , given the transmit power levels vector of the other is the maximum transmit power level in the th , which means that user's strategy space concave function of is a quasiconcave func- Then, the uniqueness of the Nash equilibrium operating point tion optimized on a convex set , and game theory results can be proved similarly to game since the best response guarantee the existence of a Nash equilibrium point In the vector of users in remainder of this section we prove the uniqueness of this Nash is also a standard function. The following Lemma then guar- equilibrium point. We first need the following result.
antees Pareto optimality (efficiency) of the equilibrium point Proposition 1: For game defined in (1), the best response of both NRGP and NPGP games of user , given the transmission rates vector of the other users Lemma 2: The Nash equilibrium point is the maximum allowed transmission rate in the th is Pareto optimal. Mathemat- user's strategy space ically speaking, for Proof: Define the best response function th user as the best action that user can take to attain the component wise. For maximum pay off given the other users' actions component wise.
where this set contains only one point From (7), Proof: We already know from (6) that unconstrained maximizer of the target function maximizer is unique. Now, assume that is not feasible, that is, , then user will get his/her maximum at the target function is increasing on the set is the best response of user , therefore (10) can be written as The following theorem, proven in guarantees the unique- ness of a Nash equilibrium operating point of game Theorem 1: If a power control algorithm with a standard best response function has a Nash equilibrium point, then this Nashequilibrium point is unique.
Without loss of generality, let See for the definition of a standard function. Theorem 1 . Then we have the following: allows us to state the following lemma, whose proof is omitted.
Lemma 1: In game , the best response vector of all users is a standard vector function. Therefore, by theorem 1, gamehas a unique Nash equilibrium point HAYAJNEH AND ABDALLAH: GAME-THEORETIC ALGORITHMS FOR WIRELESS DATA In order to find out how , we need to find the first-order derivative of . One can check easily for all users, and by this we conclude that is a Pareto optimal NE point of NRGP game Normalized equilibrium rates of the game G1() and the normalized is a Pareto optimal NE point of , it is enough to minimum required rates of the users (+) in the upper graph and the equilibrium is the minimum required transmit power powers of the game G2() in the lower graph versus the user index with pricingfactor  = 10 , utility factors u = 10 , and  = 10 , 8 i 2 N .
. By re-writing (9) as: And from (5), we conclude The additive-white-gaussian noise (AWGN) variance was set It was proven in that both synchronous and asynchronous run for different values of the minimum transmission rates for algorithms with standard best response functions converge to different users. Results show that all users were able to reach the same point. Therefore, we consider asynchronous power reasonable transmission rates with low transmit power levels and rate control algorithms which converge to the unique Nash resulting from game as shown in the lower graph of Fig. 1.
equilibrium point rithm, the users update their transmission rates and powers inthe same manner as in Assume user updates its transmis- sion rate at time instances in the set In this letter two joint game-theoretic distributed rate and power control algorithms for wireless data systems were pro- posed. We presented target functions which are composed of be the transmission rates vector picked randomly from the total the difference between a utility function and a pricing function to set the rules of the games among the users. We established Algorithm 1: Consider the game given in (1) and gen- the existence, uniqueness and Pareto optimality (efficiency) of erate a sequence of transmission rates vectors as follows: (a)
the Nash equilibrium point of both games. All 50 users in the Set the transmission rate vector at time studied example were able to attain transmission rates that are (b) For all
higher than their minimum required transmission rates at very low transmit power levels.
the transmission rate
(c) If
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Algorithm 2: The game as given in (2) generates a [5] F. Kelly, P. Key, and S. Zachary, "Distributed admission control," IEEE sequence of power vectors as follows: (a) Set the power
J. Select. Areas Commun., vol. 18, pp. 2617–2628, Dec. 2000.
. (b) For all
[6] R. D. Yates, "A framework for uplink power control in cellular radio systems," IEEE J. Select. Areas Commun., vol. 13, pp. 1341–1347, Sept.
[7] N. Feng, N. Mandayam, and D. Goodman, "Joint power and rate opti- mization for wireless data services based on utility functions," in Proc. stop and declare the Nash equilibrium power CISS, vol. 1, Mar. 1999, pp. 109–113.
[8] A. B. MacKenzie and S. B. Wicker, "Game theory in communications: and go to (b).
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[9] C. U. Saraydar, N. B. Mandayam, and D. J. Goodman, "Efficient power IV. SIMULATION RESULTS control via pricing in wireless data networks," IEEE Trans. Commun., We consider a wireless data system with vol. 50, pp. 291–303, Feb. 2002.
[10] A. B. MacKenzie and S. B. Wicker, "Game theory and the design of self- ceiver/transmitter pairs. The path gains configuring, adaptive wireless networks," IEEE Commun. Mag., vol. 39, from a uniform distribution on pp. 126–131, Nov. 2001.

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