2010 European Wireless Conference
A Coalitional Game-Inspired Algorithm for Resource Allocation in Orthogonal Frequency Division Multiple Access Farshad Shams
Giacomo Bacci, Marco Luise
Dept. Computer Science and Engineering IMT Institute for Advanced Studies Piazza San Ponziano, 6, 55100, Lucca, Italy Email:
[email protected]
Dipartimento di Ingegneria dell’Informazione University of Pisa Via G. Caruso, 16, 56122 Pisa, Italy Email: {giacomo.bacci, marco.luise}@iet.unipi.it
Abstract—This work investigates the problem of resource allocation (in terms of transmit powers and subchannel assignment) in the uplink channel of an orthogonal frequency division multiple access (OFDMA) network, populated by mobile users with constraints in terms of target transmit data rates. The optimization problem is tackled with the analytical tools of coalitional game theory, and a simple and practical algorithm based on Markov modeling is introduced. The proposed algorithm allows the mobile devices to fulfill their data rate demands with minimum utilization of the network resources. Simulation results are provided to validate the theoretical analysis for practical OFDMA network parameters.
I. I NTRODUCTION Orthogonal frequency-division multiple access (OFDMA) represents a broadband digital modulation and access scheme frequently used in modern wireless communication standards, such as 4th generation cellular systems and the IEEE 802.16e/m WiMAX standard. OFDMA allows multiple mobiles to simultaneously access the wireless channel through different sets of orthogonal subcarriers, so as to combat the effects of multipath propagation and to deliver broadband signals over highly frequency-selective channels. Due to the concurrent access to the channel by different users in the network, resource allocation for OFDMA is a challenging problem in the field of wireless communications, especially in the uplink scenario. In this work we present an iterative technique based on coalitional game theory [1], [2] for efficient resource allocation in terms of transmit power and bandwidth assignment for OFDMA systems. The analytical tools that we use to tackle this optimization problem are based on coalitional game theory. Although coalitional game theory play a substantial role in economics and political science, its applications to wireless communications are far less popular. When a common resource, such as the wireless medium, must be shared between several agents, such as the mobile devices, coalitional game solutions like bargaining games and auctions prove to be useful [3]. For instance, the Nash bargaining solution is one of the best schemes for fair multiuser resource allocation in wireless networks [4], and autonomous auctions and negotiations among software agents
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can provide smart distributed intrusion detection techniques for virus and intruder detection [5]. In the last few years coalitional games have been considered to address resource allocation issues in OFDMA networks. Han et al. in [6] introduce a Nash bargaining solution for fair resource allocation in the uplink scenario in OFDMA to maximize the overall system rate under a power and rate constraint. Zhang and Zhang in [7] propose a Nash bargaining solution to achieve an optimal power allocation in a downlink OFDMA scheme to guarantee the quality-of-service (QoS) demands of the users. Both [6] and [7] show a good tradeoff between overall system rate and fairness. The game-theoretic framework proposed in [6], [7] outperforms other approaches, such as that presented in [8], that aims at maximizing the overall system rate based on a min-max criterion. A common problem of the algorithms available in the literature (e.g., [6]– [9]), is their high complexity. In fact, although the proposed solutions achieve good performance, they cannot satisfy the requirements of modern cellular network, with thousands of subcarriers and hundreds of users. The main novelty of this paper is twofold: i) we allow every subcarrier to be assigned to more than one user; and ii) we design a low complexity algorithm that allows practical solutions to be achieved in a few steps using typical network parameters. It is worth noting that we do not aim at maximizing the channel capacity of mobile devices under a power constraint. Channel capacity is in fact increased when every subcarrier is assigned to the user with the best path gain, and the power is distributed according to the waterfilling (WF) criterion. With WF, only users with the best channel gains receive an acceptable channel capacity, while users with bad channel conditions achieve very low data rates. In this work, we aim at fulfilling each user’s QoS requirement in terms of target transmit rates exactly. The remainder of the paper is structured as follows. Sect. II introduces the basics of coalitional game theory. We formulate the resource allocation game in the uplink OFDMA scenario in Sect. III, and we introduce an algorithm based on Markov process and coalitional game theory in Sect. IV. Sect. V shows
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the experiment results, and some conclusions are drawn in Sect. VI. Notation: For the reader’s convenience, Sect. VII reports the list of symbols used throughout the paper.
Definition 3: A core apportionment x ∈ RM is a payoff distribution with the following property: ⎫ ⎧ ⎬ ⎨ xm = max ν (S) and xm ≥ ν (S) ∀S ⊂ M x: ⎭ ⎩ ψ∈Ψ m∈M
S∈ψ
m∈S
II. C OALITIONAL GAME THEORY
A coalitional game is a synergy among individual agents to achieve a goal with notable payoffs [1], [2]. It is denoted as G = (M, ν), where M denotes the set of players and ν the coalition function. We also denote with xm the payoff of player m in M, m = 1, 2, . . . , M = |M|. If S ⊆ M is a coalition (subset) of M formed in G, then its members get an overall payoff ν (S), with ν (S) = 0 when S = ∅. In a cooperative game with transferable utility (TU), the payoff of a coalition can be expressed by a real value. A relevant issue in coalitional games is how the players make mutual binding agreements to form the coalition that provides them with the highest payoff. When the players are better off when staying together, they tend to form the grand coalition [2]. The grand coalition is formed only if the game is superadditive: Definition 1: A TU game G is superadditive if ν (S ∪ T ) ≥ ν (S) + ν (T )
∀S, T ⊂ M s.t. S ∩ T = ∅ (1)
An important issue in a coalitional TU game is how to distribute the payoff of the grand coalition among agents. The fundamental solution is the core solution. The core of a coalitional game is the set of all payoff vectors (i.e., all those vectors whose entries add up to a same amount equal to the utility of the grand coalition) such that the sum of all payoffs of the players in any existing coalition S is no smaller than the utility of the coalition: Definition 2: Let M be the set of M players of the superadditive TU-game G, and let ν be the payoff of the game. The core of G is the set xm = ν (M) and xm ≥ ν (S) ∀S ⊂ M x: m∈M
m∈S
(2) where x ∈ RM is the payoff distribution across players, and xm ∈ x if and only if no coalition can improve upon xm . For a non-superadditive coalitional game, the network formation process does not lead the players to form a grand coalition. In this case, Definition 2 does not apply. Let us redefine the core set in general (not necessarily superadditive) coalitional formation TU-game. Let ψ = [S1 , S2 , . . . , Sm ] denote a partition of the set M wherein Si ∩ Sj = ∅ for m i = j, i=1 Si = M and Si = ∅ for i = 1, . . . , m, and let Ψ denote the set of all possible partitions m ψ. Let us also define F = [S1 , S2 , . . . , Sm ], such that i=1 Si = M and Si = ∅ for i = 1, . . . , m, as a family of coalitions.
(3)
Note that, if G is superadditive, maxψ∈Ψ S∈ψ ν (S) = ν (M). The core allocation set can be found through linear programming and can also be an empty set. We can study the non-emptiness of the core without explicitly solving the core equation, using the following lemma: Lemma 1 ([1]): A necessary and sufficient condition for the core of a TU game to be non-empty is the TU game to be balanced. Definition 4: A superadditive TU game G for a family F of coalitions is balanced if, for any S ∈ F, the inequality μS · ν (S) ≤ ν (M) (4) S∈F
holds, where μS is a collection of numbers in [0, 1] (balanced weights) such that μS · 1S = 1M (5) S∈F M
with 1S ∈ R elements are
denoting the characteristic vector whose 1, (1S )i = 0,
i∈S otherwise
(6)
Definition 5: A non-superadditive TU game G for a family F of coalitions is balanced if, for every balanced collection of weights μS , and for any S ∈ F, μS · ν (S) ≤ max ν (S) (7) S∈F
ψ∈Ψ
S∈ψ
III. P ROBLEM FORMULATION Consider the uplink channel of an OFDMA system wherein K mobile terminals located in the same cell transmit over a subset of subcarriers subject to a maximum transmit power constraint on each subcarrier and a target rate to be achieved. Let K = [1, . . . , K] be the set of terminals. The base station of the cell must allocate a subset of subcarriers to each terminal among all the available subcarriers in the set N = [1, . . . , N ] in order to guarantee the target rate requirement to each terminal. Let R = [R1 , . . . , RK ] be the set of target data rates required by the mobile terminals. We suppose fulfilling such constraints simultaneously to be feasible. The transmission takes place over a total bandwidth B, and the carrier spacing of every subcarrier is Δf = B/N . In terms of coalitional game theory, each subchannel between one mobile and one subcarrier is identified as a player in the game. Since ODFMA can assign the same subcarrier to more than one terminal, we consider M = K · N players in this game, with the set of players denoted by M = K × N ,
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n∈N
where Ckn is the Shannon capacity achieved by user k on the subcarrier n: Ckn = Δf · log2 (1 + γkn )
= Δf · log2
1+ j=k
2
|Hkn | pkn 2
2 |Hjn | pjn + σw
(9)
which depends on the received signal-to-interference-plusnoise ratio (SINR) γkn at the base station. The SINR γkn is a function both of the strategy (i.e., the transmit power) chosen by player (k, n) (i.e., the nth subcarrier to be assigned to the kth terminal), of the transmit power of other terminals on the same subcarrier, and of the power of the additive white 2 . The received power depends Gaussian noise (AWGN) σw on the channel response Hkn between the kth terminal and the base station over the nth subchannel. Note also that in an OFDMA system there is no interference between adjacent subcarriers. Hence, the Shannon capacity considers only intrasubcarrier noise, that occurs when some subcarriers are shared by more than one mobile. Each player (k, n) in the coalition Sk causes interference only to the other players (j, n), with j = k (i.e., to its virtual copies). Since increasing Ck increases the interference on the other coalitions (i.e., other mobile terminals), whereas decreasing Ck may not fulfill the QoS requirement in terms of target rate, the best transmit power and bandwidth allocation is the one that ensures exactly Ck = Rk . In view of this goal, we can define a utility function for the kth coalition Sk as ν (Sk ) =
1 − α · u (1 − Ck /Rk ) |Ck /Rk − 1|
(10)
where u (·) is the step function, with u (y) = 1 if y ≥ 0 and u (y) = 0 otherwise. If Ck = Rk , Sk earns the highest possible payoff ν (Sk ) = +∞. If Ck > Rk , Sk gets a positive payoff, whereas it obtains a negative payoff if Ck < Rk . The factor α is a positive constant that ensures ν (Sk ) to be a negative payoff when Ck < Rk . This term serves as a penalty
payoff function ν (Sk )
corresponding to K virtual copies of each of the N total subcarriers. The strategy of each player (i.e., the kth virtual copy of the nth subcarrier) is represented by the amount of transmit power pkn ∈ [0, pkn ] assigned to such subcarrier. The case pkn = 0 implies that the kth copy of the nth subcarrier is assigned no power. In other words, the kth mobile does not transmit over the nth subcarrier. The coalition game forms K coalitions ψ = [S1 , . . . , SK ], to be assigned to the K terminals. Each coalition Sk , k ∈ K, contains the N players (k, n), n ∈ N = [1, . . . , N ]. The system under investigation aims at fulfilling the QoS requirement of every mobile terminal k in terms of target data rate Rk . For the sake of analytical convenience, we estimate the achieved data rate as the Shannon capacity [10] of terminal k. As a consequence, when terminal k is assigned the coalition Sk , it achieves a rate Ck = Ckn (8)
0
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Utility as a function of the Shannon capacity.
to encourage the players belonging to Sk to increase Ck . The shape of ν (Sk ) as a function of Ck is shown in Fig. 1. In the game G with utility function (10), each player (k, n) tries to find its best power amount pkn to increase its coalition payoff ν (Sk ). To this aim, pkn can also be 0. In this case, the kth terminal does not transmit over the nth subcarrier. To provide further insight into the problem, in the remainder of the section we investigate some properties of the proposed game G. As a first step, we note that the players in G do not tend to form the grand coalition. This is because every player (k, n) does not leave its coalition Sk . The members of every coalition are then fixed and do not change during the game. This means that creating the grand coalition is impossible. This appears to be inappropriate to the notion of a coalitional game. As suggested by the title of the paper, we resort anyway to the theoretical framework of coalitional games to derive a simple and stable solution to our resource allocation problem. In the proposed game structure, the most important parameter is the gain of each coalition, whereas the outcome of each player does not matter at all. Therefore, the proposed game is a TU game. Theorem 1: The core of the game G = (M = K × N , ν) with utility function (10) is not empty. Proof: The number of coalitions and the number of players in each coalition are both fixed. Since each player belongs just to one coalition, the unique balanced collection μS = 1 ∀S ∈ ψ. To conclude of weights (μS )S∈ψ is the proof, ν (S) ≤ max we must verify that ψ∈Ψ S∈ψ S∈ψ ν (S). Since the target rates of all terminals are assumed to be feasible, then every coalition can expect Ck to approach Rk . Therefore, every coalition is allowed to earn the highest possible payoff. IV. T HE BEST- RESPONSE ALGORITHM When analyzing a coalition formation (non-superadditive) game, an important question is to be answered: How to form the coalitions to minimize the cost and to maximize the payoff? To address this question, we propose a best-response iterative
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and dynamic algorithm [11], [12] based on Markov modeling of the coalitional TU game. At each (discrete) time step of the algorithm, the autonomous players simultaneously adjust their transmit powers based on a model to increase the payoff of their own coalitions. Although this leads to interference when copies of the same subcarriers simultaneously change their powers, we show that this dynamic myopic procedure guarantees the maximum payoff to each coalition. The process starts up at time step t = 0 with an arbitrary assignment of the transmit powers pt=0 kn to all K · N players in the game, grouped in K coalitions with player (k, n) belonging to coalition Sk for n = 1, . . . , N . At time step t, our finite Markov chain is in the state ω t = (ψ t , ν t ), where ψ t is the set t t ], and ν t = [ν (S1t ) , . . . , ν (SK )] ∈ RK contains [S1t , . . . , SK t the payoffs of the coalitions in ψ . The strategy of each player (k, n) is to find the best power amount ptkn to increase the payoff ν (Skt ) of its own coalition Sk . In the practice, player (k, n) decides whether to change its power allocation, making its coalition better off, or to keep transmitting at the same power level (e.g., when its coalition’s payoff is infinite). The following snippet pseudocode shows how each player (k, n) takes its decision during time step t. t if ν (Skt ) = +∞, then pt+1 kn = pkn , exit; else if ν (Skt ) ≤ 0, then p˜kn = ptkn , p˜max kn = pkn ; t else p˜kn = 0, p˜max kn = pkn ; repeat pˆkn = p˜kn ; compute ν(S˜k ); pkn ; p˜kn = p˜kn + Δ˜ pkn > p˜max until (ν(S˜k ) > ν (Skt )) or (˜ kn ) if (ν(S˜k ) > ν (Skt )), then pt+1 ˆkn ; kn = p t else pt+1 kn = pkn ;
In this algorithm, ν(S˜k ) is the “trial” value of the current payoff of the coalition: it is computed with pjn = ptjn for all n ∈ N and for any j = k, and pkn = p˜kn . For every step of the update process, the power step Δ˜ pkn is the realization of a random variable uniformly distributed between 0 and Δpkn , with Δpkn pkn . If ν (Skt ) ≤ 0, then Ck < Rk , and the best strategy for player (k, n) is to increase its current transmit power so as to increase its coalition’s payoff. As a result of the random power stepping, the tentative power is a random number in the interval [ptkn , pkn ]. Player (k, n) accepts this value if and only if the coalition payoff ν (Skt ) increases, otherwise it ends up transmitting at its previous value. If 0 < ν (Skt ) < ∞, player (k, n)’s best strategy is on the contrary to decrease ptkn , and thus the tentative (random) transmit power belongs to the interval [0, ptkn ]. As a matter of fact, the convergence speed of the algorithms depends not only on the parameters of the network, but also on the choice of the maximum update step Δpkn . As already stated, two copies (k, n) and (j, n) of the same subcarrier n may happen to adjust their transmit powers in a conflicting (and thus incompatible) way. For instance, if
both raise their transmit powers, then they interfere with each other, thus affecting the payoff of both coalitions Sk and Sj . We assume that each player follows the decision rules listed in the pseudocode above. As is, this provides a high probability of conflicting decisions. To reduce the occurrence of this event, we modify our algorithm by allowing each player not to update its transmit power at every step of the game with a probability λ ∈ [0, 1]. At each time step t, every player t uniformly distributed in (k, n) selects a random number ξkn t [0, 1]. If ξkn > λ, then the player applies the algorithm and t+1 t (possibly) update pt+1 kn , otherwise pkn = pkn (i.e., during time step t, it skips the update process). If λ is close to 1, then the probability of conflicting decisions tends to 0, but high values of λ imply higher convergence time for the algorithm, since the probability of updates is low. As better detailed in Sect. V, the optimal value of λ must be selected as a suited trade-off. Note that the value of λ is common knowledge among the players at every step of the algorithm. In the remainder, we show that our proposed algorithm reaches a stable state, which corresponds to the core apportionment of the game. We model the evolution of the algorithm as the output of a finite-state Markov chain with state space Ω = {ω = (ψ, ν )|ψ ∈ Ψ, ν ∈ RK }. At each time step t, ψ t belongs to the subset of all coalitions with exactly N members. As time step increases, the states of the Markov process tend towards a stable coalition structure state, where no player has any incentive to change its power. In other words, all coalitions get their maximum payoffs. Our algorithm guarantees that, when t → ∞, this Markov chain tends towards a singleton steady state with probability 1. Definition 6 ([13]): A set Φ ⊂ Ω is an ergodic set if, for / Φ, the probability of reaching the state any ω ∈ Φ and ω ∈ ω starting from ω is zero. Once the Markov chain falls into a state belonging to an ergodic set, it never leaves that set, and it wavers between the states in that ergodic set from then on. The probability of reaching every state in the ergodic set is positive. Lemma 2 ([13]): In any finite Markov chain, no matter which state the process starts from, the probability of ending up into an ergodic set tends to 1 as time tends to infinity. Definition 7 ([13]): Singleton ergodic sets are called absorbing states. Lemma 3: The state ω = (ψ, ν ) is an absorbing state of the best-response process if and only if ν (Sk ) = +∞
∀Sk ∈ ψ
(11)
Proof: This condition ensures that no player has incentive to change its power amount. If this condition is met, then no coalition can get a higher payoff by deviating from state ω = (ψ, ν ). Since all the target rates are feasible, this condition is also necessary. Theorem 2: The best-response process has at least one absorbing state. Proof: Since the best-response algorithm is a Markov process, Lemma 2 ensures that the best-response process reaches an ergodic set Φ. To conclude the proof, it is enough
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target rate Rk [kb/s]
achieved rate Ckt [kb/s]
to show that Φ is singleton. Suppose that the number of states in the ergodic set is |Φ| > 1. Then all players revise their strategies without conflicting decisions with a non-null probability. As a consequence, the Markov process moves to a new state, in which all coalitions’ payoff are higher than those achieved in the previous state. This means that the probability of going back to the previous state is null, which contradicts the notion of an ergodic set. Note that Theorem 2 does not ensure the uniqueness of the one ergodic set in the best-response process. There may exist some different combinations of the power allocation for the players to reach to a steady state. It means that the game shows multiple equilibria. The major finding of Theorem 2 is that, according to the way the players adjust their strategies, the best-response process leads to one of the steady states, in which no player has any incentive to revise its power allocation. Theorem 3: The set of payoffs associated to an absorbing state of the best-response process coincides with the set of core allocation: i. if ω = (ψ, ν ) is an absorbing state, then ν is a core allocation. ii. if ν is a core allocation, then all ω = (ψ, ν ) are absorbing states. Proof: Part i) Suppose ω = (ψ, ν ) is an absorbing state but ν is not a core allocation. In this case, there exist some coalitions that can obtain a higher payoff. This is contradictory, since the game reaches an absorbing state when every coalition gets the maximum payoff. Part ii) If ν is a core allocation, then no coalition can earn by letting its member change their powers. This implies that the state will not move to a new state, and thus the current state is absorbing. Coalitional games aim at identifying the best coalitions of the agents and a fair distribution of the payoff among the agents. Interestingly, in this game the absorbing state coincides with one of the Nash equilibria [1] of the game. Suppose there are K = 2 mobiles connected to a base station with N = 1 subcarrier only. In this case, the M = K · N = 2 copies of the subcarrier, each constituting a coalition, are engaged in a 2 × 2 game. Every player has two strategies: either pk = 0 or pk = pk . It is straightforward to verify that, in this game, a mixed (vs. pure) Nash equilibrium exists which satisfies the stability of the static game. With due attention to the notation, we can extend this result to a general case. Theorem 4: The set of absorbing states in the best-response process and the set of Nash equilibria of the static game are equivalent in the long run. Proof: Let us consider the coalitions in the best-response process as players in a static game. Lemma 2 ensures that this process reaches an ergodic set in the long run. According to Theorem 2, this set is singleton, and thus its member is an absorbing state. Hence, no coalition (i.e., no player in the static game) has any incentive to revise its strategy. In static games, this is the definition of a Nash equilibrium. We can now conclude that the absorbing state is an extension of the Nash equilibrium, since the coalitions bind agree-
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ments with each other as economic agents and earn a vector value rather than a real number. Once the coalitions reach the absorbing state, their payoff is the highest possible (+∞), and no coalition is willing to revise its current strategy. In general, as follows from Theorem 4, the Nash equilibrium of the game is Pareto-optimal (efficient), since no other strategy can achieve a payoff greater than +∞. V. N UMERICAL RESULTS In this section we evaluate the performance of the bestresponse algorithm presented in Sect. IV. We consider some cases with different numbers of mobile terminals and subcarriers, showing that our suggested scheme reaches a steady state after a few steps only. To increase the convergence speed of the algorithm, we introduce a tolerance parameter ε in our utility function, in the sense that, if |Ck /Rk − 1| < ε is verified, then the payoff is +∞. We can possibly set an asymmetric range [−ε1 , ε2 ] such that −ε1 ≤ (Ck /Rk − 1) ≤ ε2 , so as to favor solutions with Ck > Rk . We consider the following parameters for our simulations: the maximum power of each terminal k on each subcarrier n is pkn = p = 10 nW; the power of the ambient AWGN 2 noise on each subcarrier is σw = 50 pW, and the constant number in (10) is α = 5000. The path coefficients Hkn , corresponding to the frequency response of the multipath wireless channel at the carrier frequency f = nΔf , are modeled using the ITU modified vehicular-B channel model with 24 paths suggested by the IEEE 802.16m standard [14]. The target rates are assigned randomly to each terminal using a uniform distribution in the range [100, 250] kb/s. Based on a numerical optimization, the parameter λ that serves to reduce the probability of conflicting decisions among members of different coalitions for different values of mobile terminals, subcarriers, and signal bandwidth, is λ = 0.87. The initial power allocation is pkn = 0 ∀(k, n) ∈ M. This experimentally provides the minimal power consumption at the steady state, and in most cases the minimum number of steps of the algorithm.
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Fig. 3. Average transmit power per terminal at the equilibrium as a function of the number of terminals.
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Fig. 4. Average number of steps of the algorithm as a function of the number of terminals. 1000
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Fig. 2 reports the behavior of the achieved rate Ck as a function of the time step t in a network with K = 10 terminals, N = 512 subcarriers, and a bandwidth B = 15 MHz. Other parameters are the tolerance ε1 = ε2 = 0.01 and the power update step Δpkn = pkn /5000 = 2 pW. Numerical results show the convergence of Ck to the target rate Rk after 31 steps of the best-response algorithm. The following simulations report the average values for 500 random realizations of a network with power update step Δpkn = pkn /10000 = 1 pW. Solid lines represent the case ε1 = ε2 = 0.02, whereas dashed lines depict the case ε1 = 0.02 and ε2 = 0.07. Square markers correspond to B = 20 MHz, N = 1024, and circles report the case B = 15 MHz, N = 512. Fig. 3 shows the average transmit power per terminal at the equilibrium n pkn as a function of the number of terminals in the network K. It is interesting to note that the average total power increases with K, since a larger K yields a larger amount of interference. The target rates are then met at the expense of a significant increase in the transmit powers. Note that using different power update steps yields a maximum difference in the order of 2% in terms of the average total power expenditure. Fig. 4 reports the average number of steps required for the best-response algorithm to converge in the same scenarios described above. Note that even in the case K = 100, the algorithm converges after a small number of steps. Fig. 5 represents the average number of subcarriers employed by each terminal in the network. Interestingly, the number of used subcarriers shows a linear relationship with K. This implies that, for larger K, every subcarrier is shared by a number of terminals. By using the results of Figs. 3 and 5, we can also derive the average amount of transmit power per subcarrier at the equilibrium, experiencing again an increase as K increases. This can be justified by using a higher amount of network resources, which calls for a larger power expenditure. Finally, it is interesting to investigate the impact of the
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Fig. 5. Average number of used subcarriers per terminal as a function of the number of terminals.
power update on the convergence speed of the algorithm assuming ε1 = ε2 = 0.02 (Fig. 6). Three different cases are considered: Δpkn = pkn /5000 = 2 pW (solid gray lines); Δpkn = pkn /10000 = 1 pW (solid black lines); and Δpkn = pkn /20000 = 0.5 pW (dashed black lines). As can be seen, the number of steps is not necessarily an increasing function of the number of mobile terminals, as occurs for Δpkn = 2 pW, since the power updates are the results of random realizations. An analogous investigation is provided in Fig. 7 for the average number of used subcarriers per terminal. VI. C ONCLUSION In this paper we introduced a coalitional game-theoretic framework to implement the function of resource allocation in the uplink of an infrastructure OFDMA wireless network. To solve this problem, we formulated a coalitional game in which the players are virtual copies of each OFDMA subcarrier, to be assigned to each mobile terminal in the
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VII. L IST OF SYMBOLS
F G Hkn k K K m M M
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Fig. 7. Average number of used subcarriers per terminal as a function of the number of terminals for different values of the power update step.
network. Each coalition is made up of the subcarriers to be assigned to each terminal. We have also derived a myopic and iterative procedure in which each player (i.e., each copy of the subcarrier) individually tries to increase its coalition’s payoff (i.e., each terminal’s payoff) in each time step. The payoff function is measured in terms of the Shannon capacity achieved on each subcarrier. To include the QoS requirements of the users in the network, each mobile terminal is allowed to set its own target rate. The proposed algorithm can be analyzed as a Markov process converging to an absorbing state with unit probability in the long run. We have also shown that the absorbing state coincides with both the core apportionment and the Nash equilibrium. Although this game (Markov chain) admits multiple absorbing states, we showed that the players’ autonomous decisions select only one of them. Experimental results show that our scheme achieves the target rates with a low-complexity procedure. Very quick convergence to a steady state is also guaranteed even using typical parameters for practical networks.
1S B Ck Ckn
20
characteristic vector of coalition S OFDM signal bandwidth Shannon capacity achieved by terminal k Shannon capacity achieved by terminal k on the carrier n family of coalitions coalitional game channel response of the channel between terminal k and the base station over carrier n generic index for a terminal number of terminals set of terminals generic index for a player number of players set of players
244
n N N pkn ptkn p˜kn pˆkn pkn p˜max kn R Rk S St t T xm x α γkn Δf Δ˜ pkn Δpkn λ ε μS ν ν νt t ξkn
generic index for a subcarrier number of carriers set of carriers transmit power of terminal k over carrier n transmit power of terminal k over carrier n at time step t tentative transmit power of terminal k over carrier n previous tentative transmit power of terminal k over carrier n maximum transmit power of terminal k over carrier n maximum tentative transmit power of terminal k over carrier n set of target data rates target data rate of terminal k coalition (subset) of players coalition at time step t generic time step generic coalition of players payoff of player m payoff distribution across players generic positive constant received signal-to-interference-plus-noise ratio of terminal k over carrier n carrier spacing power step to update the tentative transmit power of terminal k over carrier n maximum power step to update the tentative transmit power of terminal k over carrier n probability of transmit power update tolerance parameter balanced weight of coalition S coalition utility function set of coalition utilities set of coalition utilities at time step t uniformly-distributed random variable
2 σw Φ ψ ψt Ψ ω ωt Ω
AWGN power ergodic set set of disjoint coalitions set of disjoint coalitions at time step t set of all possible ψ state of the Markov chain state of the Markov chain at time step t state space of the Markov chain
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