Joint utility-based uplink power and rate allocation in wireless networks: A non-cooperative game theoretic framework

Physical Communication 9 (2013) 299–307

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Joint utility-based uplink power and rate allocation in wireless networks: A non-cooperative game theoretic framework Eirini Eleni Tsiropoulou, Panagiotis Vamvakas, Symeon Papavassiliou ∗ School of Electrical and Computer Engineering, National Technical University of Athens (NTUA), 9 Iroon Polytechniou str. Zografou 15773, Athens, Greece

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Article history: Received 2 January 2012 Received in revised form 23 April 2012 Accepted 29 April 2012 Available online 7 May 2012 Keywords: Wireless networks Joint power and rate allocation Utility function Nash equilibrium Convergence

abstract In this paper a novel utility-based game theoretic framework is proposed to address the problem of joint transmission power and rate allocation in the uplink of a cellular wireless network. Initially, each user is associated with a generic utility function, capable of properly expressing and representing mobile user’s degree of satisfaction, in relation to the allocated system’s resources for heterogeneous services with various transmission rates. Then, a Joint Utility-based uplink Power and Rate Allocation (JUPRA) game is formulated, where each user aims selfishly at maximizing his utility-based performance under the imposed physical limitations, and its unique Nash equilibrium is determined with respect to both variables, i.e. uplink transmission power and rate. The JUPRA game’s convergence to its unique Nash equilibrium is proven and a distributed, iterative and low complexity algorithm for computing JUPRA game’s equilibrium is introduced. The performance of the proposed approach is evaluated in detail and its superiority compared to various state of the art approaches is illustrated, while the contribution of each component of the proposed framework in its performance is quantified and analyzed. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Wireless communication networks are unprecedented in their impact on the world community, industry, and individuals. Along with the number of users that daily utilize wireless networks, the consumed resources as well as the Quality of Service (QoS) performance expectations of the requested services and applications increases dramatically as well. Thus considerable research efforts have been devoted to the resource allocation problem in wireless cellular networks aiming at maximum efficiency. Among the key elements that need to be considered and controlled in such environments are users’ transmission power and rate. In the literature, several approaches have been proposed that treat the power control problem separately, in order to determine the minimum feasible

∗

Corresponding author. E-mail addresses: [email protected] (E.E. Tsiropoulou), [email protected] (P. Vamvakas), [email protected] (S. Papavassiliou). 1874-4907/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.phycom.2012.04.006

required power level (e.g. [1,2]), while satisfying users’ QoS requirements. Among others, game theoretic approaches have emerged as promising alternatives in formulating the distributed transmission power control problem via the adoption of proper users’ utility functions. Nevertheless, the independent allocation of users’ transmission power does not consider the scarcity of network’s bandwidth, as well as the necessity of next generation wireless networks to support multimedia services with various transmission rates and QoS requirements. Our main objective in this paper is to provide a robust and unified methodology and framework that considers the joint treatment of transmission power and rate allocation via a game theoretic approach, by confronting the resulting optimization problem as a two-variable problem, where its energy-efficient stable outcome, i.e. Nash equilibrium, is determined simultaneously with respect to both transmission power and rate. Initially, Section 1.1 provides a comprehensive description of related state of the art work which allows us to better motivate our approach and position our work within the literature, while the key el-

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ements and contributions of our proposed approach are highlighted in Section 1.2. 1.1. Related work and motivation Towards addressing this problem several works have attempted to consider both transmission power and rate allocation. The basic approaches that have been proposed in the recent literature are mainly the following two: 1. The joint rate and power control is approached from a game theoretic perspective, and modeled as two distinct games, i.e. an uplink transmission rate allocation and an uplink transmission power allocation problem. All users determine first their uplink transmission rate and then given their uplink transmission rate, they apply power control to allocate their uplink transmission powers. 2. The joint rate and power control problem is amended in a single-variable problem of the ratio of uplink transmission rate to the uplink transmission power. Specifically following the first approach in [3] the authors model and solve separately and in sequential manner the problem of transmission power and rate control. The first game allocates the optimal transmission rates for all users, which in turn provides the second game with a vector of constants c to be used to evaluate the optimal transmission power levels that support the resulting Nash equilibrium transmission rates of the first game. However users’ power and rate allocation at two different steps introduces high processing delay resulting in slow convergence and waste of transmission power at the corresponding signal transferring process. This burden has been eliminated in [4], where the authors examine specifically only cases of the values of users’ transmission power and rate, and they conclude to a stable point where users closer to the base station benefit with respect to power and rate. In [5,6], authors have chosen to represent users’ satisfaction via a specific utility function, i.e. the ratio of the reliably transmitted bits to the base station divided by their corresponding transmission power. Due to the nature of the selected utility function, the separate and sequential solution of power and rate control problems is feasible, and the solution of the joint allocation problem consists of the two individual solutions where the output of the uplink transmission rate allocation problem serves as input to the uplink transmission power allocation problem. In brief, the main drawback of the first approach is that the optimization problem is solved asynchronously and separately considering the two system’s resources, thus the combined outcome of the two distinct optimization problems is less efficient than jointly solving the problem. More specifically the inefficiency of the solutions achieved by this approach is mainly due to the following reasons: (a) the combined outcome depends on the start point of the uplink transmission power, (b) the users do not update their uplink transmission rate and power in the same step, thus inducing additive delay to the convergence of the joint problem and (c) there is no guarantee that the obtained stable point is as efficient as the one achieved if solving the actual joint uplink transmission rate and power

control problem where the two resources are updated simultaneously in the same step. Representative of the second approach is the work introduced in [7] where the same specific form of utility function as described above is adopted (i.e. the ratio of the reliably transmitted bits to the base station divided by their corresponding transmission power), which facilitates the authors to change the two-variable optimization problem to a single-variable problem, via substituting the ratio of user’s transmission rate to the corresponding transmission power with a new variable, and solve the corresponding single-variable optimization problem. It is noted however that the application of this approach in realistic cases is limited and can be applied only in specific studies where simplified forms of utility functions are assumed (that is where the ratio of uplink transmission rate to power appears), and as a result its use strongly depends on the formulation of the problem. In this approach the single variable problem is solved with respect to the substituted ratio and in order to determine users’ optimal pair of uplink transmission rate and power the maximum value of the one resource is assumed and the other one is determined so as the ratio is equal to the optimal determined one. Even though the users update their uplink transmission rate and power in the same step, it still remains the drawback of the inferiority of the obtained solution when compared to the corresponding one of the actual joint two-variable optimization problem as shown in this paper. Moreover, additional efforts are reported in the literature to maximize the system throughput via either distributed resource allocation approaches such as the one in [8] where a game theoretic distributed resource allocation approach is proposed by constructing two interrelated games (a power-control game at the user-level and a throughput game at the system level), or heuristic ones as the approach in [9] where a simple heuristic rate allocation scheme which can be interpreted as a practical form of water-filling method is introduced and afterwards an iterative power control algorithm is proposed. 1.2. Paper contribution and approach In the previously mentioned research works, even in the case where user’s transmission power and rate are allocated at the same step, the stable outcome is extracted independently or semi-jointly, resulting in high transmission rates and low power consumption for users close to the base station and very low rate and maximum transmission power for the distant users. The basic characteristics, contributions and differences of our proposed approach and framework can be summarized as follows: 1. We introduce a more holistic approach in the formulation and treatment of the actual joint uplink transmission rate and power optimization problem. Specifically, instead of separately solving the uplink transmission rate and power problem or converting it to a single-variable optimization problem, under specific and restrictive assumptions, we treat the joint resource allocation problem as a two-variable problem,

E.E. Tsiropoulou et al. / Physical Communication 9 (2013) 299–307

2.

3.

4.

5.

we determine simultaneously users’ optimal rate and power vector and their values are updated at the same step. The outcome of our analysis verifies our intuition that solving the actual joint uplink transmission rate and power problem results in a more efficient stable point with respect to users’ satisfaction. We consider and adopt more general and realistic users’ utility functions [10,11] which instead of simply expressing the tradeoff between the number of a user’s reliably transmitted bits and corresponding consumed power, represents the tradeoff between utility-based actual throughput performance and the corresponding energy consumption. Thus the adopted utility functions reflect users’ service QoS-aware performance efficiency as a function of their achieved goodput per joule of consumed energy (Section 2). The proposed framework includes: (a) the joint utilitybased uplink rate and power control game (JUPRA) formulation as a non-cooperative game, where each user aims selfishly at maximizing his utility-based performance under the imposed physical limitations (Section 3); (b) its solution and unique Nash equilibrium point are determined simultaneously with respect to both types of resources, i.e. uplink transmission rate and power (Section 4); (c) an iterative and distributed algorithm that determines JUPRA game’s unique Nash equilibrium (its convergence is proven — (Section 5)). Based on existing proposals in the literature for incorporating pricing mechanisms in order to improve the achievable solution in terms of social welfare [12,13,10,14,15], the role and impact of various pricing schemes (i.e. linear and non-linear pricing) in obtaining a more energy efficient outcome under the proposed framework is discussed and further quantified in detail (Section 6). The performance of the proposed approach is evaluated in detail and its superiority compared to various state of the art approaches (i.e. [5–7,7]) is illustrated (Section 7). More importantly the contribution of each component of the proposed framework in its performance is quantified and analyzed. Thus the benefits of (a) jointly allocating transmission power and rate in the same step (Section 7), (b) properly formulating users’ QoS prerequisites in an appropriate and more general utility function and (c) imposing an appropriate pricing policy, are explicitly presented.

2. System model The uplink of a single-cell time slotted CDMA infrastructure wireless network, consisting of N (t ) continuously backlogged users at time slot t is considered. Let us denote by S (t ) the corresponding set of users and by Gi (t ), pi (t ) and ri (t ) user’s i path gain, uplink transmission power and rate at time slot t respectively. Due to mobile node’s physical limitations user’s uplink transmission power and rate are upper bounded, i.e. pi (t ) ≤ pMax and ri (t ) ≤ riMax . i Throughout the rest of our analysis, for simplicity in the presentation, we omit the notation of the time slot t. The intercell interference, the thermal noise components and

301

the users’ signals can be regarded together as an additive white Gaussian noise (AWGN) process, with constant power spectral density Io . Therefore, the overall sensed interference by user i ∈ S consists of the network interference and the AWGN background noise, given by: I−i (p−i ) =

N 

Gj pj − Gi pi + I0

(1)

j =1

where in the rest of our analysis p−i = (p1 , p2 , . . . , pi−1 , pi+1 , . . . , pN ) and r−i = (r1 , r2 , . . . , ri−1 , ri+1 , . . . , rN ) denotes the power and rate vector of all users except the ith user. The corresponding received bit energy to interference density ratio at the base station γi = Eb /Io for each user i ∈ S is given by:

γi (ri , pi , p−i ) =

W

Gi pi

(2)

ri I−i (p−i )

where W [Hz] denotes the available spread-spectrum bandwidth. Aiming at aligning users’ various services flow characteristics under a common optimization framework, while considering their specific QoS prerequisites, each mobile user is associated with a utility function Ui , which represents his degree of satisfaction in relation to the consumed uplink transmission power and the achieved uplink transmission rate. Typically, a user would like to achieve high bit energy to interference ratio, γi , via consuming low power pi and transmitting at high data rates ri . Therefore, the above characteristics of user’s utility function are formulated in a continuous and C (n) differentiable function Ui with respect to both user’s uplink transmission power pi and rate ri , as follows: Ui (ri , pi ) =

Qi (ˆri , pi ) pi

=

Qi (ri · fi (γi ), pi ) pi

(3)

where rˆi ≡ ri · fi (γi ) denotes user’s reliably transmitted bits to the base station, fi (γi ) is the efficiency function, representing the probability of a successful packet transmission [16] and Qi is the actual throughput utility function, representing user’s satisfaction in accordance to his service actual throughput expectations and QoS requirements fulfillment. User’s i efficiency function fi is an increasing function of his bit energy to interference density ratio γi at any time slot t. A user’s function for the probability of a successful packet transmission at fixed data rates depends on the transmission schemes being used, i.e. modulation and coding schemes, and can be represented by a sigmoidallike function of his bit energy to interference density ratio. Therefore, the efficiency function fi yields the following desirable properties: 1. fi is an increasing function of γi , 2. fi is a continuous, twice differentiable sigmoidal function with respect to γi , 3. fi (0) = 0 and fi (∞) = 1. In most of the recent literature [2,4–7], the common utility function that has been used simply reflects the tradeoff between the reliably transmitted bits to the base station and the corresponding consumed transmission power, without considering a user’s specific QoS characteristics.

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Fig. 1. Actual uplink throughput utility as a function of user’s actual uplink transmission rate for different types of service.

On the contrary, the adopted utility function in our work properly reflects a user’s personal preferences with respect to the service class (i.e. real and non-real time services) that belongs to [10,17]. It is true however that users’ perceived satisfaction conceptually differentiates the formulation of their actual uplink throughput utility function Qi (ˆri , pi ) and calls for enhanced approaches to the radio resource problem in cellular networks. Specifically, aiming at the fulfillment of non-real time data users’ greedy and high throughput performance expectations, Qi is assumed as a strictly convex or a strictly concave function of user’s i achieved actual transmission rate rˆi ≡ ri · fi (γi ). On the other hand, aiming at fulfilling real time users’ QoS performance requirements, which primarily consist of short-term throughput and delay constraints, Qi is properly selected as a sigmoidal actual throughput utility (Fig. 1). Regardless of the service a user desires, user’s actual throughput utility function Qi (ˆri , pi ) has the following common fundamental properties: 1. Qi is an increasing, continuous and C (n) differentiable function of actual uplink transmission rate rˆi ≡ ri · fi (γi ), and 2. Qi (ˆri = 0, pi ) = 0 to ensure that Ui (ri = 0, pi ) = 0. In brief, each user’s objective is to control and adjust his uplink transmission power and rate in a distributed manner, in order to maximize his sensed satisfaction, i.e. utility. 3. Joint utility-based uplink power & rate allocation (JUPRA) non-cooperative game The non-cooperative uplink transmission power and rate allocation game is formulated as a distributed utility maximization problem, in which each user updates his power and rate in a selfish manner, aiming at maximizing his received satisfaction. max Ui (pi , ri ) =

Qi (ˆri = ri · f (γi ), pi ) pi

pi ∈Pi ri ∈Ri

s.t. 0 ≤ ri ≤

riMax

,

0 ≤ pi ≤

,

i∈S (4)

pMax i

.

The above joint utility-based uplink power and rate allocation non-cooperative game (JUPRA) is denoted by G = [S , {Pi , Ri }, {Ui (pi , ri )}], where S = {1, 2, . . . , N } is the index set for the active mobile users residing in the cell, Pi is the strategy set of user powers, Ri , is the strategy set of user rates and Ui represents the objective function of the optimization problem as defined in (3). We assume that the strategy spaces Pi and Ri are compact and convex sets with maximum and minimum constraints. Thus, for each user i ∈ S we consider as respective strategy spaces the closed intervals: Pi = [0, pMax ] and Ri = [0, riMax ]. i Each user i ∈ S selects jointly a rate ri ∈ Ri and a power pi ∈ Pi to maximize utility function Ui . Let the rate vector r ∗ = (r1 , r2 , . . . , rN )T ∈ R = R1 × R2 × · · · × RN and power vector p∗ = (p1 , p2 , . . . , pN )T ∈ P = P1 × P2 × · · · × PN (where T represents the transpose operator) denote the outcome of the game in terms of selected rates and powers for all users. In our proposed framework the non-cooperative JUPRA game consists of two separate variables, i.e. uplink transmission power pi and rate ri , which should be optimized in order to maximize user’s utility. The fundamental attribute of the non-cooperative game (4), which calls for special treatment, is the simultaneous and synchronous optimization of user’s utility with respect not only to battery consumption, (i.e. user’s uplink transmission power), but to user’s uplink transmission rate as well. The noncooperative game is solved, considering both variables and the outcome of the game is the optimal pair of two vectors: (p∗ , r ∗ ). The Nash equilibrium approach has been adopted towards seeking analytically the solution of the distributed non-cooperative JUPRA game, which is most widely used for game theoretic problems. 4. Towards a Nash equilibrium for the JUPRA game The concept of Nash equilibrium offers a stable, predictable and definable solution of a non-cooperative game, where multiple users with potentially conflicting interests compete through self-optimization and reach a point where no user wishes to deviate. Specifically, at the equilibrium point no user has the incentive to change his power and rate levels since his utility can not be further improved by making any individual changes on these values, given the power and rate vectors of all other users, i.e. p∗−i and ∗ r− i , respectively. Definition 1. A pair of vectors (p∗ , r ∗ ) = (p∗1 , p∗2 , . . . , p∗N , r1∗ , r2∗ , . . . , rN∗ ) in the strategy sets p∗i ∈ Pi and ri∗ ∈ Ri is a Nash equilibrium of the JUPRA game G, such that no user can unilaterally improve his own utility, if for every i ∈ S the following condition holds: Ui (p∗i , ri∗ , p−i , r−i ) ≥ Ui (pi , ri , p−i , r−i ), for all pi ∈ Pi , ri ∈ Ri . However, due to users’ selfish behavior, a non-cooperative game does not always and necessarily have a Nash equilibrium, because it is likely that users will not compromise with a stable outcome. The absence of a Nash equilibrium means that the system is inherently unstable. Thus, the existence of a Nash equilibrium in JUPRA game is firstly investigated.

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Theorem 1. A Nash equilibrium in both uplink transmission power and rate exists (p∗ , r ∗ ) = (p∗1 , p∗2 , . . . , p∗N , r1∗ , r2∗ , . . . , rN∗ ), p∗i ∈ Pi and ri∗ ∈ Ri , ∀i ∈ S in the JUPRA game G = [S , {Pi , Ri }, {Ui (pi , ri )}]. Proof. In order to study the existence of Nash equilibrium in the JUPRA game, the following necessary conditions hold true. 1. Pi and Ri are non-empty, convex and compact subsets of some Euclidean spaces P N and RN respectively. 2. Ui (p, r ) is continuous and C (n) differentiable function with respect both uplink transmission power p ∈ P = [P1 × P2 × · · · × PN ] and rate r ∈ R = [R1 × R2 × . . . × RN ] [18]. Thus, given the above properties and based on the (Weierstrass) Extreme Value Theorem (EVT) [19], we conclude that the utility function, as defined in (3), has at least one global maximum (p∗ , r ∗ ) = (p∗1 , p∗2 , . . . , p∗N , r1∗ , r2∗ , . . . , rN∗ ) which is a Nash equilibrium of the JUPRA game, where each user’s utility is maximized and cannot be further improved by making any individual changes. According to the EVT, for a continuous function, i.e. Ui (ri , pi ) : {Ri , Pi } ⊆ R2 → R2 , where pi ∈ Pi and ri ∈ Ri , such that 0 ≤ pi ≤ pMax and 0 ≤ ri ≤ riMax there exist at least i two pairs (r1 , p1 ), (r2 , p2 ) ∈ {Ri , Pi } such that Ui (r1 , p1 ) = min pi ∈Pi Ui (ri , pi ) and Ui (r2 , p2 ) = max pi ∈Pi Ui (ri , pi ). Thus, ri ∈Ri

ri ∈Ri

EVT enables the proof of the existence of at least one Nash equilibrium in the JUPRA game.  Theorem 1 establishes and proves the existence of a Nash equilibrium in the JUPRA game. Thus, as it is identified by Theorem 1, the JUPRA game has a nonempty set of Nash equilibria, which conclude to a stable point of the system. However, considering an energyefficiency perspective and targeting to the joint uplink transmission power and rate allocation while satisfying users’ QoS prerequisites, we aim at maximizing their utility with the minimum feasible uplink transmission power for each user. The following theorem proves the uniqueness of Nash equilibrium when considering the objective of energy-efficiency (from the corresponding non-empty set of the JUPRA game’s set of Nash equilibria) and obtain the specific values of the stable outcome (p∗ , r ∗ ). Theorem 2. The unique Nash equilibrium of the JUPRA noncooperative game G from the energy-efficiency perspective is given by (p∗i , ri∗ ), pi ∈ Pi and ri ∈ Ri :





(p∗i , ri∗ ) = min min arg max Λk,i pi ∈Pi pi ∈Pi ,ri ∈Ri  Max × Ui (pi , ri ), pi Λk,i = (p∗1,i , p∗2,i , . . . , p∗k,i ),  ∂ Ui (pi , ri )  s.t.  ∗ =0 ∂p i

and

i

(5)

k = 1, 2, . . . , K , i ∈ S

i

i

i

i

i

i

i

i

∂ p i ∂ ri

i

i

i

∂ ri2

matrix at point (pi , ri ) and Λk,i = (p∗1,i , p∗2,i , . . . , p∗k,i ), k = 1, 2, . . . , K , i ∈ S is the non-empty set of the JUPRA game’s Nash equilibria in the under consideration compact, convex and closed interval Pi = [0, pMax ]. i Proof. The conditions of the first-order partial derivatives (5a) determine the stationary points of user’s utility function Ui (pi , ri ), which can either be a maximum, a minimum or a saddle point. Aiming to determine the global maximum of user’s utility, the (5b) condition is necessary and sufficient, i.e. the Hessian matrix must be negative definite. The validity of the (5b) condition is guaranteed for a pair (pi , ri ), pi ∈ Pi , ri ∈ Ri , due to the continuity and C (n) differentiability of the user’s utility function. Furthermore, the selection of the minimum uplink transmission power, between arg maxpi ∈Pi , ri ∈Ri Ui (pi , ri ) and pMax is necessary, i due to the physical limitation of the user’s uplink . Moreover, towards transmission power, i.e. pi ≤ pMax i achieving energy-efficiency the minimum feasible uplink transmission power among the Nash equilibria is selected, which is the unique Nash equilibrium achieving energyefficiency.  Finally, it is noted that the problem has been formulated as a joint utility-based uplink power and rate allocation game, where each user aims selfishly at maximizing his utility-based performance under the imposed physical limitations in a distributed manner. Therefore, the above theorem determines the stable outcome (p∗ , r ∗ ) by optimizing w.r.t. the i-th user’s energy efficiency, and does not necessarily lead to rate optimization or to the overall power optimization from a system’s point of view. 5. JUPRA algorithm & convergence In this section, the convergence of JUPRA game is shown and an iterative, distributed and low complexity algorithm that determines its unique Nash equilibrium point (p∗ , r ∗ ) is introduced. 5.1. Convergence of JUPRA game Based on Theorem 2, user’s i ∈ S best response BRi is denoted as follows:

   = min min arg max Ui (pi , ri ), pMax i Λk,i

pi ∈Pi

(6)

pi ∈Pi ,ri ∈Ri

Λk,i = (p∗1,i , p∗2,i , . . . , p∗k,i ), k = 1, 2, . . . , K , i ∈ S   ∂ Ui (pi , ri )  ∂ Ui (pi , ri )  = 0 and s.t.  ∗ =0 ∂ pi pi =p∗ ∂ ri ri =r i

i

and (pi , ri )T H (pi , ri )(pi , ri ) ≺ 0,

=0

(5a)

ri =ri∗

and (pi , ri )T H (pi , ri )(pi , ri ) ≺ 0,

∀pi ∈ Pi , ∀ri ∈ Ri

i

∂ pi ∂ ri   ∂ p2 where H (pi , ri ) =  2 i  is the Hessian ∂ U (p , r ) ∂ 2 U (p , r )

BRi (pi , ri )

pi =pi

 ∂ Ui (pi , ri )   ∂r

303

 ∂ 2 U (p , r ) ∂ 2 U (p , r ) 

(5b)

∀pi ∈ Pi , ∀ri ∈ Ri . Theorem 3. The JUPRA game converges to its unique Nash equilibrium, starting from any initial point.

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Proof. By definition, the Nash equilibrium of a noncooperative game has to satisfy: (p∗ , r ∗ ) = BR (p, r ). The fundamental point in order to prove the JUPRA game’s convergence to its unique Nash equilibrium is to show that a users’ utility function is a standard function [20]. In general, a function f is said to be standard, if it satisfies the following three properties: (i) Positivity: f (x) ≻ 0. (ii) Monotonicity: if x ≥ x′ , then f (x) ≥ f (x′ ). (iii) Scalability: for all a > 1, af (x) ≥ f (ax), for all x ≥ 0, where = (x1 , x2 , . . . , xN ) is a Nash equilibrium and f (x) is the best response function. Considering the JUPRA game, the above properties can be easily verified for BR (p, r ), since: (i) (p, r ) ≻ 0, thus BR (p, r ) ≻ 0, via relation (6), (ii) if (p, r ) ≻ (p′ , r ′ ), then via (6) we conclude that BR (p, r ) ≻ BR (p′ , r ′ ), (iii) for all a > 1, since BR (p, r ) is a strictly increasing function with respect to (p, r ), then we conclude that aBR (p, r ) ≻ BR (ap,ar ). 

has been argued in the recent literature [2,12,13,10,14], the Nash equilibria stemming from distributed decision taking are generally inefficient from a social point of view, due to users’ selfish behavior. Therefore, pricing system’s resources has emerged as a powerful tool for achieving a more socially desirable point. Typically, the main reasons that motivate the adoption of pricing system’s resources are twofold: the better revenue for the system that is generated via pricing mechanisms, and the encouragement of the users to use system resources more efficiently. Therefore, based on the above observations [10,14], we can enhance the performance of the proposed framework by combining existing pricing mechanisms with the proposed joint utility-based uplink transmission power and rate allocation non-cooperative game formulation. In that case each user adopts a proper utility function, which is associated with his resource requirements, while the price reflects relations between the current users’ demand and availability of resources. Thus, the non-cooperative joint uplink power and rate allocation game with pricing (JUPRA-P game) can be formulated as follows:

Therefore, JUPRA game’s convergence to its unique Nash equilibrium, starting from any initial feasible point is guaranteed.

max UiP (pi , ri ) =

5.2. JUPRA algorithm

s.t. 0 ≤ ri ≤ riMax ,

Qi (ˆri = ri · f (γi ), pi ) pi

pi ∈Pi ri ∈Ri

i∈S

The fundamental attribute of the designed algorithm is that users update their uplink transmission power and rate at the same iteration/step, thus the unique optimal solution is derived simultaneously with respect to both power and rate. Step 1. At the beginning of time slot t, each user i, i ∈ S, (0) randomly selects a feasible uplink transmission power pi (0)

(0)

(0)

(i.e. 0 ≺ pi ≤ pMax ) and rate ri (i.e. 0 ≺ ri ≤ riMax ). Set i k = 0, where k denotes the number of iterations. Step 2. Users’ overall interference I (k) (p(k) ) is broadcast by the base station, given the uplink transmission power of (k) (k) each user, thus the user computes I−i (p−i ) and refines (k+1)

his uplink transmission power, pi accordance to (5).

(k+1)

and rate ri (k+1)

Step 3. If the powers and rates converge (i.e. |pi (k+1)

ε and |ri

(k)

, in

−p(i k) | ≤

− ri | ≤ ε , ε = 10 ) then stop. −5

Step 4. Otherwise, set k = k + 1 and go to step 2. The JUPRA algorithm is characterized as a single-valued low-complexity best response distributed algorithm for every user starting from any randomly selected feasible values of power and rate of his non-empty orthogonal strategic spaces Pi = [0, pMax ] and Ri = [0, riMax ] (i.e. pi(0) i (0)

and ri , ∀i ∈ S). 6. Enhancing energy-efficient joint resource allocation via pricing In Section 4, the unique Nash equilibrium of the JUPRA game from the energy-efficiency point of view has been determined and established. However, as it

− ci (pi ), (7)

0 ≤ pi ≤ pMax i

where ci (Pi ) denotes the pricing function. In the following section for demonstration purposes we consider two different variation of the proposed framework with respect to the resource pricing mechanism incorporated: we use either linear function of user’s uplink transmission power, i.e. ci (pi ) = ci · pi (in the following we refer to it as the JUPRA-LP game), or convex pricing such as exponential function, i.e. ci (pi ) = ci · (epi − 1) (in the following we refer to it as the JUPRA-CP game). It is noted that traditionally most of the pricing mechanisms that have been applied in wireless uplink power control are linear with respect to user’s uplink transmission power [12,13]. The idea of non-linear pricing mechanism derives from the observation that the harm a user imposes on other users is not equivalent within the whole range of transmission power, as the linear pricing mechanism admits. Therefore, non-linear pricing mechanisms have also been used extensively in the literature in several fields [21] such as economics, energy, etc. The benefits of adopting such pricing mechanisms in our framework are presented in Section 7 via an overall comparative study. 7. Numerical results In this section, we present some numerical results with respect to the operation of the proposed framework that demonstrate the performance improvements that can be achieved in a single cell time-slotted cellular system. Throughout our evaluation several variations of the proposed framework are examined in order to better quantify the contribution of each component of the proposed framework in its performance. Thus the

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benefits of (a) jointly allocating transmission power and rate at the same step, (b) properly formulating users’ QoS prerequisites in an appropriate and more general utility function and (c) imposing an appropriate pricing policy, are explicitly presented. For demonstration purposes only, it is assumed that N = 9 continuously backlogged users reside in the cell, at distances d = [310 460 570 660 740 810 880 940 1000] m from the base station. Users’ path gains are modeled via the path loss model Gi = Ki /dai , where a denotes the distance loss exponent (e.g. a = 4) and Ki is a lognormal distributed random variable with mean 0 and variance σ 2 = 8 (db). Each user adopts a sigmoidal actual throughput utility function Qi (ri , pi ) = (1 − e(ri −A) )M , where A and M are positive constant values regulating function’s slope and a common efficiency function, fi (γi ) = (1 − e−3.7·γi )80 , Io = 5 ∗ 10−6 , W = 106 Hz. Users’ physical limitations, considering their transmission powers are set up to pMax = 2 Watts and their transmission rate riMax = i 64 Kbps (i.e. voice users). Each simulations lasts 10.000 time slots. The adopted pricing functions are ci (pi ) = ci · pi for JUPRA-LP and ci (pi ) = ci · (epi − 1) for JUPRA-CP game, where ci is the pricing factor broadcast by the base station, which imposes the pricing policy of the system. Figs. 2 and 3 illustrate users’ time average transmission power and rate respectively for the nine different users considered in this study which are placed (and identified accordingly) at different distances from the base station as detailed above, under seven different scenarios. Specifically, in order to allow for a detailed evaluation of the proposed framework and fair comparative analysis, the following seven different scenarios are considered: (i) the JUPRA algorithm, (ii) the algorithm presented in [6] enhanced with the utility function (3) considered here, (iii) the algorithm in [7], (iv) the JUPRA algorithm adopting the utility function used in [5–7], (v) the JUPRA-LP game, (vi) the JUPRA-CP game and (vii) the algorithm in [6] with linear pricing. It is noted that as explained in Section 1.1, in [6] the authors address the problem of power and rate allocation in the uplink of a wireless cellular network, by initially solving the rate control problem and afterwards the resulting power control problem, while in [7] the authors convert the problem to a single-variable one, via substituting the ratio of user’s uplink transmission rate to his corresponding uplink transmission power with a new variable. Considering all the scenarios, the results reveal that as users’ mean channel conditions become worse (i.e. as their distance from the base station increases) and in order to satisfy their QoS prerequisites, their average transmission power increases, while their uplink transmission rate decreases with different slope, depending on the scenario. The benefit in power saving and users’ QoS satisfaction considering the simultaneous allocation of transmission power and rate at the same step and treating the joint allocation problem as a two-variable problem (the JUPRA approach), is observed via comparing the results of the JUPRA algorithm (scenario i) and the corresponding results via the algorithm in [6] (scenario ii), while both adopt the same utility function (3). The JUPRA algorithm presents superior performance in power savings while meeting users’ rates requirements, while the algorithm in [6]

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Fig. 2. Users’ average transmission power at equilibrium under various scenarios.

Fig. 3. Users’ average transmission rate at equilibrium under various scenarios (legend used as in Fig. 2).

treats the users closer to the base station (i.e. low power consumption and high rates) in preferential way compared to distant users. Furthermore, the benefits that stem from properly formulating users’ QoS prerequisites in an appropriate and more general utility function, compared to the simple ratio of reliably transmitted bits to users’ power consumption commonly used in the literature [5–7], is observed via comparing the results of applying the JUPRA algorithm (scenario i), and the corresponding results achieved under JUPRA with the adoption however of the utility function proposed in [5–7] (scenario iv). The results reveal that the utility function (3) formulates better users’ QoS requirements, and therefore lower power consumption is necessary in order for users to achieve higher transmission rate. Furthermore, the overall and combined benefit of the appropriate formulation of the utility function and the actual joint transmission power and rate allocation in the same step can be further illustrated via the comparison of the JUPRA algorithm results (scenario i) and the corresponding results of the algorithm in [7] (scenario iii). We specially note the critical power savings

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achieved under the proposed framework and the unified way of treating users considering their obtained uplink transmission rate, which remains high, even for the distant users. Moreover, the benefits that stem from imposing and incorporating an appropriate pricing policy are observed via comparing the results of the JUPRA, JUPRA-LP and JUPRA-CP game (i.e. scenarios i, v, vi). It is illustrated that introducing the appropriate pricing mechanism to the users, forces them to compromise to a more energyefficient stable point, i.e. consume lower energy without significantly reducing their uplink transmission rate. Additionally, comparing the results of the JUPRA-LP game (scenario v) to those obtained via the algorithm in [6] with linear pricing (scenario vii), we conclude that the users at the Nash equilibrium of the JUPRA-LP game transmit at significantly lower power levels and achieve higher uplink transmission rates. Concluding the comparison of the seven different scenarios, it is observed that the JUPRACP game (scenario vi) achieves the most energy-efficient stable point. Fig. 4 illustrates all users’ uplink transmission power evolution as a function of the iterations required for the JUPRA algorithm to converge at game’s G equilibrium point (p∗ , r ∗ ). The results reveal that the convergence of the JUPRA algorithm is fast, as less than eighty iterations are required to reach equilibrium for all users, while in less than forty iterations the power values have converged very close to the final values. It is noted that with respect to the results presented in Fig. 4 random initial (feasible) values have been assumed for the users’ powers, and therefore some spikes are observed in the corresponding curves mainly during the initial iterations, till the powers converge to their final values according to JUPRA operation. Finally, a comparative study of the convergence of the three basic algorithms proposed in the literature and considered in this work for solving the power and rate control problem is provided in Fig. 5. Specifically, for demonstration purposes Fig. 5 illustrates user 1’s (i.e. user closest to the base station) uplink transmission power evolution as a function of the iterations required for convergence at their final equilibrium points for the following cases: (a) the algorithm in [6], (b) the JUPRA algorithm and (c) the algorithm in [7]). It is noted that a similar behavior is observed for the rest of the users as well. The results reveal that the JUPRA algorithm, by solving jointly the two-variable problem, converges faster to the corresponding game’s equilibrium point, thus outperforming the other approaches. 8. Conclusion The introduced joint utility-based power and rate allocation framework with the adopted utility function and the proposed approach of determining the optimal power and rate allocation, provides us with an enhanced flexible framework which can be used to uniformly treat multiple user services’ diverse expectations in the next generation wireless networks. The fundamental benefit of our approach is the solution of the optimization problem with respect to both variables, i.e. uplink transmission

Fig. 4. Different users’ transmission power convergence under JUPRA.

Fig. 5. Convergence time of various joint power and rate allocation algorithms for user 1.

power and rate and the allocation of them at the same step. The comparisons to other approaches in the recent literature verified the significant performance benefits that can be obtained by our framework. Taking into account that the future Internet envisions the synergy among heterogeneous wireless communications, the study and analysis of heterogeneous wireless networks’ joint resource allocation and QoS provisioning mechanisms, under a common framework as the one introduced and evaluated in this paper, becomes of high research and practical importance. References [1] S. Gunturi, F. Paganini, Game theoretic approach to power control in cellular CDMA, in: Proc. of 58th IEEE Vehicular Technology Conference, VTC, Orlando, FL, Oct. 2003, pp. 2362–2366. [2] C.U. Saraydar, N.B. Mandayam, D.J. Goodman, Efficient power control via pricing in wireless data networks, IEEE Transactions on Communications 50 (2002) 291–303. [3] M. Hayajneh, C.T. Abdallah, Distributed joint rate and power control game-theoretic algorithms for wireless data, IEEE Communications Letters 8 (8) (2004) 511–513. [4] W. Zhao, M. Lu, Distributed rate and power control for CDMA uplink, in: Proceedings 2004 IEEE Wireless Telecommunications Symposium, Pomona, CA, May 2004, pp. 9–14. [5] M.R. Musku, A.T. Chronopoulos, D.C. Popescu, Joint rate and power control using game theory, IEEE CCNC’2006 2 (2006) 1258–1262.

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Eirini Eleni Tsiropoulou received her diploma Degree in electrical and computer engineering (summa cum laude) and her M.Sc. Degree in techno-economics from the National Technical University of Athens (NTUA), Greece, in 2008 and 2010, respectively. She is currently working towards the Ph.D. Degree in the ECE Department of NTUA, and as a research assistant in the Network Management and Optimal Design Laboratory. Her main research interests lie in the area of wireless and heterogeneous networks with emphasis on optimization and resource allocation.

Panagiotis Vamvakas received his diploma Degree in electrical and computer engineering from the National Technical University of Athens (NTUA), Greece, in 2011. His main scientific interests lie in the area of mobile communications and computer networking. Currently he is a research assistant in the Network Management and Optimal Design Laboratory, National Technical University of Athens.

Symeon Papavassiliou is an Associate Professor with the Faculty of Electrical and Computer Engineering, National Technical University of Athens, since 2005. Before joining NTUA he was an Associate Professor with the ECE Department at the New Jersey Institute of Technology (NJIT), USA, and a Senior Technical Staff Member at AT&T Laboratories, New Jersey, USA. His research interests lie in the area of design and optimization of communication networks where he has more than one hundred and seventy technical journal and conference published papers. He is a senior member of IEEE, Associate Editor for IEEE Transactions on Parallel and Distributed Systems and Technical Editor for IEEE Wireless Communications Magazine.