Crosslayer distributed power control: a repeated game formulation to improve the sum energy efficiency
 Mariem Mhiri^{1}Email author,
 Vineeth S. Varma^{2},
 Karim Cheikhrouhou^{1},
 Samson Lasaulce^{3} and
 Abdelaziz Samet^{4}
https://doi.org/10.1186/s136380150486z
© Mhiri et al. 2015
Received: 18 May 2015
Accepted: 25 November 2015
Published: 8 December 2015
Abstract
The main objective of this work is to improve the energy efficiency (EE) of a multiple access channel (MAC) system, through power control, in a distributed manner. In contrast with many existing works on energyefficient power control, which ignore the possible presence of a queue at the transmitter, we consider a new generalized crosslayer EE metric. This approach is relevant when the transmitters have a nonzero energy cost even when the radiated power is zero and takes into account the presence of a finite packet buffer and packet arrival at the transmitter. As the Nash equilibrium (NE) is an energyinefficient solution, the present work aims at overcoming this deficit by improving the global energy efficiency. Indeed, as the considered system has multiple agencies each with their own interest, the performance metric reflecting the individual interest of each decisionmaker is the global energy efficiency defined then as the sum over individual energy efficiencies. Repeated games (RG) are investigated through the study of two dynamic games (finite RG and discounted RG), whose equilibrium is defined when introducing a new operating point (OP), Paretodominating the NE and relying only on individual channel state information (CSI). Accordingly, closedform expressions of the minimum number of stages of the game for finite RG (FRG) and the maximum discount factor of the discounted RG (DRG) were established. Our contributions consist of improving the system performances in terms of powers and utilities when using the new OP compared to the NE and the Nash bargaining (NB) solution. Moreover, the crosslayer model in the RG formulation leads to achieving a shorter minimum number of stages in the FRG even for higher number of users. In addition, the social welfare (sum of utilities) in the DRG decreases slightly with the crosslayer model when the number of users increases while it is reduced considerably with the Goodman model. Finally, we show that in real systems with random packet arrivals, the crosslayer power control algorithm outperforms the Goodman algorithm.
Keywords
1 Introduction
1.1 Motivation
The design and management of green wireless networks [1–3] has become increasingly important for modern wireless networks, in particular, to manage operating costs. Futuristic (beyond 5G) cellular networks face the dual challenges of being able to respond to the explosion of data rates and also to manage network energy consumption. Due to the limited spectrum and large number of active users in modern networks, energyefficient distributed power control is an important issue. Sensor networks, which have multiple sensors sending information to a common receiver with a limited energy capacity have also recently surged in popularity. Energy minimization in sensor networks has been analyzed in many recent works [4–6].
 1.
Multiple transmitters connected to a common receiver
 2.
Lack of centralization or coordination, i.e., a distributed and decentralized network
 3.
Relevance of minimizing energy consumption or maximizing energy efficiency (EE)
 4.
Transmitters that have arbitrary data transmission
These features are present in many modern systems like a sensor network which has multiple sensors with limited energy connected in a distributed manner to a common receiver. These sensors do not always have information to transmit, resulting in sporadic data transmission. Another example would be several mobile devices connected to a hotspot (via wifi or even Bluetooth). Due to these features of the network, intertransmitter communication is not possible and the transmitters are independent decisionmakers. Therefore, implementing frequency or time division multiple access becomes harder, and a MAC protocol (with single carrier) is often the preferred or natural method of channel access.
1.2 Novelty
In many existing works, both networkcentric and usercentric approaches have been studied. In a networkcentric approach, the global energy efficiency (GEE) is defined as the ratio between the system benefit (sumthroughput or sumrate) and the total cost in terms of consumed power [7, 8]. However, when targeting an efficient solution in an usercentric problem, the GEE becomes not ideal as it has no significance to any of the decisionmakers. In this case, other metrics are required to reflect the individual interest of each decisionmaker. Therefore, we redefine the GEE to be the sum over individual energy efficiencies as a suitable metric of interest [9].
The major novelty of this work is in improving the sum of energy efficiencies for a communication system with all the listed features above. In such a decentralized and distributed network, as each transmitter operates independently, implementing a frequency division or a time division multiple access is not trivial. Therefore, we are interested in looking at a MAC system where all transmitters operate on the same band. Additionally, EE will be our preferred metric due to its relevance. This metric has been defined in [10] as the ratio between the average net data rate and the transmitted power. In [11, 12], the total power consumed by the transmitter was taken into account in the EE expression to design distributed power control which is one of the most wellknown techniques for improving EE. However, many of the works available on energyefficient power control consider the EE defined in [10] where the possible presence of a queue at the transmitter is ignored. In contrast with the existing works, we consider a new generalized EE based on a crosslayer approach developed recently in [13, 14]. This approach is important since it takes into account: (1) a fixed cost in terms of power namely, a cost which does not depend on the radiated power and (2) the presence of a finite packet buffer and sporadic packet arrival at the transmitter (which corresponds to including the fourth feature mentioned above). Although providing a more general model, the distributed system in [14] may operate at a point which is energyinefficient. Indeed, the point at which the system operates is a Nash equilibrium (NE) of a certain noncooperative static game. The present work aims at filling this gap by not only considering a crosslayer approach of energyefficient power control but also improving the system performance in terms of sum of energy efficiencies.
1.3 State of the art
 1.
Determine the closedform expressions of the minimum number of stages for the FRG and the maximum discount factor for the DRG. These two parameters identify the two considered RG.
 2.
Determine a distributed solution Paretodominating the NE and improving the system performances in terms of powers and utilities compared not only to the NE but also to the NB solution even for high number of users.
 3.
Show that the RG formulation when using the new EE and the new OP leads to significant gains in terms of social welfare (sum of utilities of all the users) compared to the NE
 4.Show that the following aspects of the crosslayer model improve considerably the system performances when comparing to the Goodman model even for large number of users:

The minimum number of stages in the crosslayer EE model can always be shorter than the minimum number of stages in the Goodman EE formulation.

The social welfare for the DRG in the crosslayer model decreases slightly when the number of users increases while it decreases considerably in the Goodman model.

 5.
Show that in real systems with random packet arrivals, the crosslayer power control algorithm outperforms the Goodman algorithm and then the new OP with the crosslayer approach is more efficient.
1.4 Structure
This paper is structured as follows. In section 2, we define the system model under study, introduce the generalized EE metric, and define the noncooperative static game. This is followed (section 3) by the study of the NB solution. In section 4, we introduce the new OP, give the formulation of both RG models (FRG and DRG), and determine the closedform expressions of the minimum number of stages and the maximum discount factor as well. Numerical results are presented in section 5, and finally, we draw several concluding remarks.
2 Problem statement
2.1 System model
where p=(p _{1},p _{2},…,p _{ N }) defines the power vector of all users and can be written as p=(p _{ i },p _{−i }) with p _{−i }=(p _{1},…,p _{ i−1},p _{ i+1},…,p _{ N }).
The purpose of this work is to determine how each user is going to control its power in an optimum way. Game theory, as a powerful mathematical tool, helps to solve such an optimization problem where the utility function is the EE which is a function of the users powers. Since the system under study has multiple agencies each with individual interest, the sum over individual energy efficiencies will be considered as the performance metric reflecting the individual interest of each decisionmaker.
2.2 Energy efficiency metric
It is important to highlight that this new generalized EE given by (5) includes the conventional case of (4) when making q→1.
2.3 Static crosslayer power control game
The static crosslayer power control game is a noncooperative game which can be defined as a strategic form game [17].
Definition 1.
where χ _{ i }(p) is given by Eq. (5).
where \(\gamma _{i}^{\prime } = {\frac {\mathrm {d} \gamma _{i}}{\mathrm {d} p_{i}}} = {\frac {\gamma _{i}}{p_{i}}}, f^{\prime } = {\frac {\mathrm {d} f}{\mathrm {d} \gamma _{i}} }\), and \(\Phi ^{\prime } = {\frac {\mathrm {d} \Phi }{\mathrm {d} \gamma _{i}}}\).
3 Nash bargaining solution
3.1 Compactness and convexity of the achievable utilities region
We define \(\bar {\mathcal {R}}^{\ast }\) the Pareto boundary (the outer frontier) of the convex hull of \(\bar {\mathcal {R}}\). Figure 1 shows the convexified achievable utilities region with the NE point, the NB solution, and the Nash curve (both will be defined next).
3.2 Existence and uniqueness of the NB solution
Since the NE can always be reached and the achievable utility region is a compact convex set, the NB solution exists. It is unique since it verifies certain axioms: individual rationality and feasibility, independence of irrelevant alternatives, symmetry, Pareto optimality (efficiency), and independence of linear transformations [21]. The NB solution results from the intersection of the Pareto boundary (\(\bar {\mathcal {R}}^{\ast }\)) with the Nash curve whose form is \(m=\prod _{i=1}^{N}\left (u_{i}u_{i}^{NE}\right)\) where m is a constant chosen such that there is precisely one intersection point [22] (see Fig. 1). Although the NB solution is Paretoefficient, it generally requires global CSI at the transmitters due to the Nash product (m) introducing all the users utilities [15]. For this reason, we are looking for another efficient solution through the study of the dynamic RG.
4 Repeated game formulation
RG consist in their standard formulation, in repeating the same static game at every time instance and the players seek to maximize their utility averaged over the whole game duration [16]. Repetition allows efficient equilibrium points to be implemented and which can be predicted from the oneshot static game according to the Folk theorem, which provides the set of possible Nash equilibria of the repeated game [18, 23]. In a repeated game, certain agreements between players on a common cooperation plan and a punishment policy can be implemented to punish the deviators [16]. In what follows, we introduce the new OP and characterize the two RG models.
4.1 New OP
Accordingly, each transmitter needs only its individual SINR and the constant α (depending only on p _{ i } and g _{ i }^{2}) to establish the received signal power P _{ y }. We assume that the data transmission is over block fading channels and that channel gains g _{ i }^{2} lie in a compact set \(\left [\nu _{i}^{\min }, \nu _{i}^{\max }\right ]\) [18]. Thus, the interval to which the received signal power belongs is \({\Delta = \left [\sigma ^{2}, \sigma ^{2} + \sum _{\substack {i=1}}^{N} p_{i}\nu _{i}^{\max }\right ]}\). Since the players detect a variation of the received signal power, a deviation from the cooperation plan has occurred. Indeed, when playing at the new OP, the received signal power is constant and equal to \({\frac {\sigma ^{2}(\tilde {\gamma }\,+\,1)}{1\,\,(N\,\,1)\tilde {\gamma }}}\). Consequently, when any player deviates from the new OP, the latter quantity changes and the deviation is then detected [18].
4.2 Repeated game characterization

The finite RG where the number of stages of the game (denoted as T≥1) during which the players interact is finite

The discounted RG where the discount factor (denoted as λ∈] 0,1[) is seen as the stopping probability at each stage
The utility function of each player results from averaging over the instantaneous utilities over all the game stages in the FRG while it is a geometric average of the instantaneous utilities during the game stages in the DRG [18, 25, 26]. We denote δ=(δ _{1},δ _{2},…,δ _{ N }) the joint strategy of all players.
Definition 2.
In RG with complete information and full monitoring, the Folk theorem characterizes the set of possible equilibrium utilities. It ensures that the set of NE in a RG is precisely the set of feasible and individually rational outcomes of the oneshot game [24, 25]. A cooperation/punishment plan is established between the players before playing [18]. The players cooperate by always transmitting at the new OP with powers \(\tilde {p}_{i}\). When the power of the received signal changes, a deviation is then detected and the players punish the deviator by transmitting with their maximum transmit power \(P_{i}^{\max }\) in the FRG and by playing at the oneshot game in the DRG. In what follows, we give the equilibrium solution of each repeated game model and mention the corresponding algorithm [27–29]. It is important to note that in contrast with iterative algorithms (e.g., iterative waterfilling type algorithms), there is no convergence problem in repeated games (FRG and DRG). Indeed, the transmitters implement an equilibrium strategy (referred to as the operating point) at every stage of the repeated game.
4.2.1 Finite RG
The FRG is characterized by the minimum number of stages (T _{min}). If the number of stages in the game T verifies T>T _{min}, a more efficient equilibrium point can be reached. However, if it is less than T _{min}, the NE is then played. Assuming that channel gains g _{ i }^{2} lie in a compact set \(\left [\nu _{i}^{\min }, \nu _{i}^{\max }\right ]\) [18], we have the following proposition [19]:
Proposition 1 (FRG equilibrium).
The quantities A, B, C, D, E, F, G, and H are defined in Appendix and \(\gamma _{i}^{\ast }\) is the SINR at the NE while \(\bar {\gamma }_{i}\) and \(\widehat {\gamma _{i}}\) are the SINRs related to the maximal utility and the utility minmax respectively (the proof of this proposition is detailed in [19]). The corresponding algorithm is as follows.
4.2.2 Discounted RG
In the DRG, the probability that the game stops at stage t is λ(1−λ)^{ t−1} with λ∈] 0,1[ defines the discount factor [17]. Accordingly, we can express the analytic form of the maximum discount factor in a DRG when assuming that channel gains g _{ i }^{2} lie in a compact set \(\left [\nu _{i}^{\min }, \nu _{i}^{\max }\right ]\) [18].
Proposition 2 (DRG equilibrium).
For the proof, see Appendix. The corresponding algorithm is as follows.
5 Numerical results
In this section, we consider the efficiency function f(x)=e ^{−c/x } with \(c=2^{\frac {R}{R_{0}}}1\). It has been proven in [30, 31] that such a function is sigmoidal as it is convex on the open interval (0,c/2] and concave on (c/2,+∞). The throughput R and the used bandwidth R _{0} are equal to 1 Mbps and 1 MHz, respectively. The maximum power P ^{max} is set to 0.1 Watt while the noise variance is set to 10^{−3} W. The buffer size K, the packet arrival rate q and the consumed power b are fixed to 10, 0.5 and 5×10^{−3} W, respectively. We consider Rayleigh fading channels and a spreading factor L introducing an interference processing (1/L) in the interference term of the SINR.
In Fig. 6, we highlight that the minimum number of stages is an increasing function of the packet arrival rate q according to the crosslayer model while it is a constant function for the Goodman model since the latter does not take into account the packet arrival process. One can confirm that the minimum number of stages is an increase function of the number of users as deduced previously. Simulations show that it exists a packet arrival rate q _{0} before which T _{min} of the crosslayer model is much lower than T _{min} of the Goodman model for different number of users. Simulations show that q _{0}≈0.6 and for q≥q _{0},T _{min} of the crosslayer model converges to T _{min} corresponding to the Goodman model. It is important to highlight that when N=3 and q≥q _{0},T _{min} of the crosslayer model takes higher values than T _{min} corresponding to the Goodman model but values are quite similar. With the increase of the number of users, the difference between the minimum number of stages for both models becomes noticeable. According to Figs. 5 and 6, one can conclude that the crosslayer model can be exploited for short games.
The study of the variation of λ _{max} versus the packet arrival rate q (in Fig. 9) shows that the maximum discount factor λ _{max} decreases with the number of users and with the packet arrival rate q as well. Simulations show that it exists a packet arrival rate q _{1} before which the λ _{max} corresponding to the crosslayer model takes higher values than the maximum discount factor of the Goodman model for different numbers of users. We notice that starting from q _{1}, the maximum discount factor of the crosslayer model converges to λ _{max} corresponding to the Goodman model.
6 Conclusions
In this paper, we have investigated RG for distributed power control in a MAC system. As the NE is not always energyefficient, the NB solution might be a possible efficient solution since it is Paretoefficient. However, the latter, in general, requires global CSI at each transmitter node. Thus, we were motivated to investigate using the repeated game formulation and develop a new OP that simultaneously is both more efficient than the NE and achievable with only individual CSI being required at the transmitter. Also, we consider a new EE metric taking into account the presence of a queue at the transmitter with an arbitrary packet arrivals.
Cooperation plans are proposed where the new OP is considered and closedform expressions of the minimum number of stages for the FRG and the maximum discount factor for the DRG have been established. The study of the social welfare (sum of utilities of all the users) shows that considerable gains are reached compared to the NE (for the FRG and DRG). Moreover, our model proves that even with a high number of users, the FRG can always be played with a minimum number of stages shorter than when using the Goodman model. In addition, the social welfare in the DRG decreases slightly with the number of users with the crosslayer approach while it decreases considerably with the Goodman model. Finally, the comparison of the crosslayer algorithm versus the Goodman algorithm shows that in real systems with random packet arrivals, the crosslayer power control algorithm outperforms the Goodman algorithm. Thus, the new OP with the crosslayer approach is more efficient. An interesting extension to this work would be to consider the interference channel instead of the MAC channel and generalize the framework applied here. Another possible extension would be to consider the multicarrier case and the resulting repeated g ame.
7 Appendix
7.1 Proof of λ _{ max }
7.1.1 Determination of the maximal utility
with: \({\gamma _{i} = \frac {p_{i}g_{i}^{2}}{\sigma ^{2}}}\).
7.1.2 Determination of λ _{max}
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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