Achieving QoS in Virtual MIMO systems: a satisfaction equilibrium analysis

Many studies have been published which used multiple approaches using the power control algorithms and game theory. In [1], the authors present the power control in cellular radio systems from an economic point of view and study the uplink power control problem as a non-cooperative N-person game. Goodman et al. [2] implement a new algorithm network assisted power control, in order to study the impact of effective power control on the system quality and efficiency in wireless communications. The authors in [3], study the effect of pricing transmit power in a multi-cell wireless data system, and introduce pricing of transmit power as a mechanism for influencing data user behavior. One of the challenges of wireless communications systems is the management and the efficient use of radio resources. In [4], the authors present a power control solution for wireless data in the analytical setting of a game theoretic framework and introduce pricing of transmit powers in order to obtain Pareto improvement of the non-cooperative power control game.

T. Alpcan et al. [5] present a new game theoretic treatment of distributed power control in CDMA wireless systems in order to study power control problem, pricing and allocation of a single resource among several users. In [6], a game theoretic approach in multirate code-division multiple access system, in order to maximize the total system throughput by means of power control is presented. Objectives of most radio resource-management schemes can be classified as either user-centric or network-centric. Authors in [7] consider the joint optimization of both user-centric and network-centric metrics. They use a utility function as the user-centric metric and introduce an explicit pricing mechanism to mediate between the user-centric and network-centric resource-management problems. F. Meshkati et al. propose in [8] a power control game for MC-CDMA systems; it is shown that the proposed approach results in a significant improvement in the total utility achieved at equilibrium. Cui et al. [9] presents an analysis of the best modulation and transmission strategy to minimize the total energy consumption in a radio application in sensor networks. By considering different MIMO systems’ schemes, it is shown that tremendous energy saving is possible for transmission distances larger than a given threshold, and over some distance, cooperative MIMO can achieve both energy and delay.

S. K. Jayaweera et al. [10] proposed a VMIMO scheme for distributed and cooperative wireless sensor networks, where the energy and delay efficiencies are derived using semi-analytic techniques. The author shows in this study that with judicious choice of design parameters the VMIMO technique can be made to provide significant energy and delay efficiencies. In [11], a new multi-hop VMIMO communication protocol is proposed by the cross-layer design to jointly improve the energy efficiency, reliability, and end-to-end (ETE) QoS provisioning in wireless sensor network (WSN). Simulation results of this work show the effectiveness of the proposed protocol in energy saving and QoS provisioning. Traditional VMIMO transmission schemes principally focus on maximizing the throughput of grouped mobile terminals without taking into account the quality-of-service (QoS) provisioning. The authors in [12] propose the optimal power allocation schemes with statistical QoS provisioning to maximize the effective capacity of non-collaborative/collaborative VMIMO wireless networks, respectively. Xu Hongli et al. study in [13] the problem of constructing an energy-efficient topology in wireless sensor networks using VMIMO communication. The simulation results show that the VMIMO topology control helps to decrease the power consumptions by approximately 32 % compared to the existing algorithms.

In [14], the author examine a tax inspired mechanism design to Achieve QoS in VMIMO Systems using a game theoretic approach. The utility function used in the study is defined so that the throughput used by the user is divided into two components: the throughput received from cellular Network, and throughput received from Virtual MIMO System. The study has shown that there is a unique constrained Nash equilibrium for the proposed game when the sum of demanded QoS for all users is bounded.

MIMO and smart antenna techniques have been extensively used as promising schemes to improve the spectrum efficiency. The authors in [15] propose a VMIMO scheme for direction of arrival estimation in which a set of user equipments are grouped together to simultaneously communicate with the base station on a given resource block and propose an automatic weighted subspace fitting algorithm that can detect the number of independent signals automatically and show accurate direction of arrival estimation. VMIMO systems present several challenges in order to improve the spatial diversity gain and spectrum efficiency. Karimi et al. [16] presents a novel solution that decomposes the VMIMO user grouping. The results of this solution under different network configuration demonstrate that it achieves much higher data throughput as compared to existing solutions.

Massive MIMO technology has received great attention recently. The main benefit of this technology can be realized only when the quality of estimated channel is ensured at both transmitter and receiver, by using a large number of antennas into the wireless transceiver. Sunho Park et al. [17] present an alternative channel estimation technique dealing with the pilot shortage in the massive MIMO systems. It is shown that the proposed method achieves substantial performance gain over conventional approaches employing pilot signals exclusively. The authors in [18] study energy-efficient data gathering in wireless sensor networks using VMIMO. The theoretical analysis and simulations show that the energy consumption decreases by 81 and 36 % compared to existing solutions. Wei Lu et al. [19] propose a compressed sensing feedback scheme for zero-forcing beam-forming in order to enhance the throughput in MIMO broadcast channel. Simulation-based results show that the proposed scheme has good performances compared to existing feedback schemes. In systems based on network MIMO, the interference is one of the major performance limiting factors to form a multi-cell virtual MIMO system. The authors in [20] develop a scalable interference coordination strategy of combining clustered network MIMO with fractional frequency reuse. It is shown that the cell rate performance of this new approach is better than that of the two existing strategies. The authors in [21] investigate the problem of distributed channel selection using a game-theoretic stochastic learning solution in an opportunistic spectrum access and propose a genie-aided algorithm to achieve the NE points under the assumption of perfect environment knowledge. The study shows that the SLA based learning algorithm achieves high system throughput with good fairness. In [22], the authors examine the problem of achieving global optimization for distributed channel selections in cognitive radio networks (CRNs), using game theoretic approach. The authors propose two special cases of local interaction game to study this problem. The results of this study shown that with the proposed games, global optimization is achieved with local information. In [23], the authors propose a scheme for channel access self-regulation. The main idea is to introduce hierarchy among mobile users. This scheme seems to behave well and succeeds to increase the bandwidth utilization, which may increase the number of mobile users to serve and therefore to enlarge the stability region.

Consider a wide geographical area served by a cellular network (3G, 4G, or 5G). We assume that mobile users are randomly distributed on a plane following a Poisson point process (PPP) with density λ. Each terminal is equipped by at least two separated network interfaces¹ (e.g., LTE and WLAN). This allows simultaneous or alternative connections to the two technologies according to the coverage criterion and offered services as well. Now, each mobile user could transmit/receive data over the two interfaces. Thereby, the average throughput of user i could be divided into two parts: the throughput allocated by the base station and the perceived throughput from other users over the WLAN radio. A key assumption of our analysis is that each user should cooperate with the others users. This induces a behavioral issue of cooperation among users. Indeed, if nodes refuse to cooperate, then our VMIMO-based scheme is just impossible to implement. The research community is invited to gather their input in order to design efficient incentive mechanisms for cooperation. A simple incentive mechanism is to simply say that if I cooperate today with others, I should almost surely get help from others tomorrow.

In this section, we seek to develop a fully distributed learning scheme to be implemented on each mobile terminal. This scheme will help to automatically learn the satisfactory solution derived above. We propose a fully distributed inspired from the well-known Banach-Picard learning algorithm [24]. Our algorithm is fully distributed in the sense that no external information is needed. Yet, a player does not need to observe the actions chosen by the other players in order to update its strategy iteratively. It only observes the realization of its own reward function that can be injected in a rule to predict its own strategy for next round.

The operator \({[\!\cdot ]_{0}^{1}}\) denotes the projection over the nonempty interval and continuous [ 0,1]. Denotes by f=(f _i ) _i∈N the right-hand side of Eq. 17. Then, the set of fixed points of f, denoted by f i x(f), belongs to the set of satisfactory solutions. The function f represents the response function associated to the payoff Θ. It is clear that, if the algorithm converges to some interior point x ^∗, then x ^∗ is a satisfactory solution. To prove this statement, consider a converging sequence to x ^∗, then the continuity of the projection map and the continuity of the payoff function implies the continuity of the quantity \(x_{i,t}\frac {d_{i}}{\Theta _{i,{t}}}\). The stationary equation is \(x_{i}^{*}=x_{i}^{*}\frac {d_{i}}{\Theta _{i}^{*}}\), i.e., the payoff of player i is \(\Theta _{i}^{*}=\Theta _{i}(x^{*})=d_{i}\). This means that every player i is satisfied at an interior steady state.

N is the processing gain and h _i the channel gain of mobile user i. Without loss of generality and only for simplicity purpose, we consider that h _i =h _j ∀i,j.

In the scenarios, we examine the impact of the sharing function in VMIMO system in three cases. In first case, the users have the same quality of service and the same WLAN radio condition (same success probability). In the second case, the users have the same quality of service but different WLAN radio condition (different success probabilities). In the third case, the users have different quality of service and the same WLAN radio condition.

Case 1: The users have the same quality of service and the same probability of success:

- ρ _i =0.5,d _i =70 Kbps for all users i.

In Fig. 3, we notice that users have the same self-proportion and the same sharing-proportion in the two scenarios. This is reasonable since that all users have the same quality of service and the same probability of success. Concerning the distribution of the sharing function, we notice that in the first scenario where the sharing proportion is divided equally between users, users prefer to share than to use the self-proportion. However, in the second scenario where the sharing proportion is divided in terms of the needs of each user, users prefer to use the self-proportion than to share.

Case 2 : The users have the same quality of service but different probability of success:

- ρ ₁=0.1,ρ ₂=0.3,ρ ₃=0.6,ρ ₄=0.9.

- d _i =70 Kbps for all users i.

In Figs. 4 and 5, we notice that the probability of success has an impact on the distribution of the proportion even if users have the same quality of service. This is justified by the fact that the user who has a high probability of success will tend to use the self-proportion more than to use the sharing-proportion and vice versa. In the first scenario where the sharing proportion is divided equally between users, users prefer to share than to use the self-proportion. In the second scenario where the sharing proportion is shared on the basis of the need of each user, we notice that users prefer to use the self-proportion than sharing proportion.

Case 3 : The users have different quality of service and the same probability of success:

- ρ _i = 0.5 for all users i.

- d ₁=40Kbps, d ₂=80kbps, d ₃=60Kbps, d ₄=100kbps.

In Figs. 6 and 7 we notice that the quality of service has great impact on the distribution of the proportion even if users have the same probability of success. This is justified by the fact that the user who needs a high quality of service will not share much and vice versa. In the first scenario where the sharing proportion is divided equally between users, users prefer to share than to use the self-proportion. In the second scenario where the sharing proportion is shared on the basis of the need of each user, we notice that users prefer to use the self-proportion than sharing proportion.

6.1 Results and analysis

As depicted in Fig. 8, the evolution of the cellular proportion of all users and convergence to stable proportion when they meet Nash equilibrium after less than 10 iterations.

Figure 9 shows the evolution of the demanded throughput (utility) per user.

The condition of Nash equilibrium has been meeting in iteration which is nearly 5, and each user has no intention to change his strategies to improve his throughput. This result observed after the iteration 5 is the best that each user can achieve.

¹ Nowadays, smart devices are usually equipped with several interfaces, e.g., 2G, 3G, 4G, WLAN, Bluetooth, NFS, and Ir).

² Note that power allocation problem is outside of the scope of this paper. However, fortunately and since our proposed learning algorithm presented in Section 5 is very fast, one could efficiently/dynamically combine the VMIMO game and the poser control game.