Achieving QoS in Virtual MIMO systems: a satisfaction equilibrium analysis
 Hassan Bennani^{1},
 Essaid Sabir^{2}Email author,
 Abdellatif Kobbane^{1},
 Jalel BenOthman^{3} and
 Mohamed Elkoutbi^{1}
https://doi.org/10.1186/s1363801606126
© Bennani et al. 2016
Received: 12 March 2015
Accepted: 13 April 2016
Published: 6 May 2016
Abstract
This paper presents a game theoretic framework to analyze a cellularWLAN heterogeneous network from a virtual multipleinput multipleoutput (VMIMO) perspective. Namely, we restrict to the uplink case and consider a noncooperative game where each user seeks to meet some quality of service (QoS). Moreover, mobile users are allowed to reinject, through their WLAN interfaces, a part of their throughput in the induced game. Thus, the interaction among users defines a distributed VMIMO with a possibly throughput exchange. This mechanism will help all users to meet their respective QoS expressed by the perceived average throughput. The average throughput is considered to be the utility function used in this framework. Now, each mobile user experiences a certain throughput composed of two parts: (1) the throughput received from cellular subsystem and (2) the throughput received from WLAN subsystem. Naturally, we use the concept of satisfaction equilibrium to predict the behavior of the network. Indeed, we provided a sufficient condition for the existence of such an equilibrium. Next, we prove the uniqueness of the equilibrium and compute it explicitly. Afterward, we propose a fully distributed algorithm inspired from the wellknown BanachPicard learning algorithm. Our scheme has many good features facilitating its implementation and usability. Indeed, it accurately converges to the equilibrium (if exists), it is very fast, and it requires no external information. Simulation results validate the algorithm and show its robustness and illustrate numerically the proposed learning scheme for qualityofservice management in such a heterogeneous network.
Keywords
1 Introduction
Nowadays, users increasingly need higher throughput for better use of applications and new services such as streaming videos, files transfer, and 3D gaming. Through time, many techniques have been developed to satisfy that need. Among the techniques developed are frequency division multiple access (FDMA) and time division multiple access (TDMA) in the 2G, code division multiple access (CDMA) in 3G, and orthogonal frequency division multiple access (OFDMA) and multipleinput multipleoutput (MIMO) in 4G.
MIMO technology requires two or more antennas at transmission and two or more antennas at reception. This mode is not just a combination between MISO and SIMO because multiple data streams are transmitted simultaneously in the same frequency and time, taking full advantage of the different paths in the radio channel. MIMO have been amply recognized as promising techniques in order to enhance and manage effectively the spectral and resources mobile communication systems. Several international standards such as LTEA (LTEadvanced), 3GPP LTE (thirdgeneration partnership project longterm evolution), and IEEE 802.16 e/m have supported the MIMO techniques. However, some challenges such as implementing a conventional antenna seem a problematic challenge due to very small size mobile transmitter/receiver. The spatial diversity can be realized by cooperation between mobile terminals, to form a relaying system through a multiple single antenna mobile. This mechanism is called a virtual MIMO system (VMIMO). In this system, the terminal antennas form a virtual antenna array with a virtual joint. Nowadays, the research area around VMIMO know a significant grow for the purpose to study and understand it. For the fourthgeneration and fifthgeneration mobile communications technologies, VMIMO is considered one of the most crucial technologies enabling new paradigms such Internet of Things and devicetodevice communications.
The rest of this manuscript is organized as follows: Section 2 discusses and presents a detailed review on related literature. Section 3 presents a simple noncooperative game for the VMIMO system with possible throughput reinjection. Next, we provide a complete analysis of the induced game and characterize the equilibria points (existence, uniqueness, and computation) in Section 4. A fully distributed algorithm to discover the satisfaction equilibrium is presented in Section 5. Extensive numerical examples and simulations are given in Section 6. We finally draw a conclusion and give some future directions in Section 7.
2 Related work
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 noncooperative Nperson 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 multicell 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 noncooperative 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 codedivision multiple access system, in order to maximize the total system throughput by means of power control is presented. Objectives of most radio resourcemanagement schemes can be classified as either usercentric or networkcentric. Authors in [7] consider the joint optimization of both usercentric and networkcentric metrics. They use a utility function as the usercentric metric and introduce an explicit pricing mechanism to mediate between the usercentric and networkcentric resourcemanagement problems. F. Meshkati et al. propose in [8] a power control game for MCCDMA 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 semianalytic 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 multihop VMIMO communication protocol is proposed by the crosslayer design to jointly improve the energy efficiency, reliability, and endtoend (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 qualityofservice (QoS) provisioning. The authors in [12] propose the optimal power allocation schemes with statistical QoS provisioning to maximize the effective capacity of noncollaborative/collaborative VMIMO wireless networks, respectively. Xu Hongli et al. study in [13] the problem of constructing an energyefficient 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 energyefficient 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 zeroforcing beamforming in order to enhance the throughput in MIMO broadcast channel. Simulationbased 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 multicell 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 gametheoretic stochastic learning solution in an opportunistic spectrum access and propose a genieaided 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 selfregulation. 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.

We design a new taxinspired mechanism appealing for cooperation in heterogeneous networks.

We build a simple game theoretic framework to analyze the behavior of mobile users in VMIMO systems.

To the best of our knowledge, this paper is one of the first work to deal with autonomic QoS provisioning in VMIMO systems.

Our framework is quite simple but is adapted for devicetodevice, machinetomachine and autonomic communications; yet, both links device/basestation and device/device are modelled in a simple way.

A full characterization (existence, uniqueness, computation, and convergence) of the satisfaction equilibrium has been provided.

We proposed a fully distributed (no external information is required) algorithm to discover the satisfaction algorithm.
3 Problem formulation
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 interfaces^{1} (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 VMIMObased 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.
3.1 Throughput received from cellular network
where L and M denote, respectively, the number of information bits and the total number of bits per packet. R _{ i } is the transmission rate and γ _{ i } denotes the signal to interference ratio (SIR)^{2} of user i. Whereas f(·) stands for the efficiency function representing the packet success rate (PSR), i.e., the probability that a packet is received without an error. We suppose that if a packet has one or more bit errors, it will be retransmitted entirely. The efficiency function, f(γ _{ i }), is assumed to be increasing and Sshaped (sigmoidal) with f(∞)=1. We also require that f(0)=0 to ensure that \({T_{i}^{C}}=0\) when γ _{ i }=0.
3.2 Throughput received from VMIMO (through WLAN)
3.3 Average throughput
In our VMIMO scheme, we consider that each user may experience a different allocated throughput by the base station. These differences are due to the running service, geographic position of mobile users, channel conditions, transmit power, and many other reasons. We assume along this paper that each user runs a service with some QoS requirements. For simplicity, let us consider that the QoS metric is the throughput demand that should be met in order for the service to have an acceptable quality of service and an acceptable quality of experience as well. Obviously, our study holds for other QoS metrics such as delay or jitter.
When negotiating transmission parameters needed to achieve a good service, the base station will compute the transmission power and the resource scheduling. Let d _{ i } be the throughput demand required by user i, and let D be the vector of all throughput demands. After being accepted in the network, user i would be informed by appropriate parameters’ values to use, e.g., the transmit power for uplink. In reality, it is almost impossible to achieve exactly the required throughput d _{ i }. Thus, user i would receive average throughput d _{ i }±Δ d _{ i }. Thereby, the received throughput would be either less than or greater than what we asked for. The main two ideas of this work are (1) to give the throughput surplus to other users that need a bit and (2) to ask others to give us throughput when needed.
where \((1x_{j}){T_{j}^{C}}\) is the proportion of cellular throughput allocated by user j to its neighboring users. But, the effective throughput reinjected by user j in the system is only equal to \((1x_{j}){T_{j}^{W}} = (1x_{j})\rho _{j} {T_{j}^{C}}\) (collisions effect). The function π _{ j,i }(·) defines how much virtual throughput offered by user j should be allocated to its neighboring node i. To finetune the virtual throughput allocation, we shall assume that the sharing distribution depends on the whole demand vector D=(d _{1},d _{2},⋯,d _{ m }). Other criteria could apply as well to the sharing function.
4 VMIMO as a noncooperative game
4.1 The noncooperative game
We consider the simple form of the interaction among mobile users, where they only can choose the amount of throughput to keep (i.e., x _{ i }) and thereafter the amount of throughput to inject in the system (i.e., 1−x _{ i }). Power control is assumed to be run by the base station. Each mobile user will then face the problem of deciding the appropriate amount of throughput to allocate to others. We propose a noncooperative game where each user (selfishly) decides the amount of throughput to yield to others in order to satisfy its throughput requirement d _{ i }.
We denote the VMIMO game on its normal form as \(\mathcal {G}=[\Omega,\mathcal {A},\mathbf {\Theta }, \mathbf {D}]\), where Ω={1,..,m} is the set of players (mobile users), \(\mathcal {A}=[0,1]^{m}\) is the space of strategies set for all users, and Θ=[Θ _{1},Θ _{2},⋯,Θ _{ m }] is the vector of payoff functions. In our setting, the average throughput will be considered as the utility function. D=(d _{1},d _{2},⋯,d _{ m }) denotes the throughput demand vector which defines the set of objectives (QoS) to meet by mobile users.

Step 1 : Mobile user i receives an average throughput \({T_{i}^{C}}\) from the cellular networks. This received throughput may be greater than, less than, or equal to its required throughput demand d _{ i }.

Step 2 : Mobile user i decides the amount of throughput x _{ i } to keep for itself. Then 1−x _{ i } represents the amount of throughput to reinject into the system through WLAN interface. We refer to this as a virtual throughput since it will be shared according to some sharing rule by the other mobile users.

Step 3 : Each mobile user will finetune unilaterally the amount of throughput to keep (alternatively to reinject into the WLAN) until its average total throughput is equal to its throughput demand.
4.2 Feasible strategy and satisfactory solution
When one imposes constraints over the payoff functions that each mobile user obtains or over the action that a player can choose in the game \(\mathcal {G}\), it becomes plausible to replace the Nash equilibrium concept by a constrained (Nash) equilibrium. Here, in the presence of QoS constraints, the set of individual actions reduces to the set of actions which verifies the constraints given the actions adopted by the other mobile users.
Definition 1.
From now on, each mobile user will seek to achieve its throughput demand by finetuning the amount of throughput to reinject in the WLAN subsystem.
Definition 2.
 (a)All individual throughput demand are met. Namely,$$ \mathbf \Theta_{i}(\mathbf X^{*}) \geq d_{i}, \qquad\forall i, $$(8)
and
 (b)
X ^{∗} is a feasible strategy profile.
In the remainder, we will be interested in the case of strict equality. This is not a restriction but a plausible point that has sense if the required demand is equal to the rate at which packets are generated. Moreover, energy efficiency considerations can also be a good reason to only address the case of strict equality.
4.3 Existence and uniqueness of a SNEP
Lemma 1.
If A is a Cramer invertible matrix and \(\mathbf {A}^{1}\cdot \underline {\mathbf {D}} \in [\!0, 1]^{m}\). Then, the game \(\mathcal {G}\) has a unique pure satisfaction equilibrium given by \(\mathbf {X}^{*}=\mathbf {A}^{1}\underline {\mathbf {D}}\).
Proof.
It is straightforward that the column and the row vectors of the matrix A are noncollinear, therefore it is a Cramer matrix with nonzero determinant. Henceforth, the system has a unique solution given by \(\mathbf {X}^{*}=\mathbf {A}^{1}\underline {\mathbf {D}}\).
4.4 A sufficient condition for existence of a symmetric SNEP
Now, we seek to derive a sufficient condition for existence of a satisfaction equilibrium. For that, we imagine an equivalent network with symmetric setting, with same radio conditions and with same parameters for all mobile users. This symmetric network has the same number of users and will only be used to derive an upper bound on the total demand such that an satisfaction equilibrium still exists. We set for the equivalent network R _{ i }=R _{ j }=R, L _{ i }=L _{ j }=L, d _{ i }=d _{ j }=d, ρ _{ i }=ρ _{ j }=ρ, f(γ _{ i })=f(γ _{ j })=f(γ), and π _{ i,j }(·)=π _{ j,i }(·)=π(·)∀i,j. Here, one consider the worst case where all mobile users ask for a throughput demand \(d=\max \limits _{i} d_{i}\). Note that the total demand generated by the symmetric network is m·d.
The idea behind derivation of our sufficient condition is the following simple reasoning: A unique satisfaction equilibrium will exist if and only if the total throughput demand \(\sum \limits _{i=1}^{i=m}d_{i}\) does not exceed the achievable throughput (total capacity) of the network. We have the following result:
Proposition 1.
Proof.

Case 1 : (1−(m−1)ρπ(d))≥0

In this case Ξ _{ sym } reaches its maximum when x is equal to 1, i.e., all the users will satisfy their throughput demand from the base station. Here, no throughput reinjection is required to achieve the satisfaction equilibrium. Replacing x by 1 in (13) yields$$ \Xi_{\text{sym}}^{1}=\frac{mLR}{M}f(\gamma). $$(14)

Case 2 : (1−(m−1)ρπ(d))≤0

Now the total throughput Ξ _{sym} is maximized when x is set to 0. All mobile users will satisfy their whole throughput demand from the WLAN subsystem, i.e., after reinjecting their whole throughput received first from the cellular subsystem. We have the following$$ \Xi_{\text{sym}}^{0}=\frac{m(m1)LR}{M}\rho f(\gamma)\pi(q). $$(15)
which completes the proof.
5 BanachPicard learning algorithm
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 wellknown BanachPicard 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 righthand 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.
6 Simulation and results
Simulation parameters
Parameter  Value 

p _{ i } (power of user i)  7 W 
σ ^{2} (Gaussian noise)  25×10^{−16} W 
R  100 Kbps 
M  100 
L  100 
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.
 1.Flatrate sharing: Here, the amount of throughput to be reinjected in the system through WLAN is equally shared among the other users. The explicit formula of the sharing function can write$$ \pi_{i,j}(d_{1}, d_{2}, \cdots, d_{m})=\frac{1}{m1}. $$(19)
 2.Proportional sharing : Now, the throughput to be reinjected through WLAN will be shared among other users taking into account their respective throughput demands. The expression of the sharing function in this scenario is the following:$$ \pi_{i,j}(d_{1}, d_{2}, \cdots, d_{m})=\frac{d_{j}}{\sum\limits_{k\neq i}d_{k}}. $$(20)
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.
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.
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.
Simulation parameters
Parameter  Value 

p _{ i } (power of user i)  7 W 
σ ^{2} (Gaussian noise)  25 × 10^{−16} W 
R  100 Kbps 
M  100 
L  100 
d1  60 Kbps 
d2  20 Kbps 
d3  40 Kbps 
d4  75 Kbps 
ρ _{1}  0.2 
ρ _{2}  0.2 
ρ _{3}  0.2 
ρ _{4}  0.2 
6.1 Results and analysis
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.
7 Conclusions
In this work, we studied a constrained noncooperative game analysis to achieve QoS in VMIMO systems using a approach. A noncooperative game is proposed in order to study the interaction among users, where each user (selfishly) decides the appropriate amount of throughput to reinject in the WLAN network in order to satisfy its throughput requirement. Yet, the equilibrium is achieved when demanded QoS is met for all users. We showed that there is a unique satisfaction equilibrium for the proposed game; we characterized it and computed it explicitly for general case. Next, we proposed a fully distributed algorithm to learn the satisfaction equilibrium. Then, using the proposed scheme over an uplink virtual MIMO system with m users, we discussed the effect of asked QoS and the WLAN radio condition that have a substantial impact on the amount of throughput to be reinjected in the system.
Next, our analytical results are corroborated with extensive simulation results. We considered many schemes regarding the sharing function. This latter function shows how reinjected throughput must be shared among users inside WLAN range of injecting user. In particular, we analyzed the case of flatrate sharing and the case of proportional sharing. We found that the results achieved in the proportional sharing scenario are more interesting compared to those achieved in the flatrate scenario. Moreover, proportional sharing scheme presents better fairness properties; this is because the distribution of the sharingrate depends on the demand of each user. Finally, simulations show that our BanachPicardinspired algorithm converges well and super fast to the satisfaction equilibrium which is very appreciated in this kind of communications.
As a future direction, we are working towards adapting our model to support dense and ultra dense networks such as Massive MIMO and ultra dense devicetodevice communications using evolutionary game theory and mean field dynamic game theory.
8 Endnotes
^{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.
Declarations
Acknowledgements
We would like to thank so much the anonymous referees for their effort, their time, and their valuable inputs that invariably improve this article. We also would like to thank Professor Tembiné Hamidou (from New York University) for his very constructive comments and suggestions.
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
References
 H Ji, CY Huang, Noncooperative uplink power control in cellular radio systems. Wireless Netw. 4:, 233–240 (1998).View ArticleGoogle Scholar
 DJ Goodman, NB Mandayam, in Vehicular Technology Conference, 2001. VTC 2001 Spring. IEEE VTS 53rd, 2. Network Assisted Power Control for Wireless Data, (2001), pp. 1022–1026. doi:http://dx.doi.org/10.1109/VETECS.2001.944533 ISSN:10903038.
 CU Saraydar, NB Mandayam, DJ Goodman, Pricing and power control in a multicell wireless data network. IEEE J. Selected Areas Commun. 19:, 1883–1892 (2001).View ArticleGoogle Scholar
 CU Saraydar, NB Mandayam, DJ Goodman, Efficient power control via pricing in wireless data networks. IEEE Trans. Commun. 50:, 291–303 (2002).View ArticleGoogle Scholar
 T Alpcan, T Basar, R Srikant, E Altman, in Decision and Control, 2001. Proceedings of the 40th IEEE Conference on, 1. CDMA Uplink Power Control as a Noncooperative Game, (2001), pp. 197–202. doi:http://dx.doi.org/10.1109/.2001.980097.
 CW Sung, WS Wong, A noncooperative power control game for multirate CDMA data networks. IEEE Trans. Wireless Commun. 2:, 186–194 (2003).View ArticleGoogle Scholar
 N Feng, SC Mau, NB Mandayam, Pricing and power control for joint networkcentric and usercentric radio resource management. IEEE Trans. Commun. 52:, 1547–1557 (2004).View ArticleGoogle Scholar
 F Meshkati, M Chiang, SC Schwartz, HV Poor, NB Mandayam, in IEEE Wireless Communications and Networking Conference, 2005, 1. A Noncooperative Power Control Game for Multicarrier CDMA Systems (New Orleans, LA, USA, 2005), pp. 606–611. doi:http://dx.doi.org/10.1109/WCNC.2005.1424570 ISSN:15253511.
 S Cui, AJ Goldsmith, A Bahai, Energyefficiency of MIMO and cooperative MIMO techniques in sensor networks. IEEE J. Select. Areas. Commun. 22(6), 1089–1098 (2003).View ArticleGoogle Scholar
 SK Jayaweera, Virtual MIMObased cooperative communication for energyconstrained wireless sensor networks. IEEE Trans. Wireless Commun. 5(5), 984–989 (2008).View ArticleGoogle Scholar
 Y Yuan, ZH He, M Chen, Virtual MIMObased crosslayer design for wireless sensor networks. IEEE Trans. Vehicular Technol. 55(3), 856–864 (2009).View ArticleGoogle Scholar
 W Cheng, X Zhang, H Zhang, QoSaware power allocations for maximizing effective capacity over virtualMIMO wireless networks. Selected Areas Commun. IEEE J. 31:, 2043–2057 (2013).View ArticleGoogle Scholar
 X Hongli, H Liusheng, Q Chunming, W Xinglong, S Yue, Topology control with vMIMO communication in wireless sensor networks. Wireless Commun. IEEE Trans. 12:, 6328–6339 (2013).View ArticleGoogle Scholar
 H Bennani, E Sabir, A Kobbane, A Walid, J BenOthman, in 2014 International Wireless Communications and Mobile Computing Conference (IWCMC). A TaxInspired Mechanism Design to Achieve QoS in VMIMO systems: Give to Receive (Nicosia, Cyprus, 2014), pp. 56–62. doi:http://dx.doi.org/10.1109/IWCMC.2014.6906332 ISSN:23766492.
 H Chen, Z Pan, L Tian, J Shi, G Yang, M Suzuki, A novel AWSF algorithm for doa estimation in virtual MIMO systems. Selected Areas Commun. IEEE J. 31:, 1994–2003 (2013).View ArticleGoogle Scholar
 OB Karimi, MA Toutounchian, J Liu, C Wang, Lightweight user grouping with flexible degrees of freedom in virtual MIMO, selected areas in communications. IEEE J. 31:, 2004–2012 (2013).Google Scholar
 S Park, JW Choi, JY Seol, B Shim, in Information Theory and Applications Workshop (ITA), 2014. Virtual pilot signal for massive MIMOOFDM systems, (2014). doi:http://dx.doi.org/10.1109/ITA.2014.6804285.
 H Xu, L Huang, C Qiao, W Dai, S Yue, Joint virtual MIMO and data gathering for wireless sensor networks. Parallel Distributed Syst. IEEE Trans. 99:, 1 (2014).Google Scholar
 W Lu, Y Liu, D Wang, Efficient feedback scheme based on compressed sensing in MIMO wireless networks. J. Comput. Electr. Eng. 39:, 1587–1600 (2013).View ArticleGoogle Scholar
 A Thampi, S Armour, Z Fan, D Kaleshi, in European Wireless 2014; 20th European Wireless Conference; Proceedings of. Clustered Network MIMO and Fractional Frequency Reuse for the Downlink in LTEA Systems, (2014), pp. 1–6.Google Scholar
 Y Xu, J Wang, Q Wu, A Anpalagan, YD Yao, Opportunistic spectrum access in unknown dynamic environment: A game theoretic stochastic learning solution. IEEE Trans. Wireless Commun. 11(4), 1380–1391 (2012).View ArticleGoogle Scholar
 Y Xu, J Wang, Q Wu, A Anpalagan, YD Yao, Opportunistic spectrum access in cognitive radio networks: Global optimization using local interaction games. IEEE J. Selected Topics Signal Process. 6(2), 180–194 (2012).View ArticleGoogle Scholar
 E Sabir, R ElAzouzi, Y Hayel, Hierarchy sustains partial cooperation and induces a Braesslike paradox in slotted alohabased networks. Comput. Commun. 35(3), 273–286 (2012).View ArticleGoogle Scholar
 H Tembine, R Tempone, P Vilanova, in 52nd IEEE Conference on Decision and Control. MeanField Learning for Satisfactory Solutions, (2013), pp. 4871–4876. doi:http://dx.doi.org/10.1109/CDC.2013.6760653 ISSN:01912216.