@article {Bennani20161,
	title = {Achieving QoS in Virtual MIMO systems: a satisfaction equilibrium analysis},
	journal = {Eurasip Journal on Wireless Communications and Networking},
	volume = {2016},
	number = {1},
	year = {2016},
	note = {cited By 0},
	pages = {1-13},
	abstract = {This paper presents a game theoretic framework to analyze a cellular-WLAN heterogeneous network from a virtual multiple-input multiple-output (VMIMO) perspective. Namely, we restrict to the uplink case and consider a non-cooperative game where each user seeks to meet some quality of service (QoS). Moreover, mobile users are allowed to re-inject, 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 well-known Banach-Picard 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 quality-of-service management in such a heterogeneous network. {\textcopyright} 2016, Bennani et al.},
	doi = {10.1186/s13638-016-0612-6},
	url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-84966280664\&doi=10.1186\%2fs13638-016-0612-6\&partnerID=40\&md5=e67ac437a619c2d9ac7539dace4426e6},
	author = {Bennani, H.a and Sabir, E.b and Kobbane, A.a and Ben-Othman, J.c and El-koutbi, M.a}
}
