A game theoretic optimization of the multi-channel ALOHA protocol

In this paper we consider the problem of distributed throughput maximization of networks with multi-channel ALOHA medium access protocol. In the multi-channel ALOHA protocol, each user tries to randomly access a channel using a probability vector defining the access probability to the various channels. First, we characterize the Nash Equilibrium Points (NEPs) of the network when users solve the unconstrained rate maximization. We show that in this case, for any NEP, each user's probability vector is a standard unit vector (i.e., each user tries to access a single channel with probability one and does not try to access other channels). Specifically, when the number of users, N, is equal to the number of channels there are N! NEPs. However, when the number of users is much larger than the number of channels, most of the users get a zero utility (due to collisions). To overcome this problem we propose to limit each user's total access probability and solve the problem under a total probability constraint. We characterize the NEPs when user rates are subject to a total transmission probability constraint. We propose a simple best-response algorithm that solves the constrained rate maximization, where each user updates its strategy using its local channel state information (CSI) and by monitoring the channel utilization. We prove that the constrained rate maximization can be formulated as an exact potential game. This implies that convergence of the proposed algorithm is guaranteed. Finally, we provide numerical examples to demonstrate the algorithm's performance.