Nonhomogeneous Place-dependent Markov Chains, Unsynchronised AIMD, and Optimisation

A stochastic algorithm is presented for a class of optimisation problems that arise when a group of agents compete to share a single constrained resource in an optimal manner. The approach uses intermittent single-bit feedback, which indicates a constraint violation and does not require inter-agent communication. The algorithm is based on a positive matrix model of AIMD, which is extended to the nonhomogeneous Markovian case. The key feature is the assignment of back-off probabilities to the individual agents as a function of the past average access to the resource. This leads to a nonhomogeneous Markov chain in an extended state space, and we show almost sure convergence of the average access to the social optimum.