Generalized Max-Weight Policies in Stochastic Matching

We consider a matching system where items arrive one by one at each node of a compatibility network according to Poisson processes and depart from it as soon as they are matched to a compatible item. The matching policy considered is a generalized max-weight policy where decisions can be noisy. Additionally, some of the nodes may have impatience, that is, leave the system before being matched. Using specific properties of the max-weight policy, we construct a simple quadratic Lyapunov function. This allows us to establish stability results, to prove exponential convergence speed toward the stationary measure, and to give explicit bounds for the stationary mean of the largest queue size in the system. We finally illustrate some of these results using simulations on toy examples.