MaxWeight scheduling: Smoothness of the service process

The model is a generalized switch, serving multiple traffic flows in discrete time. The switch uses MaxWeight algorithm to make a service decision (scheduling choice) at each time step, depending on the current queue lengths. In some applications, it is not important to keep the queue lengths/delays small (e.g., when queues are virtual, rather than physical), but is important that the service processes provided to each flow remains smooth (i.e., without large gaps in service) even when the switch is heavily loaded. Addressing this question reduces to the analysis of the asymptotic behavior of the unscaled queue-differential process in heavy traffic. We prove that the stationary regime of this process converges to that of a positive recurrent Markov chain, whose structure we explicitly describe. This in turn implies smoothness of the service processes.