Large Deviations of Max-Weight Scheduling Policies on Convex Rate Regions

We consider a single server discrete-time system with a fixed number of users where the server picks operating points from a compact, convex, and coordinate convex set. For this system, we analyse the performance of a stabilising policy that at any given time picks operating points from the allowed rate region that maximise a weighted sum of rates, where the weights depend on the work loads of the users. Assuming a large deviations principle (LDP) for the arrival processes in the Skorohod space of functions that are right continuous with left-hand limits, we establish an LDP for the work load process using a generalised version of the contraction principle to derive the corresponding rate function. With the LDP result available, we then analyse the tail probabilities of the work loads under different buffering scenarios.