Large deviations of max-weight scheduling policies on convex rate regions

We consider a single server discrete-time system with K users where the server picks operating points from a compact, convex and co-ordinate convex set in R₊ ^K. For this system we analyse the performance of a stablising policy that at any given time picks operating points from the allowed rate region that maximise a weighted sum of rate, where the weights depend upon the workloads 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 workload 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 workloads under different buffering scenarios.