Large Deviations Of Max-Weight Scheduling Policies

We consider a single-server discrete-time system with $K$ users where the server picks operating points from a compact, convex and coordinate convex set in $\Re_+^K$. 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. In particular, we are interested in a Large Deviations based analysis of this policy, and under both the ``large-buffer" and ``many-sources" regimes. The unifying theme of this work is to prove a Large Deviations Principle (LDP) for the queueing process using an appropriate generalization of the contraction principle, namely, Puhalskii's extended contraction principle and Garcia's extended contraction principle.