Optimality and Large Deviations of Queues under the pseudo-Log Rule Opportunistic Scheduling

We consider a wireless node shared by multiple user flows where the channel capacity available to each user varies randomly with time. A scheduling rule in this context selects which flow to serve based on the current channel state and user queues. This involves a tradeoff between maximizing current service rate (being opportunistic) versus balancing unequal queues (enhancing user-diversity to enable future high capacity opportunities). We propose a throughput-optimal scheduling rule, called the pseudo-Log (p-Log) rule, and show that, in the case of two users, it maximizes the asymptotic exponential decay rate of the sum-queue distribution. The proof relies on the radial sum-rate monotonicity (RSM) property satisfied by the p-Log rule, whereby as the queues scale up linearly, the scheduling rule de-emphasizes queue-balancing in favor of greedily maximizing the service rate. It also relies on refined sample path large deviation principle recently introduced by Stolyar to study such non-homogenous schedulers. In a companion paper we demonstrate via further analysis and simulations other virtues of RSM opportunistic schedulers (in particular the Log rule) in terms of minimizing overall mean delay, robustness to uncertainty in the traffic and channel statistics etc. The p-Log rule is a slight modification of the Log rule, for the sake of analytical convenience.