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We consider a discrete time stochastic queueing system where a controllermakes a 2-stage decision every slot. The decision at the first stage reveals ahidden source of randomness with a control-dependent (but unknown) probabilitydistribution. The decision at the second stage incurs a penalty vector thatdepends on this revealed randomness. The goal is to stabilize all queues andminimize a convex function of the time average penalty vector subject to anadditional set of time average penalty constraints. This setting fits a wideclass of stochastic optimization problems. This includes problems ofopportunistic scheduling in wireless networks, where a 2-stage decision aboutchannel measurement and packet transmission must be made every slot withoutknowledge of the underlying transmission success probabilities. We develop asimple max-weight algorithm that learns efficient behavior by averagingfunctionals of previous outcomes. The algorithm yields performance that can bepushed arbitrarily close to optimal, with a tradeoff in convergence time anddelay.

Max Weight Learning Algorithms with Application to Scheduling in Unknown Environments