Scheduling Servers with Stochastic Bilinear Rewards

We address a control system optimization problem that arises in multi-class, multi-server queueing system scheduling with uncertainty. In this scenario, jobs incur holding costs while awaiting completion, and job-server assignments yield observable stochastic rewards with unknown mean values. The rewards for job-server assignments are assumed to follow a bilinear model with respect to features characterizing jobs and servers. Our objective is regret minimization, aiming to maximize the cumulative reward of job-server assignments over a time horizon while maintaining a bounded total job holding cost, thus ensuring queueing system stability. This problem is motivated by applications in computing services and online platforms. To address this problem, we propose a scheduling algorithm based on weighted proportional fair allocation criteria augmented with marginal costs for reward maximization, incorporating a bandit strategy. Our algorithm achieves sub-linear regret and sub-linear mean holding cost (and queue length bound) with respect to the time horizon, thus guaranteeing queueing system stability. Additionally, we establish stability conditions for distributed iterative algorithms for computing allocations, which are relevant to large-scale system applications. Finally, we validate the efficiency of our algorithm through numerical experiments.