Towards Maximizing Nonlinear Delay-Sensitive Rewards in Queuing Systems

We consider maximizing the long-term average reward in a single server queuing system, where the reward obtained for a job is a non-increasing, possibly nonlinear function of its sojourn time. The motivation behind this work comes from delay-sensitive applications, including quantum information processing and multimedia streaming. Although the goal of optimizing the total sojourn time is well-studied, optimizing a nonlinear function of the sojourn times remains unexplored to the best of our knowledge. We consider two arrival models - the first is a ‘burst arrival’ model, wherein all jobs arrive at the server at the same instant. We show that shortest job first (SJF) maximizes the average reward for any monotonic function of the sojourn times. In the second setting, jobs arrive according to some stochastic process with i.i.d. service requirements. This setting is significantly more challenging to analyze, and identifying an optimal discipline remains elusive. We introduce a new service discipline, shortest predicted sojourn time (SPST), and provide analytical guarantees under specific settings. Numerically, we demonstrate that SPST outperforms well-known disciplines across multiple settings.