Service Scheduling for Random Requests With Deadlines and Linear Waiting Costs

We study service scheduling problems in a slotted system where jobs arrive according to a Bernoulli process and leave within two slots after arrival. Service costs are quadratic in service rates, and there is also a linear waiting cost. We frame the problems as average cost Markov decision processes. While the studied system is a linear system with quadratic costs, it has state-dependent control and a non-standard cost function structure, rendering the optimization problem complex. We obtain explicit optimal policies in the case when all the jobs are of the same size. In particular, we show that the optimal policy is linear or piece-wise linear in the system state, depending on the system parameters. We then consider a scenario where each service request comes from a rational agent interested in optimizing his/her service and waiting cost, and we obtain a symmetric Nash equilibrium. We extend our study to a scenario where job sizes can take distinct values, and job arrivals constitute a Markov chain. Here, we provide an algorithm that yields the optimal policy, but it is of exponential complexity. Finally, we propose an approximate policy of linear complexity for general job size distributions and derive its performance bound.