Stochastic Scheduling with Abandonments via Greedy Strategies

Motivated by applications where impatience is pervasive and service times are uncertain, we study a scheduling model where jobs may depart at an unknown point in time and service times are stochastic. Initially, we have access to a single server and $n$ jobs with known non-negative values: these jobs have unknown stochastic service and departure times with known distributional information, which we assume to be independent. When the server is free, we can run an available job which occupies the server for an unknown amount of time, and collect its value. The objective is to maximize the expected total value obtained from jobs run on the server. Natural formulations of this problem suffer from the curse of dimensionality. In fact, this problem is NP-hard even in the deterministic case. Hence, we focus on efficiently computable approximation algorithms that can provide high expected reward compared to the optimal expected value. Towards this end, we first provide a compact linear programming (LP) relaxation that gives an upper bound on the expected value obtained by the optimal policy. Then we design a polynomial-time algorithm that is nearly a $(1/2)\cdot (1-1/e)$-approximation to the optimal LP value (so also to the optimal expected value). We next shift our focus to the case of independent and identically distributed (i.i.d.) service times. In this case, we show that the greedy policy that always runs the highest-valued job whenever the server is free obtains a $1/2$-approximation to the optimal expected value. Our approaches extend effortlessly and we demonstrate their flexibility by providing approximations to natural extensions of our problem. Finally, we evaluate our LP-based policies and the greedy policy empirically on synthetic and real datasets.