Models and Algorithms for Stochastic Online Scheduling

Abstract

We consider a model for scheduling under uncertainty. In this model, we combine the main characteristics of online and stochastic scheduling in a simple and natural way. Job processing times are assumed to be stochastic, but in contrast to traditional stochastic scheduling models, we assume that jobs arrive online, and there is no knowledge about the jobs that will arrive in the future. The model incorporates both stochastic scheduling and online scheduling as a special case. The particular setting we consider is nonpreemptive parallel machine scheduling, with the objective to minimize the total weighted completion times of jobs. We analyze simple, combinatorial online scheduling policies for that model, and derive performance guarantees that match performance guarantees previously known for stochastic and online parallel machine scheduling, respectively. For processing times that follow new better than used in expectation (NBUE) distributions, we improve upon previously best-known performance bounds from stochastic scheduling, even though we consider a more general setting.

A Tight 2-Approximation for Preemptive Stochastic Scheduling

“…In general, all known performance guarantees for nonpreemptive policies depend on the distribution of processing times. This is also true for recent results for the online version of the stochastic scheduling model obtained by Megow et al [19], Chou et al [5], and Schulz [30]. All obtained results that include asymptotic optimality (Chou et al [5]) and approximation guarantees for deterministic as well as randomized policies (Megow et al [19], Schulz [30]) address nonpreemptive scheduling.…”

“…The performance guarantees they prove are functions of a parameter that bounds the squared coefficient of variation of processing times. For specific types of probability distributions of the processing times, stochastic policies with improved performance ratios were given in Megow et al [19] and Schulz [30]. In fact, these subsequent results also apply to the more general stochastic scheduling model with online job arrivals.…”

“…Thus, stochastic online policies are required to be online and nonanticipatory. We refer to Megow et al [19] for a more detailed discussion on stochastic online policies. As suggested in that paper, we use in this model a generalized definition of approximation guarantees from the stochastic scheduling setting by comparing the expected outcome of a nonanticipatory online policy with the expected outcome of an optimal nonanticipatory offline policy.…”

“…A full range of articles is concerned with criteria that guarantee the optimality of simple policies for special scheduling problems; see, e.g., Pinedo [26]. It is only recently that research also focuses on approximations for less restrictive problem settings (Möhring et al [24], Skutella and Uetz [37], Megow et al [19], Schulz [30], Dean et al [8]). All these results apply to nonpreemptive scheduling, and we are not aware of any approximation results when job preemption is allowed and processing times are stochastic.…”

“…The analysis of our policy is tight. Both, the policy and its analysis are also valid in the model of stochastic online scheduling (Megow et al [19], Chou et al [5], Vredeveld [39]). They rely on the celebrated Gittins index priority rule (Gittins [9,10]), which is an optimal policy for the setting when there is only one machine and there are no nontrivial release dates (Sevcik [33], Weiss [41]).…”