On the optimality of the earliest due date rule in stochastic scheduling and in queueing

Abstract

We consider an environment with m servers in parallel and n jobs. The n jobs have i.i.d. exponentially distributed processing requirements. The servers operate at different speeds and preemptions are allowed. The jobs have random release times that are not known in advance, but whenever a job is released, its due date is fixed and becomes known to the scheduler. We first consider three stochastic scheduling problems with three due date related objective functions and consider variations of the Earliest Due Date (EDD) rule, including the preemptive policy that at any point in time assigns the job with the Earliest Due Date to the Fastest Server (EDD-FS). Our optimality results turn out to be examples of stochastic scheduling problems that have relatively simple priority rules that are optimal while their deterministic counterparts do not allow such simple priority rules to be optimal. We furthermore extend our results to include a priority queueing model with m exponential servers that operate at different speeds and preemptions being allowed. The jobs arrive according to an arbitrary renewal process, and we assume that the utilization factor of the system is less than 1. Upon a job’s release the time difference between its due date and release time is set by the draw of a random variable from a given distribution. At a job’s release time, its due date is then immediately fixed and known while its actual processing time only becomes known upon its service completion. We show that for this priority queueing model the preemptive EDD-FS rule also minimizes the limit of several due date related objective functions as the number of jobs converges to infinity.