Stochastic Machine Minimization with Constant Service Times

We consider the nonpreemptive scheduling of n ≥ 1 jobs on identical, parallel machines. With job running times all given by some constant, the objective is to minimize the expected number of machines needed throughout a schedule subject to the following waiting-time constraints. At time 0, a timer with (random) initial value W is started; after time W all jobs must either be finished or running on a machine. The value of W is not known in advance, but its distribution is made available to the scheduler.

In general, if job running times are random, there might be situations in which, after running jobs on a given number of machines, it would be desirable to activate new machines, because the remaining times of unfinished jobs are stochastically too large compared to the time remaining on the timer. A principal result of this paper is that randomness of job running times is essential to such situations; if running times are deterministic, then irrespective of the distribution of W, an optimal policy initially assigns jobs to some number of machines k_n (which depends on the distribution of W), and maintains that number while there are jobs available, until the timer has expired, at which point any remaining, as yet unscheduled jobs are assigned to new, unused machines.

This paper also analyzes the dependence of k_n and the minimal expected cost on the parameters of the timer distribution when W is either exponentially distributed or uniformly distributed on a finite interval. The determination of k_n turns out to have an intriguing number-theoretic flavor.