Scheduling Jobs with Stochastic Processing Requirements on Parallel Machines to Minimize Makespan or Flowtime

A number of identical machines operating in parallel are to be used to complete the processing of a collection of jobs so as to minimize either the jobs' makespan or flowtime. The total processing required to complete each job has the same probability distribution, but some jobs may have received differing amounts of processing prior to the start. When the distribution has a monotone hazard rate the expected value of the makespan (flowtime) is minimized by a strategy which always processes those jobs with the least (greatest) hazard rates. When the distribution has a density whose logarithm is concave or convex these strategies minimize the makespan and flowtime in distribution. These results are also true when the processing requirements are distributed as exponential random variables with different parameters. DYNAMIC PROGRAMMING; FLOWTIME; MAKESPAN; MONOTONE HAZARD RATE; OPTIMAL CONTROL; SIGN-CONSISTENT FREQUENCY; STOCHASTIC SCHEDULING 1. Scheduling to minimize makespan or flowtime 1.1. Stochastic processing requirements. A number of identical machines operating in parallel are available for processing a collection of jobs. The total processing required to complete each job has the same probability distribution, but some jobs may have received differing amounts of processing prior to the start. The objective is to complete the jobs so as to minimize either their makespan or flowtime. Let Ci be the time at which job i is completed. The makespan is the time of the last job completion, max{C }, and the flowtime is the sum of the job completion times, I C. Both are random variables which depend on the strategy used to order the processing of the jobs on the machines. Preemptive scheduling is permitted, and thus any job may instantaneously be removed from a machine and another job processed instead. A single machine may process several jobs simultaneously, provided its rates of processing those jobs sum to no more than 1, the maximum rate at which a single machine can work. Received 10 September 1980. * Postal address: Queens' College, Cambridge CB3 9ET, U.K.