# Scheduling Stochastic Jobs on Parallel Machines to Minimize Makespan or Flowtime

## Abstract

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 the jobs’ makespan or flowtime. The amount of processing required to complete the jobs have known probability distributions. It has been established by several researchers that when the required amounts of processing are all distributed as exponential random variables, then the strategy (LEPT) of always processing jobs with the longest expected processing times minimizes the expected value of the makespan, and the strategy (SEPT) of always processing jobs with the shortest expected processing times minimizes the expected value of the flowtime. We prove these results and describe a more general instance in which they are also true: when the jobs have received differing amounts of processing prior to the start, their total processing requirements are identically distributed, and the common distribution of total processing requirements has a monotone hazard rate. Under the stronger assumption that the distribution of the total processing requirements has a density whose logarithm is concave or convex, LEPT and SEPT minimize the makespan and flowtime in distribution.

### Keywords

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