The "least flexible job first" rule in scheduling and in queueing

In this paper, we consider an environment with m machines (or servers) in parallel that operate at different speeds. The processing requirements of all the jobs (or customers) are independent and identically distributed according to the exponential distribution with mean 1. Job j may only be processed on a specific subset of the m machines, which is referred to as its restricted set. We assume that them restricted sets are nested. One job is said to be less flexible than another job if its restricted set is contained within the restricted set of the other job. We assume that preemptions are allowed. In this note, we show that in the preemptive setting the rule that assigns at each point in time a Least Flexible Job to the Fastest Machine (LFJ-FM) minimizes the expected makespan as well as the total expected completion time in a machine scheduling setting. Our result also implies that the LFJ-FM rule minimizes the expected waiting time of jobs in a queueing setting that allows preemptions and that is subject to an infinite arrival stream.