The “least flexible job first” rule in scheduling and in queueing

We consider an environment with m machines in parallel operating at different speeds. The processing requirements of all jobs are independent and have the same exponential distribution. Job j may only be processed on a specific subset of the m machines, referred to as its restricted set. The restricted sets are nested and preemptions are allowed. We show that the Least Flexible Job to the Fastest Machine (LFJ-FM) minimizes the expected makespan and the total expected completion time.