Scheduling with Random Arrivals and Linear Loss Functions

The problem under consideration involves scheduling of the processing of an initial queue of jobs and subsequent Poisson arrivals on a single processor. Each job to be processed incurs a loss which increases linearly with its waiting time. The scheduling algorithm is sought which minimizes the average rate of expected loss over infinite time. It is shown that if statistical equilibrium exists for the total loss of an individual arrival, the optimal schedule minimizes the expected total loss of a single arrival, and is given by the scheduling rule which applies when there are no additional arrivals.