Applications of order statistics to queueing and scheduling

Abstract

We prove several basic combinatorial identities and use them in two applications: the queue inference engine and earliest due date (EDD) scheduling. We generalize the standard order statistics result for Poisson processes, and show how to sample a busy period in the M/M/c system. We obtain simple expressions for the variance of the total waiting time in the M/M/c and M/D/1 queues given that n Poisson arrivals and departures occur during a busy period of length t. We also perform a probabilistic analysis of the EDD heuristic for a one machine scheduling problem with earliness/tardiness penalties. The schedule is without preemption and with no inserted idle time. The jobs are independent and each may have a different due date. For large n, our result shows that the variance of the performance of the EDD heuristic is linear in n. The average-case performance of the EDD heuristic is known to be proportional to the square root of n.