Order statistics applications to queueing and scheduling problems

Abstract

We prove several basic combinatorial identities and use them in two applications: the queue inference engine (QIE) and earliest due date rule (EDD) scheduling. Larson (1990) introduced the QIE. His objective was to deduce the behavior of a multiserver queueing system without observing the queue. With only a Poisson arrival assumption, he analyzed the performance during a busy period. Such a period starts once all servers are busy with the queue empty, and it ends as soon as a server becomes idle. 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 derive 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. We also perform a probabilistic analysis of the EDD for a one-machine scheduling problem with earliness and 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, we show that the variance of the total penalty costs of the EDD is linear in n. The mean of the total penalty costs of the EDD is known to be proportional to the square root of n (see Harel (1993)).