THE M/G/1 PROCESSOR-SHARING QUEUE WITH LONG and SHORT JOBS

In this paper we study the classical M/G/1 processor-sharing queue under the assumption that there tend to be some jobs much longer than others, as occurs with a heavy-tailed service-requirement distribution. We suggest analyzing the evolution of the long and short jobs in di erent time scales. To study the short jobs in a short time scale, we act as if a speci ed number of long jobs are permanently in the system. We use the steady-state results with k permanent jobs as an approximation for the shorter-term steady-state behavior that should prevail while a given number of especially long jobs are in the system. We also describe the evolution of the longer jobs in a longer time scale, assuming that the long jobs arrive in a Poisson process with small arrival rate and have large nite mean service requirements. If the arrival rate and the reciprocal of the mean service requirement of the long jobs approach zero, with the tra c intensity held xed, then in the time scale of the mean long-job service requirement it is possible to act as if the number of short jobs in the system is always equal to its conditional steady-state mean given the number of long jobs present. Hence, in the long time scale, it is possible to determine the transient and steady-state behavior of the long jobs, without considering the uctuations of the short jobs. Thus we can approximately describe the e ect of imposing an upper limit on the number of long jobs allowed in the system. The long jobs exceeding this upper limit might be blocked or delayed.