Stationary workload and service times for some nonwork-conserving M/G/1 preemptive LIFO queues

Abstract

We analyze two nonwork-conserving variations of the M/G/1 preemptive last-in first-out (LIFO) queue with emphasis on deriving explicit expressions for the limiting (stationary) distributions of service times found in service by an arrival, workload and a variety of related quantities of interest. Workload is also used as a tool to derive the proportion of time that the system is busy, and stability conditions. In the first model, known as preemptive-repeat different (PRD), preempted customers are returned to the front of the queue with a new independent and identically distributed service time. In the second, known as preemptive-repeat identical (PRI), they are returned to the front of the queue with their original service time. Our analysis is based on queueing theory methods such as the Rate Conservation Law, PASTA, regenerative process theory and Little’s Law (). For the second model we even derive the joint distribution of age and excess of the service time found in service by an arrival, and find they are quite different from what is found in standard work-conserving models. We also give heavy-traffic limits and tail asymptotics for stationary workload for both models, as well as deriving an implicit representation for the distribution of sojourn time by introducing an alternative effective service time distribution.