A 2-Class Maintenance Model with a Finite Population and Competing Exponential Failure Rates

Abstract

We investigate a maintenance system represented as a single-server polling model. Within the model, we assume two classifications for the type of failure a machine may experience. There are C total machines in the system, which at any point in time are either working, in service, or waiting to be served in one of two queues. Working machines are subject to independent and identically distributed exponential failure rates. Machines are returned to working condition after eventually receiving service according to the class of their failure. Service and switch-in time distributions for each class are assumed to be phase-type. Multiple service policies are examined, including preemptive resume priority, non-preemptive priority, and exhaustive service. We model the system as a level-dependent quasi-birth-and-death process, and use matrix analytic techniques to compute the steady-state joint queue length distribution as well as the sojourn time distribution of a broken machine. We present several numerical examples which highlight the dependency of the expected number of working machines on factors such as the service policy and the probability of a non-zero switch-in time.