A Markovian Queue with N Servers Subject to Breakdowns and Repairs

• Published in: Management science Vol. 25; no. 9; pp. 849 - 861
• Main Authors: Neuts, Marcel F, Lucantoni, David M
• Format: Journal Article
• Language: English
• Published: Linthicum INFORMS 01.09.1979; Institute for Operations Research and the Management Sciences
• Series: Management Science
• Subjects: Accuracy; algorithmic methods: server breakdowns and repairs; Algorithms; Breakdowns; Customer services; Markov analysis; Operations research; queues: multi-channel; Queuing theory; Random variables; Repair & maintenance; Repairs
• ISSN: 0025-1909; 1526-5501
• DOI: 10.1287/mnsc.25.9.849

We study a queue with N servers, who may break down and require repair at a facility, which has c repair crews. Under exponential assumptions, this model has an algorithmically tractable solution. It is then in fact a particular case of the M / M / N queue in a Markovian environment. The purpose of this paper is twofold. As a novel methodological contribution, the stationary distributions of various waiting times are discussed. Although fairly involved, these distributions may be computed by classical numerical methods. The second and primary purpose is to demonstrate the utility of interactive computation in answering questions on the behavior, design and control of certain service systems. By numerical examples, we shall show that during periods when most servers are down, large build-ups may occur which affect the queue adversely for a long time afterwards. We also find that such build-ups are aggravated by reducing the number of repair crews, but may be attenuated by reducing the arrival rates during periods when the service has deteriorated. Potential applications are in manpower planning, as in a typing pool where persons may be absent and in determining the size of a battery of machines, where machines may be inoperative due to maintenance and repair. More generally, this paper draws attention to qualitative behavior of queues with randomly varying service rates. Such queues may exhibit substantial and long-lived fluctuations, which are not always apparent from simulation runs and which are also typically not present in the more classical queueing models.