Analysis of an Unreliable Server M^[X]/G/1 System with a Randomized Vacation Policy and Delayed Repair

Abstract

This article examines an M^[x]/G/1 queueing system with a randomized vacation policy and at most J consecutive vacations. Whenever the system is empty, the server immediately takes a vacation. Upon returning from a vacation, the server will be immediately activated for service if there is at least one customer waiting in the queue. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1 − p. This pattern continues until the number of vacations taken reaches J. If the system is empty at the end of the Jth vacation, the server becomes idle in the system until at least one customer arrives at the queue. Assume that the server may suffer an unpredictable breakdown and the repair may be delayed. For such a system, we derive the distributions of some important system characteristics, such as the system size distribution at a random epoch and at a departure epoch, the system size distribution at the initiation epoch of a busy period, the distributions of idle and busy periods, and reliability indices. Finally, a cost model is developed to determine the joint parameters (p, J) that correspond to minimum cost, and some numerical examples are presented for illustrative purpose.

Keywords:

• Batch-arrival vacation queue
• Delayed repair
• Randomized control
• Reliability
• Supplementary variable technique

Mathematics Subject Classification:

• Primary 90B22
• Secondary 60K25

ACKNOWLEDGMENTS

The authors acknowledge the anonymous referees and editor for detailed report on an earlier version of this article, which contributed significantly to improvement in the presentation of this article.