A batch arrival unreliable server delaying repair queue with two phases of service and Bernoulli vacation under multiple vacation policy

Abstract

This paper deals with an M^X/G/1 unreliable queue with two phases of service and Bernoulli vacation schedule under multiple vacation policy, where after each vacation completion or service completion, the server takes sequence of vacations until a batch of new customer arrive. Further concept of the delay time is also introduced. We assume that customers arrive to the system according to a Poisson process with rate . While the server is working with any phase of service, it may breakdown at any instant and the service channel will fail for a short interval of time. After completion of both phases of service, the server either goes for a vacation with probability p(0 ≤ p ≤ 1) or may continue to serve the next unit, if any, with probability q(= 1 – p). Otherwise; it remains in the system until a customer arrives. For this model, we derive queue size distributions at various epochs, busy period distribution, waiting time distribution under the steady-state condition. Next, we derive reliability function and related reliability indices of this model. Finally, some numerical examples are presented for illustrative purpose.

Keywords:

• Stationary queue size distribution
• random breakdown
• repair time
• delay time
• Bernoulli vacation
• multiple vacation and reliability index

Acknowledgements

We would like to express our thanks to the unknown referees and the editor for their insightful comments and suggestions on improving this manuscript to its present form.