Analysis of a model of batch arrival single server queue with random vacation policy

Abstract

This article deals with batch arrival single server queue with random vacation policy, in which the server takes the maximum number of random vacations till it finds some customers waiting in a queue at a vacation completion epoch. If no arrival occurs after completing maximum number of random vacations, the server stays dormant in the system and waits for the upcoming arrival. Here, it is assumed that the customer’s arrival in batches conforms to compound Poisson process, although service time and vacation time are generally distributed. Explicit expressions are obtained for steady state queue size distribution at service completion point and steady state system size probabilities. Some significant measures such as a mean number of batches served during the busy period, Laplace-Stieltjes transform of waiting time, unfinished work and its corresponding mean values are also obtained. A cost optimal policy is developed in terms of the average cost function to determine a locally optimal random vacation policy at a lower cost. Finally, various numerical results are presented for the above system performance measures.

Keywords:

• M^X/G/1 queue
• random vacation policy
• embedded Markov chain
• Markov regenerative process
• cost optimal policy

Mathematics Subject Classification (2000):

• 60K25
• 90B22
• 60K20

Acknowledgement

Authors are thankful to the editor and two anonymous reviewers for their valuable suggestions and comments which helped improve the manuscript to great extent and gratefully acknowledges for the fnancial support received from the Department of Telecommunications (DoT), India.