Optimal ( d, c) vacation policy for a finite buffer M/M/c queue with unreliable servers and repairs

Abstract

This paper considers a finite buffer M/M/c queueing system in which servers are unreliable and follow a ( d, c) vacation policy. With such a policy, at a service completion instant, if the number of customers is reduced to c − d ( c > d), the d idle servers together take a vacation (or leave for a random amount of time doing other secondary job). When these d servers return from a vacation and if still no more than c − d customers are in the system, they will leave for another vacation and so on, until they find at least c − d + 1 customers are in the system at a vacation completion instant, and then they return to serve the queue. This study is motivated by the fact that some practical production and inventory systems or call centers can be modeled as this finite-buffer Markovian queue with unreliable servers and ( d, c) vacation policy. Using the Markovian process model, we obtain the stationary distribution of the number of customers in the system numerically. Some cost relationships among several related systems are used to develop a finite search algorithm for the optimal policy ( d, c) which maximizes the long-term average profit. Numerical results are presented to illustrate the usefulness of such a algorithm for examining the effects of system parameters on the optimal policy and its associated average profit.