On an unreliable retrial queue with general repeated attempts and J optional vacations

In this paper, we consider a single-server retrial queue with constant retrial rate and batch arrivals, in which the unreliable server has the option to take an additional vacation after the first essential vacation. Customers arriving at the system according to a compound Poisson process are served immediately as long as the server is available; otherwise, they enter a retrial orbit and form a single queue. When the orbit becomes empty, the server leaves for the first essential vacation, after which he may either remain idle within the system or take one of J optional vacations. When the server is busy, he is subject to random breakdowns and repairs. It is assumed that the service times, repair times and customer retrial times are arbitrarily distributed. The implementation of the supplementary variable technique makes it possible to derive the probability generating functions of the system size distribution at a random epoch. We also develop a variety of system performance measures as well as two reliability indices. Finally, we deal with the problem of cost optimization and provide a number of numerical examples.