An \(\text{M}^X/\text{G}/1\) retrial queue with exhaustive vacations (Q2713496)

The general queueing model for non-retrial queues assumes that customers are in continuous contact with the server. They can see when the server is busy and hence enter service as soon as the server becomes free. In contrast, in the retrial queueing systems the customer is not allowed to remain in contact with the server and can only look in to check the validity of the server. If the server is free, then the customer enters service immediately, otherwise, it joins a source of unsatisfied customers. From this source a customer must keep looking in until it finds the server free. The previous works on these types of queueing systems include a comprehensive review of the subject by \textit{G. Falin} [Queueing Syst. 7, No. 2, 127-167 (1990; Zbl 0709.60097)] and a recent article of the reviewer and \textit{N. Livingston} [Eur. J. Oper. Res. 131, No. 3, 530-535 (2001)].NEWLINENEWLINENEWLINEHowever, the present paper considers a retrial queueing system in which the arrivals can arrive in batches and the server for a random period of time can go on vacation or becomes unavailable to the customers. The time and the size of the \(n\)th arriving batch are both assumed to be random variables. The arriving batch behaves like the above single arriving customer, in that if any batch finds the server idle at the instance of arrival, then one of the customers from the batch gets served while the others enter the source of unattended customers. However, if the batch finds the server busy or on vacation, all the units in the batch join the source of unsatisfied customers. If there are any unsatisfied customers in the system, the next attempt to get service is exponentially distributed. The author derives the difference-differential equations which govern the above queueing system. It is obvious from above that these equations are complicated. However, the paper succeeds in obtaining the generating function of the number of customers in the system and those unsatisfied. Using these equations the author discusses the stochastic decomposition property and the optimal control problem of the vacation policy.