Batch arrival bulk service queue with unreliable server, second optional service, two different vacations and restricted admissibility policy

This paper is concerned with batch arrival queue with an additional second optional service to a batch of customers with dissimilar service rate where the idea of restricted admissibility of arriving batch of customers is also introduced. The server may take two different vacations (i) Emergency vacation-during service the server may go for vacation to an emergency call and after completion of the vacation, the server continues the remaining service to a batch of customers. (ii) Bernoulli vacation-after completion of first essential or second optional service, the server may take a vacation or may remain in the system to serve the next unit, if any. While the server is functioning with first essential or second optional service, it may break off for a short period of time. As a result of breakdown, a batch of customers, either in first essential or second optional service is interrupted. The service channel will be sent to repair process immediately. The repair process presumed to be general distribution. Here, we assumed that the customers just being served before server breakdown wait for the server to complete its remaining service after the completion of the repair process. We derived the queue size distribution at a random epoch and at a departure epoch under the steady state condition. Moreover, various system performance measures, the mean queue size and the average waiting time in the queue have been obtained explicitly. Some particular cases and special cases are determined. A numerical result is also introduced.