Unreliable bulk retrial queues with delayed repairs and modified vacation policy

The present investigation deals with the bulk arrival M/G/1 retrial queue with impatient customers and modified vacation policy. The incoming customers join the virtual pool of customers called orbit if they find the server being busy, on vacation or in broken down state otherwise the service of the customer at the head of the batch is started by the server. The service is provided in k essential phases to all the customers by the single server which may breakdown while rendering service to the customers. The broken down server is sent to a repair facility wherein the repair is performed in d compulsory phases. As soon as the orbit becomes empty, the server goes for vacation and takes at most J vacations until at least one customer is noticed. The incoming customers are impatient and may renege on seeing a long queue of the customers for the service. The probability generating functions and queue length for the number of customers in the orbit and queue have been obtained using supplementary variable technique. Various system characteristics viz. average number of customers in the queue and the orbit, long run probabilities of the system states, etc. are obtained. Furthermore, numerical simulation has been carried out to study the sensitivity of various parameters on the system performance measures by taking an illustration.