Steady state analysis of a non-Markovian bulk queue with restricted vacations

A MX/G(a, b)/1 queueing system with restricted number of vacations is considered here. After completing a bulk service, if the queue size is less than 'a' then the server leaves for a vacation of random length. When he returns from a vacation, if queue length is still less than 'a', he avails another vacation and so on until he completes M number of vacations in successions or he finds at least 'a' customers wait for service. After M vacations, if the queue size is less than 'a' then the server will remain in the system until he finds 'a' customers in the queue. After a vacation or service completion if the server finds at least 'a' customers waiting for service then the server serves according to Neut's bulk service rule. The probability generating function of queue size at an arbitrary time and some important system performance measures are obtained. Cost model is discussed with numerical illustration.