Steady state analysis of a non-Markovian bulk queueing system with overloading and multiple vacations

This paper analyses a non-Markovian bulk queueing model with the possibility of overloading and multiple vacations. It is considered that, on the completion of a service, if the queue length ξ is less than 'a', then the server leaves for a secondary job (vacation) of random length. On returning from this job, again if the queue length is still less than 'a', then the server repeats the secondary job until he finally finds, at least 'a' customers. After a service or a vacation completion epoch, if the server finds at least 'a' customers waiting for service, say ξ (a≤ξ<N), then he serves a batch of min (ξ, b) customers where b≥a. On the other hand, if he finds more than N customers (ξ≥N), then he increases the service capacity (overload) and serves a batch of N customers with a different service rate. For the proposed model, the probability generating function of number of customers in the queue at an arbitrary time epoch and various measures are obtained. Numerical illustrations are also presented for managerial decision to optimise the cost.