Analysis of a bulk queue with N-policy multiple vacations and setup times

We derive the system size distribution and expected length of idle and busy period of a MX/G(a, b)/1 queueing system with N-policy, multiple vacations and setup time. After finishing a service, if the queue length is less than “a”, the server leaves for a vacation of random length. When he returns, if the queue length is less than “N” (N≥b), he leaves for another vacation and so on, until he finally finds at least “N” customers waiting for service. After a vacation, if he finds more than “N” customers in the system, he requires a setup time R to start the service. After the setup time he starts the service with a batch of “b” customers. After a service, if the number of waiting customers is ξ (ξ≥a) then he serves a batch of min(ξ, b) customers, where b≥a. A cost model for the queueing system is discussed. The numerical solution for a particular case of the model is also presented.