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

Abstract

In this paper a M X / G( a, b)/1 queueing system with N-policy, multiple vacations and setup time is considered. 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. Server vacation models are useful for the systems in which server wants to utilize his idle time for different purposes. Application of vacation models can be found in production systems, designing local area networks and data communication systems. This paper is concentrated on a vacation system with setup times. In practical situations, the setup time corresponds to the preparatory work of the server before starting the service. We derive the system size distribution and expected length of idle and busy period of a M X / 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.