Queue length and waiting time analysis of a batch arrival queue with bilevel control

Scope and Purpose—This paper concerns the analysis of a production system in which an early set-up and N-policy are employed. We model the system by Mx/M/1 queue with bilevel control and obtain the queue length and waiting time distributions. In general, with more than two control policies mixed up, obtaining the waiting time distribution is extremely complicated due to the dependencies between the random variables involved. The methodology employed in this paper can be extended to other batch arrival queueing system with more complicated control policies. We consider an Mx/M/1 queue with bilevel control. As soon as the system empties, the server becomes idle until m or more customers accumulate, at which time the server starts a set-up. After the set-up, if N or more customers are in the system (N m) the server immediately start serving the customers in FCFS fashion. If less than N customers are in the system after the set-up, the server remains dormant in the system and wait until N or more customers accumulate. We derive the distributions of the number of customers in the system and the waiting time.