Working vacation queue with closed-down, set-up period and negative customers

In M/M/1 working vacation queue, the strategy of negative customers, closed-down and set-up period was introduced. The set-up period is the time needed for setting up a virtual connection according to signaling protocol. The working vacation period can be considered as a delayed period with a lower service rate than that of busy period. With a strategy of removing customers in the end (RCE), negative customers are regarded as external noise. By quasi birth-and-death (QBD) process and matrix-geometric solution method, the distributions of the stationary queue length and stationary waiting time were obtained. The stochastic decomposition structures of queue length and waiting time in stationary state were proved to achieve the distributions of additional queue length and additional delay. The results can provide basis for optimal design of asynchronous transfer mode (ATM) networks queues.