M/M/1 queueing system with delayed controlled vacation

An M/M/1 queue with delayed vacation is studied. If the server has been idle for a period of time (called the delay time), the server be-gins an exponentially distributed vacation which is repeated as long as the number of customers in the system remains less than some number K. For the cases of exponential and deterministic delay time, exact expressions for the steady state probability distribution are obtained, together with associated performance measures. Sys-tem optimization is also considered; values of K are given which minimize the average total cost per unit time, and it is shown that the optimal delay period is either 0 (no delay) or infinite (no vaca-tion), in case of Poisson arrivals