On a non-Markovian queueing problem under a control operating policy and start-up times

A non-Markovian queueing system with Poisson input is studied under a modified operating rule called "control operating policy" in which the server begins "start-up" only when the queue length reaches a fixed number $n(\geq 1)$. By using the supplementary variable technique, the distribution of the queue length (excluding those being served) in the form of a generating function is obtained. As a special case, a Markovian queueing system with exponential start-up is discussed in detail to analyse the economic behaviour of the system.