MAP/G/1 queues under N-policy with and without vacations

This paper considers MAP/G/1 queueing systems under the following two situations: (1) At the end of a busy period, the server is turned off and inspects the queue length every time a customer arrives. When the queue length reaches a pre-specified value N, the server turns on and serves customers continuously until the system becomes empty. (2) At the end of a busy period, the server takes a sequence of vacations. Ab the end of each vacation, the server inspects the queue length. If the queue length is greater than or equal to a pre-specified value N at this time, the server begins to serve customers continuously until the system becomes empty. For each case, we analyze the stationary queue length and the actual waiting time distributions, and derive the recursive formulas to compute the moments of these distributions. Furthermore, we provide a numerical algorithm to obtain the mass function of the stationary queue length. The numerical examples show that in light traffic, correlation in arrivals leads to a smaller mean waiting time.