A new method for computing the sojourn times in the MAP/M/1 queues with three types of negative customers

In this paper, we propose a new method for computing the sojourn times in the MAP/M/1 queueing systems with three types of negative customers due to three basic removal rules: the RCH, RCE and RCA. To this end, we first give a necessary and sufficient condition under which the system is stable by means of the mean drift technique. Then we obtain the queue size distributions and the average stationary queue lengths by using the matrix-analytic method. Based on this, we provide a new effective method for computing the average sojourn time of any arriving customer through using the first passage times and the PH distributions. Finally, we use some numerical examples to illustrate how the performance measures are influenced by some key system parameters.