The Discrete Time MMAP[K]/PH[K]/1/LCFS-GPR Queue and Its Variants

Abstract: In this paper, we study a discrete time queueing system with multiple types of customers and a last-come-first-served general preemptive resume (LCFS-GPR) service discipline (MMAP[K]/PH[K]/1/LCFS-GPR). When the waiting space is infinite, matrix analytic methods are used to find a system stability condition, to derive the distributions of the busy periods and sojourn times, and to obtain a matrix geometric solution of the queue string. The results lead to efficient algorithms for computing various performance measures at the level of individual types of customers. Using those algorithms, the impact of the LCFS-GPR service discipline on the corresponding queueing system can be analyzed. When the waiting space is finite, the Gaussian elimination method is used to develop an efficient algorithm for computing the stationary distribution of the queue string. The relationship between the loss probabilities of individual types of customers and the size of the waiting space is explored. This paper also serves as a brief survey of the study of the MMAP[K]/PH[K]/1 queue and its related queueing models.