Matrix Product-Form Solutions for LCFS Preemptive Service Single-Server Queues with Batch Markovian Arrival Streams

This paper considers three variants of last-come first-served (LCFS) preemptive service single-server queues, where customers are served under the LCFS preemptive resume (LCFS-PR), preemptive repeat-different (LCFS-PD), and preemptive repeat-identical (LCFS-PI) disciplines, respectively. These LCFS queues are fed by multiple batch Markovian arrival streams. Service times of customers from each arrival stream are generally distributed and their distributions may differ among different streams. For each of LCFSPR, LCFS-PD, and LCFS-PI queues, we show that the stationary distribution of the queue string representing enough information to keep track of queueing dynamics has a matrix product-form solution. Further, this paper discusses the stability of LCFS-PD and LCFS-PI queues based on the busy cycle. Finally, by numerical experiment, we examine the impact of the variation of the service time distribution on the mean queue lengths for the three variants of LCFS queues.