On the time-dependent behavior of preemptive single-server queueing systems with Poisson arrivals

We present a detailed study of the time-dependent behavior of both the queue-length process and the workload process of various types of preemptive Last-Come-First-Served queueing systems, such as the preemptive repeat different and preemptive repeat identical models recently studied in Asmussen and Glynn (Queue Syst 87:1-22, 2017) and Bergquist and Sigman (Stoch Models 38:515-544, 2022). Our main results show that various quantities that provide information about the time-dependent behavior of these processes can be expressed in terms of the Laplace-Stieltjes transform of the busy period, and we show how a natural coupling procedure can be used to establish, for each preemptive queue we consider, a recursive procedure for calculating these busy period transforms on the set of all complex numbers having positive real part.