The FIFO single-server queue with disasters and multiple Markovian arrival streams

Abstract

We consider a FIFO single-server queue with disasters and multipleMarkovian arrival streams. When disasters occur, all customers areremoved instantaneously and the system becomes empty. Both thecustomer arrival and disaster occurrence processes are assumed to beMarkovian arrival processes (MAPs), and they are governed by a commonunderlying Markov chain with finite states. There are $K$ classes ofcustomers, and the amounts of service requirements brought by arrivingcustomers follow general distributions, which depend on the customerclass and the states of the underlying Markov chain immediately beforeand after arrivals. For this queue, we first analyze the first passagetime to the idle state and the busy cycle. We then obtain twodifferent representations of the Laplace-Stieltjes transform of thestationary distribution of work in system, and discuss the relationbetween those. Furthermore, using the result on the workloaddistribution, we analyze the waiting time and sojourn timedistributions, and derive the joint queue length distribution.