Subexponential asymptotics of the waiting time distribution in a single-server queue with multiple Markovian arrival streams

Abstract

This paper considers subexponential asymptotics of the tail distributions of waiting times in stationary work-conserving single-server queues with multiple Markovian arrival streams, where all arrival streams are modulated by the underlying Markov chain with finite states and service time distributions may differ for different arrival streams. Under the assumption that the equilibrium distribution of the overall (i.e., customer-average) service time distribution is subexponential, a subexponential asymptotic formula is first shown for the virtual waiting time distribution, using a closed formula recently found by the author. Further when customers are served on a FIFO basis, the actual waiting time and sojourn time distributions of customers from respective arrival streams are shown to have the same asymptotics as the virtual waiting time distribution.