Department of Applied Mathematics & Physics, Kyoto University

Technical Report 2000-004 (August 08, 2000)

Subexponential asymptotics of the waiting time distribution in the stationary single-server queue
by Tetsuya Takine

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This paper considers the subexponential asymptotics of the tail distributions of virtual waiting times, actual waiting times and sojourn times in stationary work-conserving single-server queues. To this end, a single-server queue with a Markovian arrival stream (MAS) is first examined, where arrivals are modulated by the underlying Markov chain with finite states and service time distributions may depend on the states of the underlying Markov chain immediately before and after arrivals. Under the assumption that the equilibrium distribution of the overall (i.e., customer-average) service time distribution is subexponential, a subexponential asymptotic formula is shown for the virtual waiting time distribution. Further when customers are served on a FIFO basis, the actual waiting time and the sojourn time distributions are shown to have the same asymptotics as the virtual waiting time distribution. As you will see, a common asymptotic constant of these three distributions is given only in terms of the utilization factor. Therefore the same asymptotic formulas hold in the FIFO G/G/1 queue as well, because MAS is dense in stationary marked point processes.