Heavy traffic limits associated with M/G/∞ input processes
Queueing Systems: Theory and Applications
Invited Fluid queues with long-tailed activity period distributions
Computer Communications
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We consider a $GI/G/1$ queue in which the service time distribution and/or the interarrival time distribution has a heavy tail, i.e., a tail behaviour like $t^{-\nu}$ with $1\nu \leq 2$, so that the mean is finite but the variance is infinite. We prove a heavy-traffic limit theorem for the distribution of the stationary waiting time ${\bf W}$. If the tail of the service time distribution is heavier than that of the interarrival time distribution, and the traffic load $a \rightarrow 1$, then ${\bf W}$, multiplied by an appropriate `coefficient of contraction'' that is a function of $a$, converges in distribution to the Kovalenko distribution. If the tail of the interarrival time distribution is heavier than that of the service time distribution, and the traffic load $a \rightarrow 1$, then ${\bf W}$, multiplied by another appropriate `coefficient of contraction'' that is a function of $a$, converges in distribution to the negative exponential distribution.