Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions

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.