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, so that the mean is finite but the variance is infinite. We prove a heavy-traffic limit theorem for the distribution of the stationary actual waiting time W. If the tail of the service time distribution is heavier than that of the interarrival time distribution, we give the conditions on the traffic load under which W, multiplied by an appropriate ‘coefficient of contraction’, converges in distribution to the Kovalenko distribution. If the tail of the interarrival time distribution is heavier than that of the service time distribution, we also give the conditions on the traffic load under which W, multiplied by another appropriate ‘coefficient of contraction’, converges in distribution to the negative exponential distribution.