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 th e interarrival time distribution has a heavy tail, i.e., a tail behaviour l ike t(-nu) with 1<nu less than or equal to 2, so that the mean is finite bu t 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 se rvice time distribution is heavier than that of the interarrival time distr ibution, and the traffic load a --> 1, then W, multiplied by an appropriate 'coefficient of contraction' that is a function of a, converges in distrib ution to the Kovalenko distribution. If the tail of the interarrival time d istribution is heavier than that of the service time distribution, and the traffic load a --> 1, then W, multiplied by another appropriate 'coefficien t of contraction' that is a function of a, converges in distribution to the negative exponential distribution.