Heavy-traffic limit theorems for the haevy-tailed GI/G/1 queue

The classic $GI/G/1$ queueing model of which the tail of the service time and/or the interarrival time distribution behaves as $t^{-v {S(t)$ for ${t rightarrow infty$, $1<v<2$ and ${S(t)$ a slowly varying function at infinity, is investigated for the case that the traffic load $a$ approaches one. Heavy-traffic limit theorems are derived for the case that these tails have a similar behaviour at infinity as well as for the case that one of these tails is heavier than the other one. These theorems state that the contracted waiting time ${Delta (a) {bf w$, with ${bf w$ the actual waiting time for the stable $GI/G/1$ queue and ${Delta (a)$ the contraction coefficient, converges in distribution for ${a uparrow 1 $. Here ${Delta (a)$ is that root of the contraction equation which approaches zero from above for ${a uparrow 1 $. The structure of this contraction equation is determined by the character of the two tails. The Laplace-Stieltjes transforms of the limiting distributions are derived. For nonsimilar tails the limiting distributions are explicitly known. For the tails of these distributions asymptotic expressions are derived and compared.