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.

Queueing theory (msc 60K25), Queues and service (msc 90B22)
CWI
CWI. Probability, Networks and Algorithms [PNA]

Cohen, J.W. (1997). Heavy-traffic limit theorems for the haevy-tailed GI/G/1 queue. CWI. Probability, Networks and Algorithms [PNA]. CWI.