Heavy-traffic Limit Theorems for the Heavy-tailed Gi/g/1 Queue Heavy-traac Limit Theorems for the Heavy-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 ! 1, 1 < v < 2 and S(t) a slowly varying function at innnity, is investigated for the case that the traac load a approaches one. Heavy-traac limit theorems are derived for the case that these tails have a similar behaviour at innnity 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 (a)w, with w the actual waiting time for the stable GI=G=1 queue and (a) the contraction coeecient, converges in distribution for a " 1. Here (a) is that root of the contraction equation which approaches zero from above for a " 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. Note: work carried out under project LRD in PNA 2.1.