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

• Authors: J. W. Cohen
• Venue: Heavy-traffic limit theorems for the heavy-tailed GI/G/1 queue
• Year: 1997

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 ${t \rightarrow \infty}$, $1v2$ 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.