Explicit M/G/1 waiting-time distributions for a class of long-tail service-time distributions
The workload vt of an M/G/1 model with traac a < 1 is analyzed for the case with heavy-tailed message length distributions B(), e.g. 1 ? B() = O(?); ! 1; 1 < 2. It is shown that a factor (a) exists with (a) # 0 for a "1 such that, whenever vt is scaled by (a) and time t by 1(a) = (a)(1 ?a) then w(a) = (a)v = 1 (a) converges in distribution for a "1 and every > 0. Proper scaling of the traac load kt, generated by the arrivals in 0; t), leads to ~ w = maxH(); sup 0<u<< (H() ? H(u))]; > 0; with H() = N() ?. Here fN(); 0g with 6 = 2 is-stable L evy motion, for = 2 it is Brownian motions and ~ w has the limiting distribution of w(a) for a "1. This relation is analogous to Reich's formula for the M/G/1 model with a < 1. The results obtained are generalisations of the diiusion approximation of the M/G/1 model with B() having a nite second moment.