For the $GI/G/1$-queueing model with traffic load $a<1$, service time distribution $B(t)$ and interarrival time distribution $A(t)$ holds, whenever for $t rightarrow infty$: $$ quad 1-B(t) sim frac{c{(t/ beta)^nu + {rm O ( {rm e^{-delta t ), quad c>0, quad 1< nu < 2, quad delta > 0, $$ $$ intlimits_0^infty t^mu {rm d A(t) < infty quad for quad mu > nu , $$ that $(1-a)^{frac{1{nu-1 {bf w$ converges in distribution for $a uparrow 1$. Here ${bf w$ is distributed as the stationary waiting time distribution. The L.S.-transform of the limiting distribution is derived and an asymptotic series for its tail probabilities is obtained. The theorem actually proved in the text concerns a slightly more general asymptotic behaviour of $1-B(t)$, $t rightarrow infty$, than mentioned above.

CWI. Probability, Networks and Algorithms [PNA]

Cohen, J. W. (1997). A heavy-traffic theorem for the GI/G/1 queue with a Pareto-type service time distribution. CWI. Probability, Networks and Algorithms [PNA]. CWI.