Conditioned limit theorem for the pair of waiting time and queue line processes

Abstract

The pair (W(t), L(t)t⩾0, of the virtual waiting time and the queue line processes is considered in the GI/G/1 queueing system with the traffic intensity one. An asymptotic of $$\left( {\frac{1}{{\sqrt t }}W(t), \frac{1}{{\sqrt t }}L(t)} \right)$$ ) conditioned on the event {T>t} is given ast→∞, whereT is the length of the first busy period. A similar result is also given in the situation whent runs over the arrival moments of customers.