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# Conditioned Limit Theorem for the Pair of Waiting Time and Queue Line Processes

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## Abstract

The pair (W(t), L(t)t ( \frac1Öt W(t), \frac1Öt L(t) )\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.

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... The asymptotic formula for the virtual waiting time in a GI/M/1 queueing system in heavy traffic is presented in [33]. In [34,35], the behavior of a scaled pair of a virtual waiting time and queue length processes in heavy traffic is investigated. The author of [36] presented numerous transform formulae for a virtual waiting time in a tandem queue with identical service times at both counters. ...
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