The virtual waiting time of theGI/G/1 queue in heavy traffic

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Abstract
Investigations in the theory of heavy traffic were initiated by Kingman ([5], [6] and [7]) in an effort to obtain approximations for stable queues. He considered the Markov chains {Wni} of a sequence {Qi} of stableGI/G/1 queues, whereWniis the waiting time of thenth customer in theith queueing system, and by making use of Spitzer's identity obtained limit theorems as firstn→ ∞ and then ρi↑ 1 asi→ ∞. Here &rHiis the traffic intensity of theith queueing system. After Kingman the theory of heavy traffic was developed by a number of Russians mainly. Prohorov [10] considered the double sequence of waiting times {Wni} and obtained limit theorems in the three cases whenn1/2(ρi-1) approaches (i) - ∞, (ii) -δ and (iii) 0 asn→ ∞ andi→ ∞ simultaneously. The case (i) includes the result of Kingman. Viskov [12] also studied the double sequence {Wni} and obtained limits in the two cases whenn1/2(ρi− 1) approaches + δ and + ∞ asn→ ∞ andi→ ∞ simultaneously.

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