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Limiting diffusions for the conditioned M/G/1 queue

Published online by Cambridge University Press:  14 July 2016

Douglas P. Kennedy
Douglas P. Kennedy*
Affiliation:
University of Sheffield
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Abstract

The virtual waiting time process, W(t), in the M/G/1 queue is investigated under the condition that the initial busy period terminates but has not done so by time n ≥ t. It is demonstrated that, as n → ∞, W(t), suitably scaled and normed, converges to the unsigned Brownian excursion process or a modification of that process depending whether ρ ≠ 1 or ρ = 1, where ρ is the traffic intensity.

Keywords

QUEUESLIMIT THEOREMSDIFFUSIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 2 , June 1974 , pp. 355 - 362
Copyright
Copyright © Applied Probability Trust 1974 

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References

[1] Belkin, B. (1972) An invariance principle for conditioned recurrent random walk attracted to a stable law. Z. Wahrscheinlichkeitsth. 21, 45–64.Google Scholar
[2] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[3] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Volume II. Wiley, New York.Google Scholar
[4] Iglehart, D. L. (1974) Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab. To appear.CrossRefGoogle Scholar
[5] Ito, K. and Mckean, H. P. (1965) Diffusion Processes and their Sample Paths. Springer-Verlag, Berlin.Google Scholar
[6] Kyprianou, E. K. (1971) On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals. J. Appl. Prob. 8, 494–507.CrossRefGoogle Scholar
[7] Lamperti, J. and Ney, P. (1968) Conditioned branching processes and their limiting diffusions. Theor. Probability Appl. 13, 126–137.CrossRefGoogle Scholar
[8] Prabhu, N. (1965) Queues and Inventories. Wiley, New York.Google Scholar
[9] Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar