Abstract
We prove that in the queueing system GI/G/1 with traffic intensity one, the virtual waiting time process suitably scaled, normed and conditioned by the event that the length of the first busy period exceeds n converges to the Brownian meander process, as n →∞.
This is a preview of subscription content, log in via an institution to check access.
References
P. Billingsley,Convergence of Probability Measures, (Wiley, New York, 1968).
E. Bolthausen, On a functional central limit theorem for random walks conditioned to stay positive, Ann. Probab. 4 (1976) 480–485.
J.W. Cohen,The Single Server Queue (North-Holland, Amsterdam, 1969).
W. Feller,An Introduction to Probability Theory and Its Applications, Vol. II (Wiley, New York, 1966).
G. Hooghiemstra, Conditioned limit theorems for waiting time processes of the M/G/1 queue, J. Appl. Probab. 20 (1983) 675–688.
D.L. Iglehart, Functional central limit theorems for random walks conditioned to stay positive, Ann. Probab. 2 (1974) 608–619.
D.P. Kennedy, Limiting diffusions for the conditioned M/G/1 queue, J. Appl. Probab. 11 (1974) 355–362.
E.K. Kyprianou, On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals, J. Appl. Probab. 8 (1971) 494–507.
E.K. Kyprianou, On the quasi-stationary distributions of the GI/M/1 queue, J. Appl. Probab. 9 (1972) 117–128.
E.K. Kyprianou, The quasi-stationary distribution of queues in heavy traffic, J. Appl. Probab. 9 (1972) 821–831.
N.U. Prabhu,Stochastic Storage Processes (Springer-Verlag, New York, Heidelberg, Berlin, 1980).
W. Szczotka, Joint distribution of waiting time and queue size for single server queues, Disert. Math. 248 (1986) 5–53.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Topolski, K. Conditioned limit theorem for virtual waiting time process of the GI/G/1 queue. Queueing Syst 3, 377–384 (1988). https://doi.org/10.1007/BF01157857
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01157857