Abstract
LetW k denote the waiting time of customerk, k≥ 0, in an initially empty GI/G/1 queue. Fixa> 0. We prove weak limit theorems describing the behaviour ofW k /n, 0≤k≤n, given Wn >na. LetX have the distribution of the difference between the service and interarrival distributions. We consider queues for which Cramer type conditions hold forX, and queues for whichX has regularly varying positive tail.
The results can also be interpreted as conditional limit theorems, conditional on large maxima in the partial sums of random walks with negative drift.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Asmussen, Private Communication, 1988.
S. Asmussen, Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/G/1 queue, Advances in Applied Probability 14 (1982) 143–170.
S. Asmussen,Applied Probability and Queues (John Wiley and Sons, 1987).
H. Bergstrom,Weak Convergence of Measures (Academic Press, 1982).
P. Billingsley,Convergence of Probability Measures (John Wiley and Sons, 1968).
A.A. Borovkov, Boundary value problems for random walks and large deviations in function spaces, Theory of Probability and its Applications 12, No. 4 (1967) 575–595.
R. Durett, Conditional limit theorems for random walks with negative drift, Zeit. fur Wahr. und Ver. Geb. 52 (1980) 277–287.
W. Feller,An Introduction to Probability Theory and its Applications, Vol. II (John Wiley and Sons, 1971).
A.A. Mogulskii, Large deviations for trajectories of multidimensional random walks, Theory of Probability and its Applications XXI, No. 2 (1976) 300–315.
J. Neveu,Discrete Parameter Martingales (North-Holland, 1975).
D. Picard and J. Deshayes, Grandes et moyennes deviations pour les marches aleatories, in:Grandes Deviations et applications statistiques (Seminaire Orsay 1977–1978, Asterisque, Vol. 68, 1979) pp. 53–71.
Author information
Authors and Affiliations
Additional information
Research supported by the NSF under Grant NCR 8710840 and under the PYI Award NCR 8857731.
Rights and permissions
About this article
Cite this article
Anantharam, V. How large delays build up in a GI/G/1 queue. Queueing Syst 5, 345–367 (1989). https://doi.org/10.1007/BF01225324
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01225324