Summary
Various aspects of the equilibrium M/G/1 queue at large values are studied subject to a condition on the service time distribution closely related to the tail to decrease exponentially fast. A simple case considered is the supplementary variables (age and residual life of the current service period), the distribution of which conditioned upon queue length n is shown to have a limit as n→∞. Similar results hold when conditioning upon large virtual waiting times. More generally, a number of results are given which describe the input and output streams prior to large values e.g. in the sense of weak convergence of the associated point processes and incremental processes. Typically, the behaviour is shown to be that of a different transient M/G/1 queueing model with a certain stochastically larger service time distribution and a larger arrival intensity. The basis of the asymptotic results is a geometrical approximation for the tail of the equilibrium queue length distribution, pointed out here for the GI/G/1 queue as well.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Asmussen, S.: Conditioned Limit Theorems Relating a Random Walk to its Associate, With Applications to Risk Reserve Processes and the GI/G/1 Queue. Advances in Appl. Probability 14 (to appear) (1980)
Asmussen, S.: Time-dependent Approximations in Some Queueing Systems With Imbedded Markov Chains Related to Random Walks. Submitted for publication (1981)
Billingsley, P.: Convergence of Probability Measures. New York: Wiley 1968
Breiman, L.: Probability. Reading, Mass.: Addison-Wesley 1968
Cohen, J.W.: Extreme Value Distributions for the M/G/1 and G/M/1 Queueing Systems. Ann. Inst. H. Poincaré Sect. B 4, 83–98 (1968)
Cohen, J.W.: The Single Server Queue. Amsterdam: North-Holland 1969
Cohen, J.W.: Some Results on Regular Variation for Distributions in Queueing and Fluctuation Theory. J. Appl. Probability 10, 343–353 (1973a)
Cohen, J.W.: Asymptotic Relations in Queueing Theory. Stochastic Processes Appl. 1, 107–124 (1973b)
Cohen, J.W.: On Regenerative Processes in Queueing Theory. Lecture Notes in Economics and Mathematical Systems 121. Berlin-Heidelberg-New York: Springer 1976
Cox, D.R. and Smith, W.L.: Queues. London: Methuen 1961
Feller, W.: An Introduction to Probability Theory and Its Applications 2, 2nd Ed. New York: Wiley 1971
Gaver, D.P., Jr.: Imbedded Markov Chain Analysis of a Waiting Line Process in Continuous Time. Ann. Math. Statist. 30, 698–720 (1959)
Gnedenko, B., Kovalenko, I.N.: An Introduction to Queueing Theory. Jerusalem: Israel Program for Scientific Translations 1968
Hokstad, P.: A Supplementary Variable Technique Applied to the M/G/1 Queue. Scand. J. Statist. 2, 95–98 (1975)
Iglehart, D.L.: Extreme Values in the GI/G/1 Queue. Ann. Math. Statist. 43, 627–635 (1972)
Kingman, J.F.C.: A Martingale Inequality in the Theory of Queues. Proc. Cambridge Philos. Soc. 60, 359–361 (1964)
Le Gall, P.: Les Systemes avec on sans Attente et les Processus Stochastiques. Paris: Dunod 1962
Lindvall, T.: Weak Convergence of Probability Measures and Random Functions in the Function Space D[0, ∞). J. Appl. Probability 10, 109–121 (1973)
Miller, D.R.: Existence of Limits in Regenerative Processes. Ann. Math. Statist. 43, 1275–1282 (1972)
Neveu, J.: Processus Ponctuels. Lecture Notes in Mathematics 598, 249–445. Berlin-Heidelberg-New York: Springer 1977
Prabhu, N.U.: Queues and Inventories. New York: Wiley 1965
Reich, E.: Waiting Times when Queues are in Tandem. Ann. Math. Statist. 28, 768–773 (1957)
Saaty, T.L.: Elements of Queueing Theory. New York: McGraw-Hill 1961
Smith, W.L.: Regenerative Stochastic Processes. Proc. Roy. Soc. London Ser. A 232, 6–31 (1955)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Asmussen, S. Equilibrium properties of the M/G/1 queue. Z. Wahrscheinlichkeitstheorie verw. Gebiete 58, 267–281 (1981). https://doi.org/10.1007/BF00531567
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00531567