Abstract
This paper presents a unified sample-path approach for deriving distribution-free relations between performance measures for stochastic discrete-event systems extending previous results for discrete-state processes to processes with a general state space. A unique feature of our approach is that all our results are shown to follow from a single fundamental theorem: the sample-path version of the renewal-reward theorem (Y=λX). As an elementary consequence of this theorem, we derive a version of the rate-conservation law under conditions more general than previously given in the literature. We then focus on relations between continuous-time state frequencies and frequencies at the points of an imbedded point process, giving necessary and sufficient conditions for theASTA (Arrivals See Time Averages), conditionalASTA, and reversedASTA properties. In addition, we provide a unified approach for proving various relations involving forward and backward recurrence times. Finally, we give sufficient conditions for rate stability of an input-output system and apply these results to obtain an elementary proof of the relation between the workload and attained-waiting-time processes in aG/G/l queue.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baccelli, F., and Brémaud, 1987.Plam Probabilities and Stationary Queues. Lecture Notes in Statistics, vol. 41. Springer-Verlag: New York.
Billingsley, P. (1968).Weak Convergence. Wiley: New York
Brémaud, P. (1991). An elementary proof of Sengupta's invariance relation and a remark on Miyazawa's conservation principle.J. Appl. Prob. 28:950–954.
Brémaud, P. (1993). A Swiss army formula of Palm calculus.J. Appl. Prob. 30:40–51.
El-Taha, M. 1990. Sample-path relations between time averages and asymptotic distributions Preprint, Department of Mathematics and Statistics, USM.
El-Taha, M. (1992). On conditional ASTA: A sample-path approach.Comm. Statis.: Stoch. Models 8:157–177.
El-Taha, M., and Stidham, S., Jr. 1991. Sample-path analysis of stochastic discrete-event systems.Proc. 30th IEEE CDC Meeting.
El-Taha, M., and Stidham, S., Jr. (1992a). Deterministic analysis of queueing systems with heterogeneous servers.Theoret. Comput. Sci. 106:243–264.
El-Taha, M. and Stidham, S., Jr. (1992b). A filtered ASTA property.QUESTA 11:211–222.
Ferrandiz, J., and Lazar, A. (1991). Rate conservation for stationary processes.J. Appl. Prob. 28: 146–158.
Franken, P., König, D., Arndt, U., and Schmidt, V. (1981).Queues and Point Processes. Akademie-Verlag: Berlin.
Ghaharamani, S. (1990). On remaining full busy periods of GI/GI/c queues and their relation to stationary point processes.J. Appl. Prob. 27:232–236.
Heyman, D.P., and Stidham, S., Jr. 1980. The relation between customer and time averages in queues.Oper. Res. 28:983–994.
Mazumdar, R., Kannurpatti, R., and Rosenberg, C. 1991. On a rate conservation law for nonstationary processes.J. Appl. Prob. 28:762–770.
Melamed, B., and Whitt, W. (1990). On arrivals that see time averagesOper. Res. 38:156–172.
Miyazawa, M. (1983). The derivation of invariance relations in complex queueing systems with stationary inputs.Adv. Appl. Prob. 15:874–885.
Miyazawa, M. (1985). The intensity conservation law for queues with randomly changed service rate.J. Appl. Prob. 22:408–418.
Miyazawa, M., and Wolff, R.W. 1990. Further results on ASTA for general stationary processes and related problems.J. App. Prob. 28:792–804.
Rolski, T. 1981.Stationary Random Processes Associated with Point Processes. Lecture Notes in Statistics. Springer-Verlag: New York.
Sakasegawa, H., and Wolff, R. 1990. The equality of the virtual delay and attained waiting time distributions.Adv. Appl. Prob. 22:257–259.
Sengupta, B. 1989. An invariance relation for the G/G/1 queue.Adv. Appl. Prob. 21:956–957.
Sigman, K. 1991. A note on a sample-path conservation law and its relation withH=λG.Adv. Appl. Prob. 23:662–665.
Stidham, S. Jr. 1974. A last word onL=λW.Oper. Res. 22:417–421.
Stidham, S., Jr. 1992. A note on a sample-path, approach to Palm probabilities. Forthcoming.
Stidham, S. Jr., and El-Taha, M.: 1989. Sample-path analysis of processes with imbedded point processes.QUESTA 5:131–165.
Stidham, S., Jr., and El-Taha, M. 1993. A note on sample-path stability conditions for input-output processes.Oper. Res. Lett.
Van Doorn, E.A., and Regterschot, G.J.K. 1988 Conditional PASTA.Oper. Res. Lett. 7(5):229–232
Whitt, W. 1991. A review ofL=λW, and extensions.QUESTA 9:235–268.
Whitt, W. 1992.H=λG and the Palm transformation.Adv. Appl. Prob. 24:755–758.
Wolff, R. 1982. Poisson arrivals see time averagesOper. Res. 30:223–231.
Wolff, R. 1989.Stochastic Modeling and the Theory of Queues. Prentice-Hall: Englewood Cliffs, NJ.
Yamazaki, G., and Miyazawa, M. 1991. The equality of the worlkoad and total attained waiting time in average.J. Appl. Prob. 28:238–244.
Zazanis, M., 1992. Sample-path analysis of level crossings for the workload process. QUESTA 11:419–428.
Author information
Authors and Affiliations
Additional information
Research was partially supported by the National Science Foundation under Grant no. DDM-8719825. The government of the United States of America has certain rights in this material. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. The research of this author was also partially supported by a grant from Centre International des Étudiants et Stagiares (C.I.E.S.) while he was visiting INRIA, Sophia-Antipolis, Valbonne, France (1991–92).
Rights and permissions
About this article
Cite this article
El-Taha, M., Stidham, S. Sample-path analysis of stochastic discrete-event systems. Discrete Event Dyn Syst 3, 325–346 (1993). https://doi.org/10.1007/BF01439158
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01439158