A general workload conservation law with applications to queueing systems
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In the spirit of Little’s law \(L=\lambda W\) and its extension \(H=\lambda G\) we use sample-path analysis to give a general conservation law. For queueing models the law relates the asymptotic average workload in the system to the conditional asymptotic average sojourn time and service times distribution function. This law generalizes previously obtained conservation laws for both single- and multi-server systems, and anticipating and non-anticipating scheduling disciplines. Applications to single- and multi-class queueing and other systems that illustrate the versatility of this law are given. In particular, we show that, for anticipative and non-anticipative scheduling rules, the unconditional delay in a queue is related to the covariance of service times and queueing delays.
KeywordsConservation law Workload invariance Multi-server queues Sample-path analysis Scheduling
Mathematics Subject ClassificationPrimary 60K25 Secondary 68M20 90B36
The author wishes to thank the reviewers for their helpful comments.
- 2.Baccelli, F., Brémaud, P.: Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences, Applications of Mathematics, vol. 26. Springer, New York (1994)Google Scholar
- 3.Bartsch, B., Bolch, G.: Conservation law for G/G/m queueing systems. Acta Inform. 810, 105–109 (1978)Google Scholar
- 5.El-Taha, M.: Asymptotic time averages and frequency distributions. Int. J. Stochastic Anal. 2016, Article ID 2741214. doi: 10.1155/2016/2741214 (2016)
- 9.Gelenbe, E., Mitrani, I.: Analysis and Synthesis of Computer Systems. Academic Press, London (1980)Google Scholar
- 12.Heyman, D., Sobel, M.: Stochastic Models in Operations Research, vol. I. McGraw-Hill, New York (1982)Google Scholar
- 14.Kleinrock, L.: Queueing Systems, vol. II. Wiley Intersciences, New York (1976)Google Scholar
- 15.Kleinrock, L., Muntz, R.R., Hsu, J.: Tight bounds on average response time for time-shared computer systems. Proc. IFIP Congr. 1, 124–133 (1971)Google Scholar
- 16.Niño Mora, J.: Conservation Laws and Related Applications. Wiley Encyclopedia of Operations Research and Management Science, London (2011)Google Scholar
- 19.Tsoucas, P.: The region of achievable performance in a model of klimov. Technical report, Thomas J. Watson IBM Research Center. Research Division., T.J. Watson Research Center, Yorktown Hts., New York, NY 10598 (1991)Google Scholar
- 21.Wolff, R.: Stochastic Modeling and the Theory of Queues. Prentice Hall, Englewood Cliffs, NJ (1989)Google Scholar