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A Note on a Sample-Path Rate Conservation Law and its Relationship with H = λG

Published online by Cambridge University Press:  01 July 2016

Karl Sigman
Karl Sigman*
Affiliation:
Columbia University
*
∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, Mudd Building, New York, NY 10027, USA.
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Abstract

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We present a simple sample-path version of the rate conservation law (of Miyazawa) and then show that the H = λG law (of Heyman and Stidham) is essentially the same law, that is, either one can be derived from the other. As a final remark we illustrate the use of both laws jointly to quickly obtain a queueing result.

Information

Type
Letters to the Editor
Information
Advances in Applied Probability , Volume 23 , Issue 3 , September 1991 , pp. 662 - 665
Copyright
Copyright © Applied Probability Trust 1991 

Footnotes

Research supported in part by the Japan Society for the Promotion of Science, during the author's fellowship in Tokyo, and by NSF grant DDM 895 7825.

References

1. Brémaud, P. (1991) An elementary proof of Sengupta's invariance relation and a remark on Miyazawa's conservation principle. J. Appl. Prob. 28 (4).CrossRefGoogle Scholar
2. Brill, P. and Posner, M. (1977) Level crossings in point processes applied to queues: single-server case. Operat. Res. 25, 662–674.Google Scholar
3. Brumelle, S. L. (1971) On the relation between customer and time averages in queues. J. Appl. Prob. 8, 508–520.Google Scholar
4. Ferrandiz, J. M. and Lazar, A. A. (1990) Rate conservation law for stationary processes. J. Appl. Prob. 28, 146–158.CrossRefGoogle Scholar
5. Heyman, D. P. and Stidham, S. Jr. (1980) The relation between time and customer averages in queues. Operat. Res. 28, 983–994.Google Scholar
6. Mazumdar, R., Kannurpatti, R. and Rosenberg, C. (1991) On a rate conservation law for non-stationary processes. J. Appl. Prob. 28 (4).Google Scholar
7. Miyazawa, M. (1983) The derivation of invariance relations in complex queueing systems with stationary inputs. Adv. Appl. Prob. 15, 874–885.Google Scholar
8. Miyazawa, M. (1985) The intensity conservation law for queues with randomly changed service rate. J. Appl. Prob. 22, 408–418.Google Scholar
9. Miyazawa, M. (1991) Rate conservation law with multiplicity and its applications to queues. Submitted for publication.Google Scholar
10. Sakasegawa, H. and Wolff, R. W. (1990) The equality of the virtual delay and attained waiting time distributions. Adv. Appl. Prob. 22, 257–259.CrossRefGoogle Scholar
11. Sengupta, B. (1989) An invariance relationship for the G/G/1 queue. Adv. Appl. Prob. 21, 956–957.Google Scholar
12. Whitt, W. (1991) A review of L = ?W and extensions. QUESTA. To appear.Google Scholar
13. Wolff, R. W. (1989) Stochastic Modeling and the Theory Of Queues. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
14. Yamazaki, G., Miyazawa, M. and Sigman, K. (1991) The first few moments of work-load in fluid models with burst arrivals. Submitted for publication.Google Scholar
15. Zazanis, M. A. (1991) Sample path analysis of level crossings for the workload process. Submitted for publication.CrossRefGoogle Scholar