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A unified method to analyze overtake free queueing systems

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Dimitris Bertsimas  and
Georgia Mourtzinou
Dimitris Bertsimas*
Affiliation:
Massachusetts Institute of Technology
Georgia Mourtzinou*
Affiliation:
Massachusetts Institute of Technology
*
∗ Postal address: Sloan School of Management and Operations Research Center, MIT, Cambridge, MA 02139, USA.
∗∗ Postal address: Operations Research Center, MIT, Cambridge, MA 02139, USA.
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Abstract

In this paper we demonstrate that the distributional laws that relate the number of customers in the system (queue), L(Q) and the time a customer spends in the system (queue), S(W) under the first-in-first-out (FIFO) discipline are special cases of the H = λG law and lead to a complete solution for the distributions of L, Q, S, W for queueing systems which satisfy distributional laws for both L and Q (overtake free systems). Moreover, in such systems the derivation of the distributions of L, Q, S, W can be done in a unified way. Consequences of the distributional laws include a generalization of PASTA to queueing systems with arbitrary renewal arrivals under heavy traffic conditions, a generalization of the Pollaczek–Khinchine formula to the G//G/1 queue, an extension of the Fuhrmann and Cooper decomposition for queues with generalized vacations under mixed generalized Erlang renewal arrivals, approximate results for the distributions of L, S in a GI/G/∞ queue, and exact results for the distributions of L, Q, S, W in priority queues with mixed generalized Erlang renewal arrivals.

Keywords

PERFORMANCE ANALYSISDISTRIBUTIONAL LAWSPRIORITY QUEUESGENERALIZED VACATIONSHEAVY TRAFFICMIXED GENERALIZED ERLANG ARRIVALS

MSC classification

Primary: 60K25: Queueing theory 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 2 , June 1996 , pp. 588 - 625
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

The research of D. Bertsimas was partially supported by a Presidential Young Investigator Award DDM-9158118 with matching funds from Draper Laboratory. The research of both authors was partially supported by the National Science Foundation under grant DDM-9014751.

References

[1] Baccelli, F. and Bremaud, P. (1994) Elements of Queueing Theory: Palm-Martingale Calculus and Stochastic Recurrences. Springer, New York.Google Scholar
[2] Bertsimas, D. and Nakazato, D. (1995) The distributional Little's law and its applications. Operat. Res. 43, 298.Google Scholar
[3] Bertsimas, D. and Nakazato, D. (1992) Transient and busy period analysis of the GI/G/1 queue: The method of stages. Queueing Systems 10, 153–184.CrossRefGoogle Scholar
[4] Cox, D. R. (1962) Renewal Theory. Chapman and Hall, New York.Google Scholar
[5] Doshi, B. (1985) A note on stochastic decomposition in a GI/G/1 queue with vacations or setup times. J. Appl. Prob. 22, 419–428.Google Scholar
[6] Fuhrmann, S. W. and Cooper, R. B. (1985) Stochastic decompositions in a M/G/1 queue with generalized vacation. Operat. Res. 33, 1117–1129.Google Scholar
[7] Fuhrmann, S. W. (1985) Symmetric queues served in cyclic order. Operat. Res. Lett. 4, 139–144.Google Scholar
[8] Glynn, P. W. and Whitt, W. (1991) A new view of the heavy traffic limit theorem for the infinite-server queues. Adv. Appl. Prob. 23, 188–209.CrossRefGoogle Scholar
[9] Haji, R. and Newell, G. (1971) A relation between stationary queue and waiting time distributions. J. Appl. Prob. 8, 617–620.Google Scholar
[10] Heyman, D. and Sobel, M. (1982) Stochastic Models in Operations Research. Vol 1. McGraw-Hill, New York.Google Scholar
[11] Jaiswal, N. K. (1968) Priority Queues. Academic Press, New York.Google Scholar
[12] Keilson, J. and Servi, L. (1988) A distributional form of Little's law. Operat. Res. Lett. 7, 223–227.CrossRefGoogle Scholar
[13] Keilson, J. and Servi, L. (1990) The distributional form of Little's law and the Fuhrmann-Cooper decomposition. Operat. Res. Lett. 9, 239–247.Google Scholar
[14] Kleinrock, L. (1975) Queueing Systems. Vol. 1: Theory. Wiley, New York.Google Scholar
[15] Little, J. (1961) A proof of the theorem L = ? W. Operai Res. 9, 383–387.Google Scholar
[16] Lucantoni, D. M., Meier-Hellstern, K. and Neuts, M. F. (1990) A single-server queue with server vacations and a class of non-renewal arrival processes. Adv. Appl. Prob. 22, 676–705.CrossRefGoogle Scholar
[17] Melamed, B. and Whitt, W. (1990) On arrivals that see time averages. Operat. Res. 38, 156–172.Google Scholar
[18] Miyazawa, M. (1994) Rate conservation laws: a survey. Queueing Systems 15, 1–58.Google Scholar
[19] Neuts, M. F. (1975) Probability distributions of phase type. Liber Amicorum Prof. Emeritus H. Florin Department of Mathematics, University of Louvain, 173–206.Google Scholar
[20] Neuts, M. F. (1986) Generalizations of the Pollaczek-Khinchine integral equation in the theory of queues. Adv. Appl. Prob. 18, 952–990.Google Scholar
[21] Neuts, M. F. (1989) Structured Stochastic Matrices of M/G/1 Type and their Applications. Marcel Dekker, New York.Google Scholar
[22] Ross, S. (1985) Introduction to Probability Models. 3rd edn. Academic Press, New York.Google Scholar
[23] Smith, W. L. (1954) Asymptotic Renewal Theorems. Proc. R. Soc. Edinburgh. A 64, 9–48 Google Scholar
[24] Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
[25] Whitt, W. (1991) A review of L = ?W and extensions. Queueing Systems. Google Scholar
[26] Wolff, R. (1989) Stochastic Modeling and the Theory of Queues. Prentice Hall, New York.Google Scholar