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Extremal properties of the FIFO discipline in queueing networks

Part of: Stochastic processes Distribution theory - Probability Special processes

Published online by Cambridge University Press:  14 July 2016

Rhonda Righter  and
J. George Shanthikumar
Rhonda Righter*
Affiliation:
Santa Clara University
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
*
∗ Postal address: Department of Decision and Information Sciences, Santa Clara University, Santa Clara, CA 95053, USA.
∗∗ Postal address: Walter A. Haas School of Business, University of California, Berkeley, CA 94720, USA.
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Abstract

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:

(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.

(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.

We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.

Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

Keywords

STOCHASTIC OPTIMIZATIONSTOCHASTIC SCHEDULINGTHROUGHPUTFCFSILR DISTRIBUTIONSIHR DISTRIBUTIONS

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G17: Sample path properties 60E99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 4 , December 1992 , pp. 967 - 978
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

Supported in part by NSF grant ECS-8811234.

References

Baras, J. S., Dorsey, A. J. and Makowski, A. M. (1985a) Two competing queues with linear costs and geometric service requirements: the µc-rule is often optimal. Adv. Appl. Prob. 17, 186–209.Google Scholar
Baras, J. S., Ma, D.-J. and Makowski, A. M. (1985b) K competing queues with geometric requirements and linear costs: the µc rule is always optimal. Syst. Control Lett. 6, 173–180.Google Scholar
Buyukkoc, C., Varaiya, P. and Walrand, J. (1985) The cµ rule revisited. Adv. Appl. Prob. 17, 237–238.Google Scholar
Buzacott, J. and Shanthikumar, J. G. (1991) Stochastic Models of Manufacturing Systems. Forthcoming.Google Scholar
Chang, C.-S. and Yao, D. D. (1990) Rearrangement, majorization and stochastic scheduling. Preprint.Google Scholar
Derman, C., Lieberman, G. J. and Ross, S. M. (1978) A renewal decision problem. Management Sci. 24, 554–561.Google Scholar
Halfin, S. and Whitt, W. (1989) An extremal property of the FIFO discipline via an ordinal version of L = ?W. Commun. Statist.-Stoch. Models 5, 515–529.Google Scholar
Hirayama, T. and Kijima, M. (1989) An extremal property of FIFO discipline in G/IFR/1 queues. Adv. Appl. Prob. 21, 481–484.Google Scholar
Hirayama, T., Kijima, M. and Nishimura, S. (1989) Further results for dynamic scheduling of multiclass G/G/1 queues. J. Appl. Prob. 26, 595–603.Google Scholar
Hirayama, T. and Kijima, M. (1992) Single machine scheduling problem when the machine capacity varies stochastically. Operat. Res. 40, 376–383.Google Scholar
Ku, P. and Niu, S. (1986) On Johnson's two-machine flow shop with random processing times. Operat. Res. 34, 130–136.Google Scholar
Pinedo, M. (1983) Stochastic scheduling with release dates and due dates. Operat. Res. 31, 559–572.Google Scholar
Righter, R. and Shanthikumar, J. G. (1989) Scheduling multiclass single server queueing systems to stochastically maximize the number of successful departures. Proc. Eng. Inf. Sci. 3, 323–333.Google Scholar
Righter, R., Shanthikumar, J. G. and Yamazaki, G. (1990) On extremal service disciplines in single-stage queueing systems. J. Appl. Prob. 27, 409–416.Google Scholar
Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Shanthikumar, J. G. and Sumita, U. (1987) Convex ordering of sojourn times in single-server queues: extremal properties of FIFO and LIFO service disciplines. J. Appl. Prob. 24, 737–748.Google Scholar
Sauer, C. H. and Chandy, K. M. (1981) Computer Systems Performance Modeling. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
Towsley, D. and Baccelli, F. (1991) Comparisons of service disciplines in a tandem queueing network with real time constraints. Operat. Res. Lett. 10, 49–55.Google Scholar
Walrand, J. (1988). An Introduction to Queueing Networks. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
Yamazaki, G. and Sakasegawa, H. (1987) An optimal design problem for limited processor sharing systems. Management Sci. 33, 1010–1019.CrossRefGoogle Scholar
Yamazaki, G., Sakasegawa, H. and Shanthikumar, J. G. (1991) A conservation law for single-server queues and its applications. J. Appl. Prob. 28, 198–201.Google Scholar