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Minimizing response times and queue lengths in systems of parallel queues

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Ger Koole ,
Panayotis D. Sparaggis  and
Don Towsley
Ger Koole*
Affiliation:
Vrije Universiteit
Panayotis D. Sparaggis*
Affiliation:
University of Massachusetts
Don Towsley*
Affiliation:
University of Massachusetts
*
∗Postal address: Department of Mathematics and Computer Science, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands.
∗∗Postal address: Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003–4610, USA.
∗∗∗Postal address: Department of Computer Science, University of Massachusetts, Amherst, MA 01003–4610, USA. Email address: towsley@ca.umass.edu
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Abstract

We consider the problem of routeing customers to one of two parallel queues. Arrivals are independent of the state of the system but otherwise arbitrary. Assuming that queues have infinite capacities and the service times form a sequence of i.i.d. random variables with increasing likelihood ratio (ILR) distribution, we prove that the shortest queue (SQ) policy minimizes various cost functionals related to queue lengths and response times. We give a counterexample which shows that this result is not generally true when the service times have increasing hazard rate but are not increasing in the likelihood rate sense. Finally, we show that when capacities are finite the SQ policy stochastically maximizes the departure process and minimizes the loss counting process.

Keywords

Routeingstochastic majorizationresponse timesincreasing likelihood ratio distributions

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1185 - 1193
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

This work was partially supported by NSF under contract NCR-9116183, by an IBM Graduate Fellowship Award, and by a European Human Capital and Mobility grant.

References

Barlow, R. E., and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Ephremides, A., Varaiya, P., and Walrand, J. (1980). A simple dynamic routing problem. IEEE Trans. Automatic Control 25, 690–693.Google Scholar
Hordijk, A., and Koole, G. M. (1990). On the optimality of the generalized shortest queue policy. Prob. Eng. Inf. Sci./ 4, 477–487.Google Scholar
Liu, Z., and Towsley, D. (1994). Stochastic scheduling in in-forest networks. Adv. Appl. Prob./ 26, 222–241.Google Scholar
Marshall, A. W., and Olkin, I. (1979). Inequalities: Theory of Majorization and its Applications. Academic Press, New York.Google Scholar
Menich, R., and Serfozo, R. F. (1991). Monotonicity and optimality of symmetric parallel processing systems. Queueing Systems 9, 403–418.Google Scholar
Righter, R. (1994). Scheduling. In Stochastic Orders and their Applications, eds. Shaked, M. and Shanthikumar, J. G. Academic Press, New York, pp. 381–432.Google Scholar
Righter, R., and Shanthikumar, J. G. (1992). Extremal properties of the FIFO discipline in queueing networks. J. Appl. Prob. 29, 967–978.CrossRefGoogle Scholar
Ross, S. M. (1983). Stochastic Processes. John Wiley, New York.Google Scholar
Towsley, D., Sparaggis, P. D., and Cassandras, C. G. (1992). Optimal routing and buffer allocation for a class of finite capacity queueing systems. IEEE Trans. Automatic Control 37, 1446–1451.CrossRefGoogle Scholar
Weber, R. R. (1978). On the optimal assignment of customers to parallel queues. J. Appl. Prob./ 15, 406–413.Google Scholar
Whitt, W. (1986). Deciding which queue to join: some counterexamples. Operat Res./ 34, 55–62.CrossRefGoogle Scholar
Winston, W. (1977). Optimality of the shortest line discipline. J. Appl. Prob./ 14, 181–189.Google Scholar