Hostname: page-component-84b7d79bbc-fnpn6 Total loading time: 0 Render date: 2024-07-28T16:46:11.215Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimality of the round-robin routing policy

Part of: Operations research and management science Computer system organization Miscellaneous topics in calculus of variations and optimal control Special processes Optimality conditions

Published online by Cambridge University Press:  14 July 2016

Zhen Liu  and
Don Towsley
Zhen Liu*
Affiliation:
INRIA
Don Towsley*
Affiliation:
University of Massachusetts, Amherst
*
∗ Postal address: INRIA Centre Sophia Antipolis, 2004 Route des Lucioles, 06560 Valbonne, France. This author's research was partially supported by CEC DG-XIII under the ESPRIT-BRA grant QMIPS.
∗∗ Postal address: Department of Computer Science, University of Massachusetts, Amherst, MA 01003, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider the problem of routing customers to identical servers, each with its own infinite-capacity queue. Under the assumptions that (i) the service times form a sequence of independent and identically distributed random variables with increasing failure rate distribution and (ii) state information is not available, we establish that the round-robin policy minimizes, in the sense of a separable increasing convex ordering, the customer response times and the numbers of customers in the queues.

Keywords

OPTIMAL ROUTINGSCHEDULINGSAMPLE PATH ANALYSISSTOCHASTIC ORDERING

MSC classification

Primary: 60K25: Queueing theory
Secondary: 49K30: Optimal solutions belonging to restricted classes 49N30: Problems with incomplete information 68M20: Performance evaluation; queueing; scheduling 90B22: Queues and service 90B80: Discrete location and assignment
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 2 , June 1994 , pp. 466 - 475
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research of both authors supported in part by the National Science Foundation under grants ASC 88-8802764 and NCR-9116183.

References

[1] Baccelli, F., Liu, Z. and Towsley, D. (1993) Extremal scheduling of parallel processing systems with and without real-time constraints. J. Assoc. Comput. Mach. 40, 1209–1237.Google Scholar
[2] Chang, C. S. (1992) A new ordering for stochastic majorization: theory and applications. Adv. Appl. Prob. 24, 604–634.CrossRefGoogle Scholar
[3] Chang, C. S., Chao, X. L. and Pinedo, M. (1990) A note on queues with Bernoulli routing. Proc. 29th Conf. Decision and Control, Hawaii, December.Google Scholar
[4] Daley, D. J. (1987) Certain optimality properties of the first come-first served discipline for G/G/s queues. Stoch. Proc. Appl. 25, 301–308.Google Scholar
[5] Ephremides, A., Varaiya, P. and Walrand, J. (1980) A simple dynamic routing problem. IEEE Trans. Autom. Control. 25, 690–693.Google Scholar
[6] Foss, S. G. (1980) Approximation of multichannel queueing systems. Siberian Math. J. 21, 851–857.Google Scholar
[7] Foss, S. G. (1981) Comparison of servicing strategies in multichannel queueing systems. Siberian Math. J. 22, 142–147.Google Scholar
[8] Gün, L. and Jean-Marie, A. (1993) Parallel queues with resequencing. J. Assoc. Comput. Mach. 40, 1188–1208.Google Scholar
[9] Hajek, B. (1985) Extremal splittings of point processes. Math. Operat. Res. 10, 543–556.Google Scholar
[10] Hordijk, A. and Koole, G. (1990) On the optimality of the generalized shortest queue policy. Prob. Eng. Inf. Sci. 4, 477–487.Google Scholar
[11] Jean-Marie, A. and Liu, Z. (1992) Stochastic comparisons for queueing models via random sums and intervals. Adv. Appl. Prob. 24, 960–985.CrossRefGoogle Scholar
[12] Kingman, J. F. C. (1970) Inequalities in the theory of queues. J. R. Statist. Soc. B 32, 102–110.Google Scholar
[13] Liu, Z. and Towsley, D. (1994) Effects of service disciplines in G/G/s queueing systems. Ann. Operat. Res. 48. To appear.Google Scholar
[14] Liu, Z. and Towsley, D. (1994) Stochastic scheduling in in-forest networks. Adv. Appl. Prob. 26, 222–241.Google Scholar
[15] Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
[16] Menich, R. (1987) Optimality of shortest queue routing for dependent service stations. Proc. 26th Conf. Decision and Control, 1069–1072.Google Scholar
[17] Menich, R. and Serfozo, R. F. (1991) Optimality of routing and servicing in dependent parallel processing systems. QUESTA 9, 403–418.Google Scholar
[18] Sparaggis, P. D., Cassandras, C. G. and Towsley, D. (1993) On the duality between routing and scheduling systems with finite buffer space. IEEE Trans. Autom. Control. To appear.Google Scholar
[19] Sparaggis, P. D., Towsley, D. and Cassandras, C. G. (1993) Extremal properties of the SNQ and the LNQ policies in finite capacity systems with state-dependent service rates. J. Appl. Prob. 30, 223–236.Google Scholar
[20] Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. English translation ed. Daley, D. J. Wiley, New York.Google Scholar
[21] Towsley, D., Sparaggis, P. D. and Cassandras, C. G. (1993) Optimal routing and buffer allocation for a class of finite capacity queueing systems. IEEE Trans. Autom. Control. 37, 1446–1451.Google Scholar
[22] Towsley, D. and Sparaggis, P. D. (1993) Optimal routing in systems with ILR service time distributions.Google Scholar
[23] Vasicek, O. A. (1977) An inequality for the variance of waiting time under a general queueing discipline. Operat. Res. 25, 879–884.CrossRefGoogle Scholar
[24] Walrand, J. (1988) An Introduction to Queueing Networks. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
[25] Weber, R. R. (1978) On the optimal assignment of customers to parallel queues. J. Appl. Prob. 15, 406–413.Google Scholar
[26] Whitt, W. (1986) Deciding which queue to join: some counterexamples. Operat. Res. 34, 55–62.CrossRefGoogle Scholar
[27] Winston, W. (1977) Optimality of the shortest line discipline. J. Appl. Prob. 14, 181–189.Google Scholar
[28] Wolff, R. W. (1977) An upper bound for multi-channel queues. J. Appl. Prob. 14, 884–888.CrossRefGoogle Scholar
[29] Wolff, R. W. (1987) Upper bounds on work in system for multichannel queues. J. Appl. Prob. 24, 547–551.Google Scholar