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A join the shorter queue model in heavy traffic

Part of: Special processes Operations research and management science Markov processes

Published online by Cambridge University Press:  14 July 2016

Stephen R. E. Turner
Stephen R. E. Turner*
Affiliation:
University of Cambridge
*
∗Postal address: Statistical Laboratory, University of Cambridge, 1 Wilberforce Road, Cambridge, CB3 0WB, UK. Email address: [email protected]
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Abstract

We prove a new heavy traffic limit result for a simple queueing network under a ‘join the shorter queue’ policy, with the amount of traffic which has a routeing choice tending to zero as heavy traffic is approached. In this limit, the system considered does not exhibit state space collapse as in previous work by Foschini and Salz, and Reiman, but there is nevertheless some resource pooling gain over a policy of random routeing.

Keywords

Queueing networkjoin the shorter queueresource poolingstate space collapse

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J60: Diffusion processes 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 212 - 223
Copyright
Copyright © 2000 by The Applied Probability Trust 

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References

Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Bramson, M. (1998). State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Systems 30, 89–148.CrossRefGoogle Scholar
Dai, J. G., and Williams, R. J. (1995). Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedrons. Theory Prob. Appl. 40, 1–40. (First appeared in the Russian journal of the same name.)CrossRefGoogle Scholar
Ethier, S. N., and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.CrossRefGoogle Scholar
Foschini, G. J., and Salz, J. (1978). A basic dynamic routing problem and diffusion. IEEE Trans. Commun. 26, 320–327.CrossRefGoogle Scholar
Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.Google Scholar
Harrison, J. M. (1997). Brownian models of open processing networks: canonical representation of workload. Submitted for publication.Google Scholar
Harrison, J. M., and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Prob. 9, 302–308.CrossRefGoogle Scholar
Harrison, J. M., and van Mieghem, J. A. (1997). Dynamic control of Brownian networks: state space collapse and equivalent workload formulations. Ann. Appl. Prob. 7, 747–771.CrossRefGoogle Scholar
Harrison, J. M., and Williams, R. J. (1987). Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22, 77–115.CrossRefGoogle Scholar
Harrison, J. M., and Williams, R. J. (1987). Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Prob. 15, 115–137.CrossRefGoogle Scholar
Iglehart, D. L., and Whitt, W. (1970). Multiple channel queues in heavy traffic. I. Adv. Appl. Prob. 2, 150–177.CrossRefGoogle Scholar
Iglehart, D. L., and Whitt, W. (1970). Multiple channel queues in heavy traffic. II: Sequences, networks, and batches. Adv. Appl. Prob. 2, 355–369.Google Scholar
Kelly, F. P., and Laws, C. N. (1993). Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling. Queueing Systems 13, 47–86.CrossRefGoogle Scholar
Kent, J. (1978). Time-reversible diffusions. Adv. Appl. Prob. 10, 819–835.CrossRefGoogle Scholar
Lindvall, T. (1973). Weak convergence of probability measures and random functions in the function space D[0,∞). J. Appl. Prob. 10, 109–121.CrossRefGoogle Scholar
Reiman, M. I. (1984). Open queueing networks in heavy traffic. Math. Operat. Res. 9, 441–458.CrossRefGoogle Scholar
Reiman, M. I. (1984). Some diffusion approximations with state space collapse. In Modelling and Performance Evaluation Methodology, eds Baccelli, F. and Fayolle, G., INRIA Lecture Notes in Control and Information Sciences No. 60. Springer, New York.Google Scholar
Rogers, L. C. G., and Williams, D. (1987). Diffusions, Markov Processes, and Martingales, 1st edn, Vol. 2: Itô Calculus. John Wiley, New York.Google Scholar
Skorohod, A. V. (1961). Stochastic equations for diffusion processes in a bounded region. Theory Prob. Appl. 6, 264–274. (Translated from the Russian journal of the same name.)CrossRefGoogle Scholar
Skorohod, A. V. (1962). Stochastic equations for diffusion processes in a bounded region. II. Theory Prob. Appl. 7, 3–23. (Translated from the Russian journal of the same name.)CrossRefGoogle Scholar
Turner, S. R. E. (1996). Resource pooling in stochastic networks. Ph.D. thesis, University of Cambridge.Google Scholar
Turner, S. R. E. (1998). The effect of increasing routing choice on resource pooling. Prob. Eng. Inf. Sci. 12, 109–124.CrossRefGoogle Scholar
Turner, S. R. E. (1999). Large deviations for join the shorter queue. Res. Rept. 1999-5. University of Cambridge Statistical Laboratory. Submitted for publication. To appear in Analysis of Communication Networks: Call Centres, Traffic and performance, eds. Glynn, P. W., McDonald, D. R. and Turner, S. R. E. (Fields Institute Communications). AMS, Providence, RI.Google Scholar
Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Operat. Res. 5, 67–85.CrossRefGoogle Scholar
Williams, R. J. (1987). Reflected Brownian motion with skew symmetric data in a polyhedral domain. Prob. Theory Rel. Fields 75, 459–485.CrossRefGoogle Scholar
Williams, R. J. (1995). Semimartingale reflecting Brownian motions in the orthant. In Stochastic Networks, eds Kelly, F. P. and Williams, R. J. (IMA Volumes in Mathematics and its Applications 71). Springer, New York, pp. 125–137.CrossRefGoogle Scholar