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Resource pooling in queueing networks with dynamic routing

Published online by Cambridge University Press:  01 July 2016

C. N. Laws*
Affiliation:
University of Cambridge
*
Present address: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK.

Abstract

In this paper we investigate dynamic routing in queueing networks. We show that there is a heavy traffic limiting regime in which a network model based on Brownian motion can be used to approximate and solve an optimal control problem for a queueing network with multiple customer types. Under the solution of this approximating problem the network behaves as if the service-stations of the original system are combined to form a single pooled resource. This resource pooling is a result of dynamic routing, it can be achieved by a form of shortest expected delay routing, and we find that dynamic routing can offer substantial improvements in comparison with less responsive routing strategies.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

Supported by SERC grants 8700117X and GR/F 94194.

References

[1] Chen, H. and Mandelbaum, A. (1991) Stochastic discrete flow networks: diffusion approximations and bottlenecks. Ann. Prob. 19, 14631519.CrossRefGoogle Scholar
[2] Cohen, J. E. and Kelly, F. P. (1990) A paradox of congestion in a queueing network. J. Appl. Prob. 27, 730734.CrossRefGoogle Scholar
[3] Ford, L. R. and Fulkerson, D. R. (1962) Flows in Networks. Princeton University Press.Google Scholar
[4] Foschini, G. J. (1977) On heavy traffic diffusion analysis and dynamic routing in packet switched networks. In Computer Performance , ed. Chandy, K. M. and Reiser, M., pp. 499513. North-Holland, Amsterdam.Google Scholar
[5] Foschini, G. J. and Salz, J. (1978) A basic dynamic routing problem and diffusion. IEEE Trans. Comm. 26, 320327.CrossRefGoogle Scholar
[6] Gallager, R. G. (1977) A minimum delay routing algorithm using distributed computation. IEEE Trans. Comm. 25, 7385.CrossRefGoogle Scholar
[7] Gondran, M. and Minoux, M. (1984) Graphs and Algorithms. Wiley-Interscience, New York.Google Scholar
[8] Harrison, J. M. (1985) Brownian Motion and Stochastic Flow Systems. Wiley, New York.Google Scholar
[9] Harrison, J. M. (1988) Brownian models of queueing networks with heterogeneous customer populations. In Stochastic Differential Systems, Stochastic Control Theory and Applications , IMA Volume 10, ed. Fleming, W. and Lions, P. L., pp. 147186, Springer-Verlag, New York.CrossRefGoogle Scholar
[10] Harrison, J. M. and Wein, L. M. (1989) Scheduling networks of queues: heavy traffic analysis of a simple open network. QUESTA 5, 265280.Google Scholar
[11] Harrison, J. M. and Wein, L. M. (1990) Scheduling networks of queues: heavy traffic analysis of a two-station closed network. Operat. Res. 38, 10521064.CrossRefGoogle Scholar
[12] Harrison, J. M. and Williams, R. J. (1987) Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22, 77115.CrossRefGoogle Scholar
[13] Hu, T. C. (1969) Integer Programming and Network Flows. Addison-Wesley, Reading, Massachusetts.Google Scholar
[14] Johnson, D. P. (1983) Diffusion Approximations for Optimal Filtering of Jump Processes and for Queueing Networks. PhD thesis, Dept. of Mathematics, University of Wisconsin, Madison.Google Scholar
[15] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, Chichester.Google Scholar
[16] Kelly, F. P. (1991) Loss networks. Ann. Appl. Prob. 1, 319378.CrossRefGoogle Scholar
[17] Laws, C. N. (1990) Dynamic Routing in Queueing Networks. PhD thesis, Statistical Laboratory University of Cambridge.Google Scholar
[18] Laws, C. N. and Louth, G. M. (1990) Dynamic scheduling of a four-station queueing network. Prob. Eng. Inf. Sci. 4, 131156.CrossRefGoogle Scholar
[19] Lomonosov, M. V. (1985) Combinatorial approaches to multiflow problems. Discrete Appl. Math. 11, 193.Google Scholar
[20] Martins, L. F. and Kushner, H. J. (1990) Routing and singular control for queueing networks in heavy traffic. SIAM J. Control Optim. 28, 12091233.CrossRefGoogle Scholar
[21] Minoux, M. (1981) Optimum synthesis of a network with non-simultaneous multicommodity flow requirements. In Ann. Discrete Math. 11, Studies on Graphs and Discrete Programming , ed. Hansen, P., pp. 269277, North-Holland, Amsterdam.CrossRefGoogle Scholar
[22] Papernov, B. A. (1976) On existence of multicommodity flows. In Studies in Discrete Optimization , ed. Fridman, A. A., pp. 230261, Nauka, Moscow (in Russian).Google Scholar
[23] Peterson, W. P. (1991) A heavy traffic limit theorem for networks of queues with multiple customer types. Math. Operat. Res. 16, 90118.CrossRefGoogle Scholar
[24] Reiman, M. I. (1983) Some diffusion approximations with state space collapse. In Proc. Internat. Seminar on Modelling and Performance Evaluation Methodology , ed. Baccelli, F. and Fayolle, G., pp. 209240, Lecture Notes in Control and Information Science 60, Springer-Verlag, Berlin.Google Scholar
[25] Reiman, M. I. (1984) Open queueing networks in heavy traffic. Math. Operat. Res. 9, 441458.CrossRefGoogle Scholar
[26] Schwartz, M. (1987) Telecommunication Networks. Addison-Wesley, Reading, MA.Google Scholar
[27] Stidham, S. (1974) A last word on L = ?W. Operat. Res. 22, 417421.CrossRefGoogle Scholar
[28] Wein, L. M. (1990) Optimal control of a two-station Brownian network. Math. Operat. Res. 15, 215242.CrossRefGoogle Scholar
[29] Whitt, W. (1971) Weak convergence theorems for priority queues: preemptive-resume discipline. J Appl. Prob. 8, 7494.CrossRefGoogle Scholar
[30] Whittle, P. (1971) Optimization under Constraints. Wiley, Chichester.Google Scholar