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Network design and dynamic routing under queueing demand

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Abstract

Given a nonhierarchical network and time-varying flow requirements, the problem of determining optimal capacities is termed design; that of determining optimal flows as dynamic routing. We formulate a linear program to solve both simultaneously in the case of deterministic flow requirements. A probability distribution termed the Erlang Difference Distribution is derived from a queueing model to describe random flow requirements, and this case leads to a separable convex program that has a linear programming equivalent. Both linear programs are amenable to Dantzig-Wolfe decomposition, which reveals subproblems that yield to special techniques of solution.

Zusammenfassung

Gegeben sei ein nicht hierarchisches Netzwerk mit zeitabhängigen Flüssen. Das Problem, optimale Kapazitäten festzusetzen, wird als Netzwerk-Entwurf bezeichnet. Die Bestimmung optimaler Flüsse bezeichnet man als dynamische Flußführung. Es wird ein lineares Programm formuliert, das beide Probleme gleichzeitig löst, sofern deterministische Flüsse vorliegen. Sodann wird eine Wahrscheinlichkeitsverteilung namens „Erlang'sche Differenzen-Verteilung“ aus einem Warteschlangenmodell abgeleitet, um zufällige Flüsse zu beschreiben. Dies führt auf ein separables konvexes Programm mit einem linearen Programm als Äquivalent. Bei beiden betrachteten linearen Programmen kann die Dantzig-Wolfe Dekomposition angewendet werden, wobei die auftretenden Teilprobleme durch spezielle Techniken gelöst werden können.

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References

  • Assad, A.A.: Multicommodity Networks Flows — A Survey. Networks8, 2, Spring, 1978, 37–91.

    Google Scholar 

  • Bondy, J.A., andU.S.R. Murty: Graph Theory with Applications. American Elsevier Publishing Company, New York 1976.

    Google Scholar 

  • Cooper, R.B.: Introduction to Queueing Theory. The Macmillan Company, New York, N.Y. 1972.

    Google Scholar 

  • Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton, N.J., 1963.

    Google Scholar 

  • Ford, L.R., andD.R. Fulkerson: A Suggested Computation for Maximal Multi-Commodity Network Flows. Management Science5, 1958, 97–101.

    Google Scholar 

  • —: Flows in Networks. Princeton University Press, Princeton, N.J., 1962.

    Google Scholar 

  • Gaver, D.P., andG.L. Thompson: Programming and Probability Models in Operations Research. Brooks/Cole, Monterey, California, 1973.

    Google Scholar 

  • Gomory, R.E., andT.C. Hu: Multi-terminal Network Flows. Journal of the Society for Industrial and Applied Mathematics9, 4, December, 1961, 551–570.

    Google Scholar 

  • —: An Application of Generalized Linear Programming to Network Flows. Journal of the Society for Industrial and Applied Mathematics10, 2, June, 1962, 260–283.

    Google Scholar 

  • —: Synthesis of a Communications Network. Journal of the Society for Industrial and Applied Mathematics12, 2, June, 1964, 347–369.

    Google Scholar 

  • Gross, D., andC.M. Harris: Fundamentals of Queueuing Theory. John Wiley and Sons, New York 1974.

    Google Scholar 

  • Kall, P.: Stochastic Linear Programming. Springer-Verlag, New York 1976.

    Google Scholar 

  • Kleinrock, L.: Queueing Systems Volume 1: Theory. John Wiley and Sons, New York, N.Y., 1975.

    Google Scholar 

  • Kortanek, K.O., D.N. Lee andG.G. Polak: A Linear Programming Model for Design of Communications Networks with Time-Varying Demands. Naval Research Logistics Quarterly28, 1, March 1981.

    Google Scholar 

  • Kumin, H.: On Characterizing the Extrema of a Function of Two Variables, One of Which is Discrete. Management Science20, 1, September, 1973, 126–130.

    Google Scholar 

  • MacLane, S., andG. Birkoff: Algebra. Macmillan, New York, 1979.

    Google Scholar 

  • Messerli, E.J.: Proof of a Convexity Property of the Erlang B Formula. Bell System Technical Journal51, 1972, 951–953.

    Google Scholar 

  • Murty, K.: Linear and Combinatorial Programming. John Wiley and Sons, New York, 1976.

    Google Scholar 

  • Papadimitriou, C.H., andK. Steiglitz: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs, NJ, 1982.

    Google Scholar 

  • Polak, G.G.: Multihour Multicommodity Design Synthesis under Queueing Demand for Flows. PhD thesis, Carnegie-Mellon University, April, 1983

Download references

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Research partially supported by National Science Foundation Grant ECS-8300214.

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Kortanek, K.O., Polak, G.G. Network design and dynamic routing under queueing demand. Zeitschrift für Operations Research 29, 141–160 (1985). https://doi.org/10.1007/BF01920306

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  • DOI: https://doi.org/10.1007/BF01920306

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