Overview
We illustrate a number of optimal topological network design problems in queueing networks in this chapter. We conclude our study of this problem with a number of applications and tie together the theory of queues and algorithms for the solution of a number of TND problems. These are probably the most difficult optimization problems we examine in this volume because of the integer and nonlinear programming aspects. We break down the problems into fixed topology problems and spatially generated ones. The fixed topology problems have a given topology that must be evaluated as is. The spatially generated topologies occur when the topology is not fixed, and we use mathematical programming concepts to generate the topologies a priori. They represent fundamental queueing network design problems that are not only challenging but useful in many applications. Figure 9.1 represents all TND optimization problems.
Problems worthy of attack prove their worth by fighting back.
—Paul Erdos
If you optimize everything, you will always be unhappy.
—Donald Knuth
Topology is the science of fundamental pattern and structural relationships of event constellations.
—R. Buckminster Fuller
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For both the finite exponential case and the generalized service distribution case, the first seven equations are similar. Equations (9.1) through (9.4) deal with arrivals and the feedback problem associated with the holding node h. Equations (9.5) through (9.7) are associated with solving equation (9.4) with z being a dummy parameter utilized in simplifying the solution. r 1 and r 2 are the roots of equation (9.5). Finally, equation (9.8) is an approximation to the blocking probability derived from the exact formula for the M∕M∕1∕N case.
- 2.
Equations (9.16) through (9.20) in the generalized case are concerned with the squared coefficients of variations of the arrival and service processes in the expanded network together with the formula in Equation (20) for computing the blocking probability in the generalized network. Refer to Albin and Whitt for their derivation [7, 8, 348–350]. Equations (9.17) and (9.19) are based on the work of Labetoulle and Pujolle [198]. Refer to [175] for further details of the integration and development of these equations and how they are used in the generalized expansion method.
- 3.
Equations (9.39) to (9.42) are related to the arrivals and feedback in the holding node. The Equations (9.43) to (9.45) are used for solving Equation (9.42) with z used as a dummy parameter for simplicity of the solution. Lastly, Equation (9.46) gives the the blocking probability for the M∕G∕C∕C queue. Hence, we essentially have five equations to solve, viz., 9.39 to 9.42 and 9.46.
References
Albin, S.L., 1980. “Approximating Superposition Arrival Processes of Queues,” Bell Laboratories Holmdel, N.J.
Albin, S.L., 1982. “On Poisson Approximations for Superposition arrival processes in Queues,”Mgmt. Science 28,(2), Feb. 1982.,126-137.
Burkhard, R. and U. Derigs, 1980. Assignment and Matching Problems Springer-Verlag: Berlin.
R. E. Burkard, 1984. “Locations with spatial interaction – Quadratic Assignment Problem”, in: R.L. Francis and P.B. Mirchandani (eds.), Discrete Location Theory, Academic Press, New York.
D. Chhajed, B. Montreuil, and T. Lowe, 1992. “Flow Network Design for Manufacturing System Layout”, European Journal of Operational Research 57, pp 145-161.
Evans, J.R. and E. Minieka, 1992. Optimization Algorithms for Networks and Graphs. New York: Marcel Dekker, Inc.
Garcia, A. and J. MacGregor Smith, 2007. Facilities Planning and Design. New Jersey: Pearson.
R. J. Gaskins and J. M. A. Tanchoco, 1987, “Flow Path Design for Automated Guided Vehicle System”, International Journal of Production Research 25, No. 5, pp 667-676
D. Gross and C. Harris. Fundamentals of Queueing Theory, Second Edition. John Wiley & Sons, New York, 2008.
Kerbache, L. and J. MacGregor Smith, 1987. “The Generalized Expansion Method for Open Finite Queueing Networks.” The European Journal of Operations Research 32, 448–461.
Labetoulle, J. and Pujolle, G., 1980. “Isolation Method in a network of queues,” IEEE Trans. on Software Eng’g., SE-6 4, 373-380.
Li, Wu-Ji and J. MacGregor Smith, 1995. An Algorithm for Quadratic Assignment Problems,The European Journal of Operational Research. 81, 205-216.
Pardalos, P. and H. Wolkowicz eds., 1998. Quadratic Assignment and Related Problems, Dimacs Series in Discrete Mathematics and Theoretical Computer Science.Volume 16. American Mathematical Society.
Pritsker, A.A.B., 1979. Modeling and Analysis Using Q-GERT Networks. John Wiley and Sons, N.Y., 2nd Ed.
Sahni, S., and Gonzalez, T., 1976. “P-complete approximation problems.” J. ACM 23, 555-565.
Smith, J. Macgregor, D. T. Lee and Judith S. Liebman, 1980. An 0(NlogN) Heuristic Algorithm for the Rectilinear Steiner Minimal Tree Problem,Engineering Optimization 4, 179–192.
Smith, J. Macgregor, R.J. Graves, and L. Kerbache, 1986. “QNET: An Open Queueing Network Model for Material Handling Systems Analysis,” Material Flow Vol 3, 225-242.
Smith, J. MacGregor and Wu-Ji Li, 2001. “Quadratic Assignment Problems and State Dependent Network Flows,” Journal of Combinatorial Optimization 5, 421-443.
Smith, J. MacGregor, 2011. “Optimal Routing in Closed Queueing Networks with State Dependent Routing.” INFOR, 49 (1), 45-62.
Smith, J. MacGregor, 2015. “Queue Decomposition and Finite Closed Queueing Network Models.” Computers and Operations Research, 53, 176-193.
Tansel, B.C., R.L. Francis, and T.J. Lowe., 1983. “Location on Networks: A Survey. Part I: The p-center and p-Median Problems.” Mgmt. Sci., 29(4), 482-497.
Van Vuren, M., I. Adan, S.A. Resing-Sassen, 2005. “Performance Analysis of Multi-server Tandem Queues with Finite Buffers and Blocking.” OR Spectrum, 27, 315-338.
Whitt, W., 1981. “Approximating a Point Process by a Renewal Process: The View Through a Queue, An Indirect approach,” Mgt. Sci. 27, (6), 619–636.
Whitt, W., 1983. “The Queueing Network Analyzer.” The Bell System Technical Journal. 62 (9), 2779–2815.
Whitt, W. 1985. “The Best Order for Queues in Series.” Mgmt. Sci. 31 (4), 475-487.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Smith, J.M. (2018). Optimal Topology Problems (OTOP) G(V, E)∗ in TND. In: Introduction to Queueing Networks. Springer Series in Operations Research and Financial Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-78822-7_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-78822-7_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-78821-0
Online ISBN: 978-3-319-78822-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)