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Erschienen in:
Introduction G(V, E) Kapitel lesen Erstes Kapitel lesen

2018 | OriginalPaper | Buchkapitel

9. Optimal Topology Problems (OTOP) G(V, E)∗ in TND

verfasst von : J. MacGregor Smith

Erschienen in: Introduction to Queueing Networks

Verlag: Springer International Publishing

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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.

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Vorheriges Kapitel Optimal Routing Problems (ORTE) G(E ∗) in TND
Nächstes Kapitel Final Coda
Fußnoten
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.
 
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Metadaten
Titel
Optimal Topology Problems (OTOP) G(V, E)∗ in TND
verfasst von
J. MacGregor Smith
Copyright-Jahr
2018
Buch
Introduction to Queueing Networks

Print ISBN: 978-3-319-78821-0

Electronic ISBN: 978-3-319-78822-7

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-319-78822-7_9

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