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Single-Threshold Model Resource Network and Its Double-Threshold Modifications

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  • Liudmila Zhilyakova

    (V. A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, 65, Profsoyuznaya Street, 117997 Moscow, Russia)

Abstract

A resource network is a non-classical flow model where the infinitely divisible resource is iteratively distributed among the vertices of a weighted digraph. The model operates in discrete time. The weights of the edges denote their throughputs. The basic model, a standard resource network, has one general characteristic of resource amount—the network threshold value. This value depends on graph topology and weights of edges. This paper briefly outlines the main characteristics of standard resource networks and describes two its modifications. In both non-standard models, the changes concern the rules of receiving the resource by the vertices. The first modification imposes restrictions on the selected vertices’ capacity, preventing them from accumulating resource surpluses. In the second modification, a network with so-called greedy vertices, on the contrary, vertices first accumulate resource themselves and only then begin to give it away. It is noteworthy that completely different changes lead, in general, to the same consequences: the appearance of a second threshold value. At some intervals of resource values in networks, their functioning is described by a homogeneous Markov chain, at others by more complex rules. Transient processes and limit states in networks with different topologies and different operation rules are investigated and described.

Suggested Citation

  • Liudmila Zhilyakova, 2021. "Single-Threshold Model Resource Network and Its Double-Threshold Modifications," Mathematics, MDPI, vol. 9(12), pages 1-34, June.
  • Handle: RePEc:gam:jmathe:v:9:y:2021:i:12:p:1444-:d:578777
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    References listed on IDEAS

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    1. Dhar, Deepak, 1999. "The Abelian sandpile and related models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 263(1), pages 4-25.
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    Cited by:

    1. Liudmila Zhilyakova & Vasily Koreshkov & Nadezhda Chaplinskaia, 2022. "Some Properties of Stochastic Matrices and Non-Homogeneous Markov Chains Generated by Nonlinearities in the Resource Network Model," Mathematics, MDPI, vol. 10(21), pages 1-18, November.

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