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

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.

Some Properties of Stochastic Matrices and Non-Homogeneous Markov Chains Generated by Nonlinearities in the Resource Network Model

“…In our team, two extensions of resource networks were proposed and studied: a network with the capacity limitations of attractor vertices and a network with greedy vertices [28].…”

“…n×n is a limit flow matrix. It was proven in [28] that the limit state exists for all topologies of resource networks, excluding cyclic networks (networks in which the GCD of the lengths of all cycles is greater than 1).…”

“…However, the behavior of resource networks with a large resource, in the general case, is rather complicated. The corresponding statements can be read in [28]; examples of the operation of networks with different topologies can be found in [33].…”

“…The resource network was first proposed in [27]. Since that time, the theory of resource networks has arisen, and a brief description can be found in [28]. This article also describes two two-threshold modifications of the standard model.…”

“…After this fact is established, the proof in the case W > T is made in analogy with the case W ≤ T. The main difference is that R * 3 is not a limit of the powers of a single matrix, but a limit of the products of different matrices. However, it is known from the previous works [28] that this limit of the products of matrices exists for every initial condition. In the case where there is only one attractor vertex in the final component, this limit of products depends only on W. Thus, the statement of the theorem follows.…”