On the cascade GL-model and its properties

Abstract

The article proposes a new direction for the further development of GL-models – models on the basis of which performs the
calculation of the reliability parameters of fault-tolerant multiprocessor systems. Such models reflect the reaction of the system to the
appearance of failures of arbitrary multiplicity. The essence of the new direction is the construction of a model by composition of
several basic GL-models in such a way that the values of the edge functions of one model form the input vector of the next one. This
article shows that the model obtained in this way, which is proposed to be called cascade model, will also be basic and, in general
case, can consist of an arbitrary number of submodels. This article gives a formula that allows one to determine the value of the
degree of fault tolerance of the cascade model, depending on the values of the levels of fault tolerance of its component submodels.
This article shows that the graphs of both the cascade and regular models are cyclic and have the same number of edges. At the same
time, despite the fact that the intermediate submodels also have graphs, their presence does not increase the complexity of the model
as a whole, since only the expressions of the edge functions are used in them. This article contains examples that confirm the
correctness of the theoretically obtained results, and it also shows that the cascade model, at least in some cases, has lower
computational complexity (the total number of logical operations in the expressions of edge functions) compared to the basic model.
It was found that although the cascade model is basic, the sets of edges it loses and the regular basic GL-model on some input vectors
may differ. In certain cases, several alternative cascade models can be built, which will differ in their parameters, but will have the
same resulting value of the degree of fault tolerance. Given an example, where the properties of such alternative cascade models are
compared. It was found that such models differ both in computational complexity and, in some cases, in the sets of edges they lose on
certain input vectors. The possibility of modifying the cascade model was shown by changing the expressions of the edge functions
of its component submodels, both individually and several simultaneously. At the same time, it is possible to block vectors with an
increased multiplicity of zeros. A number of tasks for future research were formulated.