Graph-logical models for (n, f, k) – and consecutive - k-out-of-n – systems

Abstract

The article is devoted to methods of constructing graph-logical models of fault-tolerant multiprocessor systems. In particular,
systems of the type (n, f, k), linear consecutive-k-out-of-n and circular consecutive-k-out-of-n are considered, which are
characterized by the failure of the system when a certain number of consecutive processors fail. Graph-logical models can be used to
estimate the reliability parameters of fault-tolerant multiprocessor systems by conducting statistical experiments with models of their
behavior in the failure flow. The graph-logical models under construction are based on the basic models with a minimum of lost
edges. It is determined that to build a graph-logical model of systems of this type, it is sufficient to calculate the maximum possible
number of failed processors at which the system remains in operation. A graph-logical model of a basic system that can handle this
number of failures is built, without taking into account the sequence of these failures. The next step is to identify all possible
consecutive failures that cause the system to fail. Then, the base model is modified in such a way as to reflect the failure of the
system when consecutive failures occur. This means weakening the base model on the previously determined vectors. The proposed
methods of model construction can be used both for linear and circular consecutive-k-out-of-n systems and for (n, f, k) systems. A
minor difference will be in the calculation of some parameters. The paper describes the calculation of such parameters as the
maximum allowable number of failures at which the system remains in an operational state, as well as the calculation of the number
of all combinations of consecutive failures at which the system fails. Experiments have been conducted to confirm the model's
compliance with the system's behavior in the failure flow. Examples are given to demonstrate the process of building graph-logical
models for linear consecutive-k-out-of-n, circular consecutive-k-out-of-n and (n, f, k) systems using the proposed methods.