Probabilistic Reliability via Subsystem Structures of Arrangement Graph Networks

With the rapid growth of the number of processors in a multiprocessor system, faulty processors occur in it with a probability that rises quickly. The probability of a subsystem with an appropriate size being fault-free in a definite time interval is a significant and practical measure of the reliability for a multiprocessor system, which characterizes the functionality of a multiprocessor system well. Motivated by the study of subgraph reliability, as well as the attractive structure and fault tolerance properties of
$(n, k)$
-arrangement graph
$A_{n, k}$
, we focus on the subgraph reliability for
$A_{n, k}$
under the probabilistic fault model in this article. First, we investigate intersections of no more than four subgraphs in
$A_{n, k}$
, and classify all the intersecting modes. Second, we focus on the probability
$P(q, A_{n, k}^{n-1, k-1})$
with which at least one
$(n-1, k-1)$
-subarrangement graph is fault-free in
$A_{n, k}$
, when given a uniform probability
$q$
with which a single vertex is fault-free, and we establish the
$P(q, A_{n, k}^{n-1, k-1})$
by adopting the principle of inclusion–exclusion under the probabilistic fault model. Finally, we study the probabilistic fault model involving a nonuniform probability with which a single vertex is fault-free, and we prove that the
$P(q, A_{n, k}^{n-1, k-1})$
under both models is very close to the asymptotic value by both theoretical arguments and experimental results.