Impact of Common-Cause Failures on the Availability of Connected (r,s)-out-of-(m,n):F systems

There has been a renewed interest in connected bi-dimensional configurations, since they successfully describe real-life systems such as cellular networks, communication and property security solutions, etc. The computation of their availability has long been known as a delicate task, especially for large systems. Most of the published studies have considered independent failures for identical elements.
In this work, we investigate the modification of the availability of connected (r,s)-out-of-(m,n):F systems when common-cause failures or correlations are present. It turns out that the availability may be larger or smaller than its value in the case of purely independent failures; this is also true for the failure frequency. The transition between these two behaviors occurs at a component critical unavailability q_c that depends on the size of the system. Analytical expressions for the asymptotic values of q_c are given in the case of the β-factor model and binomial failure rate models, as well as for correlations described by a degenerate Gaussian copula. A comparison with the case of one-dimensional consecutive configurations is also provided.

Keywords: Connected (r,s)-out-of-(m,n):F lattice system, Network reliability, Availability, Common-cause failures, Correlations, Asymptotic expansions.