Exact and Asymptotic Results for Connected (r,2)-out-of-(m,n):F Lattice Systems

A (r,s)-out-of-(m,n):F system consists in m×n elements arranged in n rows andmcolumns; it fails if all elements in a block r×s fail. The interest in such systems has never dwindled since their introduction by Salvia and Lasher (1990), because of the ever increasing number of real-life applications: reliability of electronic devices, X-ray and disease diagnostic, security of communications and property, pattern search systems, etc. Computing their exact availability has been, in the general case, deemed a numerically complex task by Nashwan (2018) and Zhao et al. (2011). Only a few configurations have allowed simple solutions.

The special case of (2,2)-out-of-(m,n):F systems has been studied by (Malinowski and Tanguy, 2022), in which exact solutions where provided for 2 ≤ m ≤ 10 and arbitrary n through recurrence relations, the order of which increases drastically with m. Based on these results, an analytical, asymptotic expansion was given for large m and n, which was shown to be in excellent agreement for m as low as 4.

In this paper, we generalize our previous work to (r,2)-out-of-(m,n):F systems. We have obtained the exact expressions of the availability for 3 ≤ r ≤ 8 and several values of m, while n can remain arbitrary. An asymptotic expansion has again been inferred for arbitrary (large) m and n, which allows quick numerical evaluations. We have also calculated the Mean Time To Failure of such systems, assuming that all elements are identical and obey a Weibull lifetime distribution.

Keywords: Cellular network, Connected (r,s)-out-of-(m,n):F lattice system, Network reliability, Availability, Generating function, Asymptotic expansion.

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