ABSTRACT
There has been a recent interest about the availability of connected (r,s)-out-of-(m,n):F lattice systems. Computing its exact value has been deemed a numerically complex task by Zhao et al. (2011) and Nashwan (2015). This calculation could be accomplished with less effort only in special cases as shown in Nakamura et al. (2018). Exact results have been proposed by Malinowski in 2021 for (2,2)-out-of-(m,n):F lattice systems, using recursive procedures and finding the exact system availability for m = 2, 3, 4.

In the present work, an alternate derivation of these results clearly demonstrating the recursive nature of the problem is proposed, which lends itself to symbolic computation. The recursion relations, as well as the associated generating functions, have been obtained for m up to 10. As n increases, the general solution exhibits an essentially power-law behavior, making numerical calculations very quick (O(1)) and accurate.

From the obtained expressions when 2 ≤ m ≤ 10, improved upper and lower bounds to the true availability may be obtained for arbitrary m and n. Furthermore, an analytical, asymptotic expression for the availability of (2,2)-outof-(m,n):F lattice systems is provided for large values of m and n.

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