Lower and upper bounds for the reliability of connected-(r,s)-out-of-(m,n):F lattice systems

A linear (m,n)-lattice system is a system whose components are ordered like the elements of a (m,n)-matrix. A circular (m,n)-lattice system is a system whose components are represented by the junctions of m circles centered at the same point and n beams starting from that point and crossing the circles (the circles and the beams are not necessarily physical objects). It is assumed that in both linear and circular cases, the components have only two states: 1 (operating) and 0 (failed). A linear/circular connected-(r,s)-out-of-(m,n):F lattice system is a linear/circular (m,n)-lattice system that fails if at least 1 connected (r,s)-submatrix of failed components occurs. The paper gives lower and upper bounds for the reliabilities of such systems