Reliability extrema of a complex circuit on bi-variate slice classes

Within the frame of reliability models, the geometry of constant level sets of the reliability function of a complex circuit – regarded as hypersurfaces, reveals properties which provide useful information on the relation between the reliability of the circuit and its components. A special role plays the study of intersections of these hypersurfaces with 2-dimensional plane slices, which provide a foliation by pencils of algebraic curves. The present study classifies these pencils and consequently, it allows: (i) to evaluate the possible bounds of the bivariate slice-reliability in terms of the circuit components; (ii) to compensate the impact of a slice-component malfunction on the slice-reliability, by tuning the appropriate pairing slice-component; (iii) to flag out the cases when the slice-reliability is linear (mono- or bi-variate) in the slice-components, or constant along the whole slice; (iv) in the quadratic case, to make use the convex/concave mutual dependence of slice-components along the curves of constant-slice reliability, in order to maintain or increase the circuit reliability.