Multi-Dimensional Recursion Relations and Mathematical Induction Techniques: The Case of Failure Frequency of k-out-of-n Systems

The probabilities of exactly and at least k successes out of n trials represent respectively the probability mass function (pmf) and the Complementary Cumulative Distribution Function (CCDF) of the generalized Binomial distribution. These two probabilities obey the same two-dimensional recursive relations (or signal-flow-graph structure), albeit with different boundary conditions (sources) and almost identical regions of validity that cover approximately an octant of the k-n plane. Related to these is the failure frequency of a non-repairable k-out-of-n system, which equals the probability density function (pdf) for the time to failure of this system. This failure frequency is represented herein by a two-dimensional signal flow graph (SFG) whose sources emanate from one of the SFGs of the aforementioned probabilities. Therefore, the recursive relations of this failure frequency take the shape of a three-dimensional loopless or acyclic graph. Closed-form symbolic expressions for this failure frequency are obtained via Mason gain formula, i.e., simply via path enumeration. Such expressions are also deduced herein via a two-dimensional strong version of mathematical induction. An offshoot contribution of this paper is to derive expressions for the k-out-of-n failure frequency via Markov-chain modeling, and to demonstrate that the resulting expressions are equivalent to those obtained via recursive relations.