Enumeration of k-Exceedance Lattice Paths with an Application to Comparing Chains of Order Statistics

TL;DR

This paper derives formulas for counting specific lattice paths related to order statistics and applies them to compare system performances in failure analysis, providing asymptotic results and conjectures.

Contribution

It introduces closed-form formulas for enumerating k-exceedance lattice paths and applies these to analyze and compare system reliability using order statistics.

Findings

Derived explicit formulas involving Catalan, ballot, and binomial numbers.

Applied formulas to failure probability distributions of k-out-of-m systems.

Provided asymptotic results and conjectures for special cases.

Abstract

We enumerate the number of monotonic lattice paths starting at $(0,0)$ and terminating at $(m,n)$ in which $l$ of the first $k$ steps lie below the line $y=x\ (0\leq k\leq m\leq n)$. These closed formulas consist of terms which are a product Catalan numbers, ballot numbers and binomial coefficients. We then apply the combinatorial formulas to failure analysis by deriving a probability distribution that compares the performance of a $k$-out-of-$m$ system to a $k$-out-of-$n$ system of continuous, independent, and identically distributed random variables. Lastly, we provide asymptotics in a few special cases of $k,m,n$ and leave others as conjecture.