Recursive Theorems for Success Runs and Reliability of Consecutive-K-Out-of-N: F Systems

Abstract

Unless otherwise stated, in this paper k is a fixed positive integer, n_f (1 ≤i ≤k) and n are non-negative integers as specified, p and x are real numbers in the intervals (0, 1) and (0, ∞), respectively, and q = 1-p. In a recent paper, Philippou and Makri [11] studied the length Ln (n≥1) of the longest success run in n Bernoulli trials, deriving the probability function of Ln, its distribution function, and its factorial moments. In particular, they found that

$$ P({L_n} \le k) = \frac{{{p^{n + 1}}}}{q}{\sum\limits_{{n_1},...,{n_{k + 1}}} {\left( {_{}^{{n_1} + ... + {n_{k + 1}}}} \right)(\frac{q}{p})} ^{{n_1} + ... + {n_{k + 1}}}},0 \le k \le n,{n_1} + 2{n_2} + ... + (k + 1){n_{k + 1}} = n + 1 $$

((1.1))

and

$$ P({L_n} \le k) = \frac{{{p^{n + 1}}}}{q}F_{n + 2}^{(k + 1)}(q/p),0 \le k \le n $$

((1.2))

where $\left\{ {F_n^{(k)}(x))_n^\infty } \right. = 0$ are the Fibnacci-type polynomials of order k [11, 15].