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Reliability of three-state k-out-of-n:G system with non-homogeneous Markov dependent components

Abdelmoumene Boulahia, Soheir Belaloui

TL;DR

The paper addresses reliability of a three-state $k$-out-of-$n:G$ system with non-homogeneous Markov dependent components. It adopts the probability generating function method to derive closed-form state distributions for increasing and decreasing configurations, yielding explicit formulas for $R^j(n)$ and $r^j(n)$ via matrix products $\\bm{H}_c^j(t)$ and the joint generating function $\\Gamma(t_1,t_2)$. The key contributions are the PGF-based state-distribution formulas for both system orientations, including handling non-homogeneous dependence and reductions to the homogeneous case, accompanied by numerical demonstrations. This framework enables tractable reliability evaluation of multi-state systems with dependent components and can be extended to three-state $k$-out-of-$n:F$ systems and higher-state models.

Abstract

In this paper, we study the reliability of a three-state k-out-of-n:G system. We consider the situation where the system components are non-homogeneous Markov dependent, and we derive a closed-form formula for the system reliability, including increasing three-state k-out-of-n:G system and decreasing three-state k-out-of-n:G system. Our study is based on the probability generating function method. Two numerical examples are presented to demonstrate the use of the formula.

Reliability of three-state k-out-of-n:G system with non-homogeneous Markov dependent components

TL;DR

The paper addresses reliability of a three-state -out-of- system with non-homogeneous Markov dependent components. It adopts the probability generating function method to derive closed-form state distributions for increasing and decreasing configurations, yielding explicit formulas for and via matrix products and the joint generating function . The key contributions are the PGF-based state-distribution formulas for both system orientations, including handling non-homogeneous dependence and reductions to the homogeneous case, accompanied by numerical demonstrations. This framework enables tractable reliability evaluation of multi-state systems with dependent components and can be extended to three-state -out-of- systems and higher-state models.

Abstract

In this paper, we study the reliability of a three-state k-out-of-n:G system. We consider the situation where the system components are non-homogeneous Markov dependent, and we derive a closed-form formula for the system reliability, including increasing three-state k-out-of-n:G system and decreasing three-state k-out-of-n:G system. Our study is based on the probability generating function method. Two numerical examples are presented to demonstrate the use of the formula.
Paper Structure (6 sections, 4 theorems, 35 equations, 1 table)

This paper contains 6 sections, 4 theorems, 35 equations, 1 table.

Table of Contents

  1. Introduction
  2. Reliability evaluation
  3. Increasing three-state $k$-out-of-$n:G$ system
  4. Decreasing three-state $k$-out-of-$n:G$ system
  5. Numerical examples
  6. Conclusion

Key Result

Theorem 1

For an increasing three-state $k$-out-of-$n:G$ system with non-homogeneous Markov dependent components Where $\bm{H}_c^j(t)$ is a $(3\times 3)$-matrix for $c=1,2,...,n$ and $j=1,2$ as and $\bm{\bar{1}}=(1\;0\;0)$, $\bm{1}=(1\;1\;1)^{'}$.

Theorems & Definitions (12)

  • Theorem 1
  • proof : Proof
  • Proposition 1
  • proof : Proof
  • Remark 1
  • Theorem 2
  • proof : Proof
  • Proposition 2
  • proof : Proof
  • Remark 2
  • ...and 2 more