[Paper Review] Reliability properties of k-out-of-n systems with one cold standby unit

Anna Dembińska, Nikolay I. Nikolov+1 more|arXiv (Cornell University)|Nov 26, 2019

Statistical Distribution Estimation and Applications35 references4 citations

TL;DR

This paper investigates the reliability of k-out-of-n systems with one cold standby unit under discrete-time failure models, focusing on systems with component lifetimes following geometric, negative binomial, or discrete Weibull distributions. It proposes a numerical procedure to approximate expected system lifetime and mean residual life with controlled error bounds, establishing monotonicity and stochastic ordering properties under IFR/DFR conditions for heterogeneous and dependent components.

ABSTRACT

In this paper, we study reliability properties of a k-out-of-n system with a single cold standby unit. We mainly focus on the case when the system operates in discrete time. In order to describe its aging behavior we consider three different mean residual life functions. By using some properties of order statistics we present several monotonicity results associated with these reliability characteristics. Since the calculation of the described quantities requires finding sums of infinite series, we provide a procedure to approximate them with an error not greater than a desired value. As an illustration we consider three special cases when the component lifetimes have geometric, negative binomial and discrete Weibull distributions.

Motivation & Objective

• To analyze reliability properties of k-out-of-n systems with a single cold standby unit when component lifetimes are discretely distributed.
• To develop a numerical procedure for approximating the expected lifetime and mean residual life of such systems with guaranteed error bounds.
• To establish monotonicity and stochastic ordering results for system lifetime and residual lifetime under IFR and DFR conditions.
• To provide explicit formulas and convergence bounds for three types of discrete lifetime distributions: geometric, negative binomial, and discrete Weibull.

Proposed method

• Derives exact expressions for the system lifetime distribution and expectation using order statistics and permutation-based conditioning on failure sequences.
• Introduces a method to bound the error in infinite series approximations of the expected lifetime by leveraging survival functions of extreme order statistics.
• Uses stochastic orderings (usual stochastic order, hazard rate order) to compare system lifetimes under different component lifetime distributions.
• Applies the inverse survival function of the negative binomial and discrete Weibull distributions to compute convergence thresholds for numerical approximation.
• Employs the upper incomplete gamma function and beta function to derive error bounds for the discrete Weibull case.
• Validates the approach through three illustrative examples with geometric, negative binomial, and discrete Weibull component lifetimes.

Experimental results

Research questions

• RQ1 How can the expected lifetime of a k-out-of-n system with one cold standby unit be approximated when component lifetimes are discrete and the series involve infinite sums?
• RQ2 What conditions ensure monotonicity of the mean residual life function when no components are failed, particularly under IFR or DFR component distributions?
• RQ3 How do stochastic orderings between system lifetimes depend on the underlying component lifetime distributions, especially in the presence of dependence or heterogeneity?
• RQ4 What are the convergence thresholds for approximating the system's reliability function and expected lifetime with a specified error tolerance?
• RQ5 How do the choice of discrete lifetime distribution (geometric, negative binomial, discrete Weibull) affect the system's reliability and residual lifetime characteristics?

Key findings

• The expected lifetime of the system can be approximated by truncating the infinite series of the survival function at a finite time t₀, with error bounded by d > 0.
• For geometric, negative binomial, and discrete Weibull components, explicit formulas are derived to compute the required truncation point t₀ to ensure error ≤ d.
• When components are IFR, the mean residual life of the system given no failures is decreasing in time; when DFR, it is increasing.
• The system lifetime stochastically dominates that of a system with fewer active components under certain distributional assumptions.
• The method provides a practical computational framework using inverse survival functions and special functions (e.g., gamma and beta functions) to compute convergence thresholds.
• The approach is validated through three concrete examples, showing accurate and efficient approximation of system reliability metrics under various discrete lifetime assumptions.