Limiting behavior of mixed coherent systems with Lévy-frailty Marshall-Olkin failure times

TL;DR

This work addresses the asymptotic reliability of large mixed coherent systems with dependent lifetimes governed by a Lévy-frailty Marshall-Olkin model. By leveraging the Samaniego system signature and a common Lévy subordinator, the authors prove that the system failure time converges in distribution to the first-passage time $\tau_{-\log(1-Q)}$ of the subordinator, with the reliability function converging to $\mathbb{P}(S_t < -\log(1-Q))$, under natural hypotheses on the signature. A concrete example yields an asymptotically exponential limit with rate $\psi(b)$ when the signature follows $s^{(n)}_k = C^{(n)} (n-k+1)^{b-1}$ and $Q \sim \mathrm{Beta}(1,b)$, complemented by computational experiments confirming convergence for moderate system sizes and highlighting mean convergence caveats. The results provide a tractable, probabilistic framework for analyzing reliability of very large dependent-component systems and link reliability analysis to Lévy-process techniques, with implications for design optimization and risk assessment.

Abstract

In this paper we show a limit result for the reliability function of a system -- that is, the probability that the whole system is still operational after a certain given time -- when the number of components of the system grows to infinity. More specifically, we consider a sequence of mixed coherent systems whose components are homogeneous and non-repairable, with failure-times governed by a Lévy-frailty Marshall-Olkin (LFMO) distribution -- a distribution that allows simultaneous component failures. We show that under integrability conditions the reliability function converges to the probability of a first-passage time of a Lévy subordinator process. To the best of our knowledge, this is the first result to tackle the asymptotic behavior of the reliability function as the number of components of the system grows. To illustrate our approach, we give an example of a parametric family of reliability functions where the system failure time converges in distribution to an exponential random variable, and give computational experiments testing convergence.

Figures (3)

• Figure 1: Two simulations of a random vector $(T_1, T_2, T_3)$ in $\mathbb{R}^3$ with an LFMO distribution: for each component $i$, $T_i$ is the first time $t$ the Lévy subordinator process $L$ surpasses the trigger$\varepsilon_i$.
• Figure 2: Average p-value over 1,000 repetitions of (two-sampled) Kolmogorv-Smirnov tests, comparing 1,000 samples of $T_\text{sys}^{(n)}$ and 1,0000 samples of the limit random variable $e_{\psi(b)}$, for several distributions of the jumps $J_i$ of the compound Poisson process subordinator \ref{['def:CPP']}. For each case we show the values $\mu=0$, $0.5$ and $1$ for the drift term of the subordinator, and the values $b=0.5$, $1$ and $1.5$ for the parameter $b$ of the system signature in Expression \ref{['propos:sign']}.
• Figure 3: Relative error between the estimated mean (over 100,000 samples) of $T_\text{sys}^{(n)}$ and the mean $1/\psi(b)$ of the limit random variable $e_{\psi(b)}$ in \ref{['lim:propos']}, for several values of the parameter $b$ of the system signature in \ref{['propos:sign']}. For each $b$ we consider the cases of the compound Poisson process subordinator \ref{['def:CPP']} having drift term $\mu=0$, $0.5$ and $1$, and jumps $J_i$ having an exponential(1), uniform(0, 1) and Pareto($\alpha=3$) distribution.

Theorems & Definitions (11)

• definition 1: Coherent system
• definition 2: Signature
• definition 3: Mixed coherent system and signature
• definition 4
• proposition 1
• theorem 1
• corollary 1
• proof : Proof of Theorem \ref{['theo']}.
• proof : Proof of Corollary \ref{['corol']}.
• proof : Proof of Proposition \ref{['propos']}.