Simple repair policies and decompositions for semi-coherent systems with simultaneous failures

TL;DR

This work studies the reliability and maintenance of semi-coherent binary systems subject to simultaneous component failures, modeling lifetimes with the Lévy-frailty Marshall-Olkin (LFMO) distribution. It introduces the $r$-out-of-$n$:R repair policy to curb the combinatorial explosion of repair strategies and derives explicit, tractable formulas for key performance metrics (mean time-to-failure, mean time-to-repair, probability of failure before repair, and repair-related costs) as functions of the system signature $\mathbf s$ and the Laplace exponent $\Psi$ of the LFMO model. The authors show how the Samaniego system-signature decomposition extends to LFMO lifetimes, enabling a probabilistic mixture interpretation via $k$-out-of-$n$:F systems and providing corollaries for i.i.d. exponential lifetimes and special cases. Computational experiments on small and medium systems validate the analytical results and illustrate maintenance-cost trade-offs under various policies and cost structures, highlighting practical maintenance implications for complex networks subject to external shocks.

Abstract

We consider semi-coherent binary systems that are subject to simultaneous failures of its components. These are systems whose components can be either working or failed; the system can also be working or failed depending on the state of the components; and repairing a component cannot cause the system to fail. We consider that one or more components can fail simultaneously, allowing us to model external shocks and disasters. For this, we use the Lévy-frailty Marshall-Olkin (LFMO) multivariate distribution to model the failure times of the components. We aim to answer in which states of the system we should repair the components. This is a challenging question, as the number of repair policies grows super-exponentially in the number of components. To tackle this, we propose a simple family of repair policies, which we call $r$-out-of-$n$:R repair policies, where one repairs all failed components when the system fails or when there are $r$ or more failed components. Our main contribution is that we derive exact and simple expressions for key performance-evaluation quantities of the system operating under our proposed repair policies. That is, we give explicit expressions for the mean time-to-failure of the system, mean time-to-repair, probability of system-failure before repair, and component- and system-repair rate. We also give expressions for the expected cost and long-term average cost, when there are components' and system repair cost. The only relevant parameters involved in the derived expressions are the structural signature of the system, and the Laplace exponent associated to the LFMO distribution.

Figures (7)

• Figure 1: (Left) For a binary system with $n=3$ components, we show all possible combinations of working and failed components (crossed in red). The system works when there is a path of working components from left to right, and the green or red background shows if the system is working or not in that configuration. The magenta lines are all possible transitions due to failures; note that more than one component can fail at the same time. The question we aim to tackle is in which states should we repair all failed components? Crucially, there is an exponential explosion of complexity with the number $n$ of components in the system: there are $2^n=8$ states, $3^n-2^n = 19$ transitions, and $2^{2^n-1-3}=16$ possible repair policies. (Right) We propose simple repair policies that we denote $r$-out-of-$n$:R repair policies (repair all failed components when there are $r$ or more failed components or the system fails). In this case, there are $n=3$ of these policies: $r=1$ in dots, $r=2$ in dashes, and $r=3$ in dash-dots. The boundary indicates the states where, upon reaching them, all failed components should be repaired.
• Figure 2: A simulation of the random vector $(T_1, T_2, T_3)$ in $\mathbb{R}^3$ with an LFMO distribution: the failure time $T_i$ of component $i$ is the first time the Lévy subordinator process $L$ surpasses the trigger $\varepsilon_i$. Note that two components fail simultaneously at time $T_1$: due to a jump of the Lévy subordinator, components 1 and 3 fail at the same time, $T_1=T_3$. This is how the LFMO model induces simultaneous failures of components. Also, these are the failure times of the components of the system on the right, which works when there is a path of working components from left to right. Hence, the system failure time $T_{fail}$ is also $T_1=T_3$.
• Figure 3: The Markov chain $\mathbf N = (N(t) \, : \, t \geq 0)$ of number of failed components (bottom) is obtained by aggregating the original states and transitions of the system (top); this is formalized in Lemma \ref{['lemma1']}. For example, Lemma 3 states that, indeed, the rate of the blue dotted transition from 0 to 2 failed components in the bottom, is obtained by summing the $\binom{3-0}{2-0} = 3$ blue dotted transition rates in the top. Similarly, the time $T_{k:n}$ until $k$ components have failed, is just the time required to go from $0$ to $k$ in the chain at the bottom; hence, the event $N(T_{r:n}) = k$ of the system having $k$ failed components at the time of the $r$-th failure, is the event of the chain jumping from $\{ 0, \ldots, r-1\}$ to $k$ failed components, see Lemma 3 part 5.
• Figure 4: We show how to compute the signature vector $s=(s_1, \ s_2, \ s_3)$ for the system in the right that has $n=3$ components. According to Proposition \ref{['prop:signature']}, $s_k$ is the proportion, between the total number of $n!=3!=6$ sequences of failures without ties, of sequences where the system first fails at the $k$-th failure. In the figure, the red background highlights, for each sequence, the first time at which the system fails. Hence, $s_2 = 4/6$ because the system first fails at the second failure in 4 of the 6 sequences.
• Figure 5: The classical Samaniego decomposition result $\mathbb{P}(T_{fail} > t) = \sum_{k=1}^n s_k \, \mathbb{P}( T_{k:n} > t)$ in \ref{['eq:samaniego']} establishes that, probabilistically, the system failure time $T_{fail}$ behaves, a proportion $s_k$ of the times, as the $k$-th failure time $T_{k:n}$, which is actually the system failure time of a $k$-out-of-$n$:F system. In the upper-half of the figure, we show a system with $n=3$ components whose signature vector $s=(0, 2/3, 1/3)$ was computed in Figure \ref{['fig:signature']}. The Samaniego decomposition result allows us to group the states of the system according to the number of failed components (in the lower-half of the figure), and analyze the $n=3$$k$-out-of-$n$:F systems (each with only $n+1=4$ states), instead of analyzing the original system with $2^n=8$ states, $3^n-2^n=19$ transitions, and $2^{2^n-1}=128$ repair policies (see the caption of Figure \ref{['fig:basic']}). This is a significative reduction in the complexity of the analysis.

Theorems & Definitions (18)

• Definition 2.1: Marshall-Olkin distribution
• Definition 2.2: Lévy-frailty Marshall-Olkin distribution
• Lemma 2.1
• Proposition 2.1
• Definition 2.3: Monotone, semi-coherent and coherent systems
• Definition 2.4: Structural signature
• Proposition 2.2
• Proposition 2.3
• Definition 2.5
• Lemma 2.2