2. Time-homogeneous load-sharing model

In order to distinguish this special case, we denote the component lifetimes by $S_1,\ldots,S_n$. The respective order statistics $S_{1:n},\ldots, S_{n:n}$ coincide with the consecutive failure times of the system components. The joint distribution of $S_1, \ldots, S_n$ is described by the family of positive parameters

(5) \begin{equation}\mathcal{L} = \{ \lambda_j(A)\,{:}\, A \subsetneqq [n], \;j \in A^c \}\end{equation}

of cardinality $n 2^{n-1}$. Indeed,

\begin{equation*}|\mathcal{L}| = \sum_{k=0}^{n-1} \left(\begin{array}{l}n \\ k\end{array}\right) (n-k) = n \sum_{k=0}^{n-1} \left(\begin{array}{c}n-1 \\ k\end{array}\right) = n 2^{n-1}.\end{equation*}

Each $\lambda_j(A)$ represents the constant multivariate conditional failure rate,

\begin{equation*}\lambda_{j}(t|A,t_1,\ldots, t_k)=\lambda_j(A),\end{equation*}

of the jth component under the condition that all the components from the set $A \subsetneqq [n]$ have failed previously, while those from $A^c$ (which contains j) are still operating. The notation

\begin{equation*}\Lambda(A)= \sum_{j \in A^c} \lambda_j(A)\end{equation*}

will be frequently used below.

Let $I_0, \ldots, I_n$ denote the set-valued variables such that $I_k$ is the label set of the first k failed components. Certainly $I_0= \emptyset$, $I_n=[n]$, $|I_k|=k$, and $|I_k \setminus I_{k-1}|=1$. Our first claim is the following.

Theorem 1. The sequence $(S_{k:n},I_k)$, $k=1,\ldots ,n$, forms a two-dimensional finite Markov chain with the initial probability rule

(6) \begin{equation}\mathbb{P}(S_{1:n}> s, \; I_1= \{\,j_1\}) = \exp \left( -s \Lambda(\emptyset)\right) \frac{\lambda_{j_1}(\emptyset)}{\Lambda(\emptyset)},\qquad s>0, \;j_1 \in [n],\end{equation}

and the transition laws

\begin{align}& \mathbb{P}(S_{k:n}> s, \; I_k= A_{k-1} \cup \{\,j_k\}|S_{k-1:n}= s_{k-1}, \; I_{k-1}= A_{k-1} ) \nonumber \\[3pt]& = \exp \left( -(s-s_{k-1}) \Lambda(A_{k-1})\right) \frac{\lambda_{j_k}(A_{k-1})}{\Lambda(A_{k-1})},\qquad s>s_{k-1}, \; j_k \in A_{k-1}^c, \nonumber \\[3pt] &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,\, |A_{k-1}|=k-1, \; k=2,\ldots, n.\nonumber \end{align}

Proof. When all the system components are working, the component lifetimes $S_1,\ldots,S_n$ have constant failure rates $\lambda_1(\emptyset), \ldots, \lambda_n(\emptyset)$, respectively. None of the $\lambda_k(\emptyset)$ depends in any way on the failure rates of the other components. This means that $S_1, \ldots,S_n$ are independent exponential, with possibly different distributions determined by the respective failure rates. It is well known that $S_{1:n}= \min \{ S_1, \ldots,S_n \}$ is exponential with the failure rate

\begin{equation*}\Lambda(\emptyset) = \sum_{i=1}^n \lambda_i(\emptyset),\end{equation*}

while

\begin{equation*}\mathbb{P}(S_i=S_{1:n}) = \frac{\lambda_{i}(\emptyset)}{\Lambda(\emptyset)}, \qquad i=1,\ldots,n,\end{equation*}

and this holds independently of the value of $S_{1:n}$. This proves the formula (6). We may also refer to a more general result of De Santis et al. [Reference De Santis, Malinovsky and Spizzichino12, Proposition 2], which asserts that for lifetime random variables $X_1,\ldots,X_n$ with general multivariate conditional hazard rates (2) and (3) there exist independent random variables $Z_1,\ldots,Z_n$ with failure rates $r_j(t) =\lambda_j(t|\emptyset)$, $j=1,\ldots ,n$, such that the joint distributions of $(X_{1:n}, \textbf{1}_{\{X_{1:n}=X_j\}})$ and $(Z_{1:n}, \textbf{1}_{\{Z_{1:n}=Z_j\}})$, $j=1,\ldots,n$, are identical.

Assume now that the $(k-1)$th consecutive failure happened at time $s_{k-1}$, and components $j_1, \ldots,j_{k-1} \in A_{k-1}$ had failed by then. From this moment on, each of the $n+1-k$ working components from the set $A_{k-1}^c$ get constant failure rates $\lambda_i(A_{k-1})$, $i \in A_{k-1}^c$. This means that, independently of the particular value of $S_{k-1:n}=s_{k-1}>0$, the random variables $S_i-s_{k-1}$, $i \in A_{k-1}^c$, are independent exponential with corresponding failure rates $\lambda_i(A_{k-1})$, $i \in A_{k-1}^c$. It follows that their minimum $S_{k:n}-S_{k-1:n}$ is exponential with failure parameter $\Lambda(A_{k-1})= \sum_{i \in A_{k-1}^c} \lambda_i(A_{k-1})$, and it is independent of the value of $S_{k-1:n}=s_{k-1}$. The minimum is achieved by $j_k \in A_{k-1}^c$ with probability

\begin{align*}\mathbb{P}(I_k \setminus I_{k-1}= A_k\setminus A_{k-1} = \{\,j_k \}&|S_{k-1:n}=s_{k-1}, \; I_{k-1}=A_{k-1} )\\[3pt]&= \frac{\lambda_{j_k}(A_{k-1})}{\Lambda(A_{k-1})},\qquad j_k \in A_{k-1}^c.\\[-40pt]\end{align*}

Remark 1. Conditionally on $(S_{k-1:n},I_{k-1})$, the variables $S_{k:n}-S_{k-1:n}$ and $I_k$ are independent, with distributions

\begin{align*}\mathbb{P}(S_{k:n}> s|S_{k-1:n}= s_{k-1}, \! I_{k-1} = A_{k-1} ) &= \exp \!\left( -(s-s_{k-1}) \Lambda(A_{k-1})\right) \!,\; s>s_{k-1},\\[3pt]\mathbb{P}( I_k= A_{k-1} \!\cup\! \{\,j_k\}|S_{k-1:n}= s_{k-1}, \! I_{k-1}= A_{k-1} )& = \frac{\lambda_{j_k}(A_{k-1})}{\Lambda(A_{k-1})}.\end{align*}

Corollary 1. The joint density function of $(S_{1:n},\ldots, S_{k:n},I_1, \ldots,I_k)$, $k=1,\ldots,n$, with respect to the product of k-dimensional Lebesgue measure and k-dimensional counting measure is

\begin{eqnarray}& & f_{\mathcal{L};S_{1:n},\ldots, S_{k:n},I_1, \ldots,I_k}(s_1,s_2, \ldots,s_k, \{\,j_1\},\{\,j_1, j_2\} , \ldots, \{\,j_1, \ldots, j_k\}) \nonumber \\[3pt]& = & \prod_{i=1}^k\lambda_{j_i}(\{\,j_1,\ldots,j_{i-1}\})\,\exp \left( - \sum_{i=1}^k (s_i-s_{i-1}) \Lambda (\{\,j_1, \ldots, j_{i-1}\})\right) , \nonumber \\[3pt]& & \qquad \qquad s_0=0 < s_1 < \ldots < s_k, \quad \{\,j_1,\ldots,j_{k}\} \subset [n].\nonumber \end{eqnarray}

Corollary 2. The sequence $I_k$, $k=1,\ldots,n$, is a Markov chain with initial probability

\begin{equation*}\mathbb{P}(I_1= \{\,j_1 \}) = \frac{\lambda_{j_1}(\emptyset)}{\Lambda(\emptyset)}\end{equation*}

and transition probabilities

\begin{equation*}\mathbb{P}( I_k= I_{k-1} \cup \{\,j_k\}| I_{k-1}= A_{k-1} ) =\frac{\lambda_{j_k}(A_{k-1})}{\Lambda(A_{k-1})}, \qquad k=2,\ldots,n.\end{equation*}

Consequently,

\begin{equation*}\mathbb{P}( I_1= \{\,j_1\}, \ldots, I_k= \{\,j_1, j_2, \ldots, j_k\})= \prod_{i=1}^k\frac{\lambda_{j_i}(\{\,j_1, j_2, \ldots, j_{i-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{i-1}\})},\end{equation*}

and

\begin{equation*}\mathbb{P}( I_k= A_k ) = \sum_{\sigma(A_k)} \prod_{i=1}^k\frac{\lambda_{\sigma_i(A_k)}(\{ \sigma_1(A_k), \sigma_2(A_k), \ldots, \sigma_{i-1}(A_k)\})}{\Lambda(\{ \sigma_1(A_k), \sigma_2(A_k), \ldots, \sigma_{i-1}(A_k)\})},\end{equation*}

where $\sigma(A_k)$ denotes a permutation of the elements of set $A_k$, and $\sigma_i(A_k)$ stands for its ith coordinate.

It is sometimes convenient to replace a sequence of set-valued random variables $I_1,\ldots,I_n$ by an equivalent sequence of integer-valued variables $J_1, \ldots ,J_n$ defined as $J_i=j_i$ if and only if $j_i\in I_i\setminus I_{i-1}$, with the obvious convention $I_0=\emptyset$.

Corollary 3. Let $V_1,\ldots, V_k$ for some $1 \leq k \leq n$ denote independent standard exponential random variables with the common mean 1. Then

(7) \begin{align}&\quad (S_{1:n},S_{2:n}, \ldots, S_{k:n}|I_1=\{\,j_1\}, I_2=\{\,j_1,j_2\} \ldots , I_{k-1}=\{\,j_1,j_2, \ldots, j_{k-1}\}) \nonumber \\[3pt]& \stackrel{\mathrm{d}}{=} (S_{1:n},S_{2:n}, \ldots, S_{k:n}|J_1=j_1, J_2=j_2 \ldots , J_{k-1}= j_{k-1})\nonumber \\[3pt]& \stackrel{\mathrm{d}}{=} \left( \frac{V_1}{\Lambda(\emptyset)}, \frac{V_1}{\Lambda(\emptyset)} +\frac{V_2}{\Lambda(\{\,j_1\})}, \ldots,\frac{V_1}{\Lambda(\emptyset)} +\frac{V_2}{\Lambda(\{\,j_1\})} + \ldots + \frac{V_k}{\Lambda(\{\,j_1, \ldots,j_{k-1}\})} \right)\!,\end{align}

and, in particular,

(8) \begin{align}&\quad (S_{k:n}|I_1=\{\,j_1\}, I_2=\{\,j_1,j_2\} \ldots , I_{k-1}=\{\,j_1,j_2, \ldots, j_{k-1}\}) \nonumber \\[3pt]& \stackrel{\mathrm{d}}{=} (S_{k:n}|J_1=j_1, J_2=j_2 \ldots , J_{k-1}= j_{k-1})\nonumber \\[3pt]& \stackrel{\mathrm{d}}{=} \frac{V_1}{\Lambda(\emptyset)} +\frac{V_2}{\Lambda(\{\,j_1\})} + \ldots + \frac{V_k}{\Lambda(\{\,j_1, \ldots,j_{k-1}\})}.\end{align}

In other words, the conditional distribution of $S_{k:n}$ is identical with the convolution of independent exponential random variables with intensity parameters $\Lambda(\emptyset), \Lambda(I_1), \ldots, \Lambda(I_k)$. The respective density functions can be easily calculated for fixed parameters $\Lambda(\emptyset), \Lambda(\{\,j_1\}), \ldots,\break \Lambda(\{\,j_1, \ldots,j_{k-1}\})$, but their particular forms essentially depend on the multiplicities of the parameters, and so we do not present them here. For brevity of notation we denote them by $f_{\mathcal{L};S_{k:n}}(s|j_1, \ldots,j_{k-1})$. Similarly, we use the notation

\begin{equation*}f_{\mathcal{L};S_{1:n}, \ldots, S_{k:n}}(s_1, \ldots, s_k|j_1, \ldots,j_{k-1})\end{equation*}

for the joint conditional density of the first k component failure times $S_{1:n},S_{2:n}, \ldots, S_{k:n}$. The conditional dependence between $S_{i:n}$ and $S_{j:n}$, expressed by means of (7), lies in the fact that both of them depend on the same $V_m$ for $1 \leq m \leq \min \{ i,j \}$. The formula (7) can be used for calculating the joint marginal density function of several arbitrarily selected order statistics.

Proposition 1. The joint unconditional density function of $S_{1:n},S_{2:n}, \ldots, S_{k:n}$ has the form

\begin{align}f_{\mathcal{L};S_{1:n}, \ldots, S_{k:n}}(s_1, \ldots, s_k) & = \sum_{(j_1, \ldots, j_{k-1}) \in \Pi([n])}f_{\mathcal{L};S_{1:n}, \ldots, S_{k:n}}(s_1, \ldots, s_k|j_1, \ldots,j_{k-1}) \nonumber \\[3pt] &\quad \times \prod_{i=1}^{k-1}\frac{\lambda_{j_i}(\{\,j_1, j_2, \ldots, j_{i-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{i-1}\})}.\nonumber \end{align}

Similarly, the unconditional density of a single $S_{k:n}$ is

(9) \begin{equation}f_{\mathcal{L};S_{k:n}}(s)= \sum_{(j_1, \ldots, j_{k-1}) \in \Pi([n])}f_{\mathcal{L}; S_{k:n}}(s|j_1, \ldots,j_{k-1})\, \prod_{i=1}^{k-1}\frac{\lambda_{j_i}(\{\,j_1, j_2, \ldots, j_{i-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{i-1}\})}.\end{equation}

Accordingly,

(10) \begin{equation}\mathbb{E}S_{k:n} = \sum_{(j_1, \ldots, j_{k-1}) \in \Pi([n])}\left( \sum_{i=1}^k \frac{1}{\Lambda(\{\,j_1, \ldots,j_{i-1}\})}\right)\prod_{i=1}^{k-1}\frac{\lambda_{j_i}(\{\,j_1, j_2, \ldots, j_{i-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{i-1}\})}. \end{equation}

The notation $(j_1, \ldots, j_{k-1}) \in \Pi([n])$ means that the summations in the above formulae run over all $(k-1)$-permutations of the set [n], i.e., the vectors of length $k-1$ built from the different elements in [n]. We adhere to this notation further on. In this notation we do not specify the length of the vectors in $\Pi([n])$, because it always follows from the context. In other words, the distribution of the kth component failure of the system working in the time-homogeneous load-sharing model is the mixture of k-fold convolutions of independent exponential distributions. As shown in the next proposition, the same holds for the distributions of the system and all its components.

Proposition 2. The density function of the ith component lifetime $S_i$ is the following convex combination of the convolutions of exponential random variables:

(11) \begin{align}f_{\mathcal{L};S_i}(s) & = \sum_{k=1}^n \sum_{(j_1, \ldots, j_{k-1}) \in \Pi([n]\setminus \{i\})}f_{\mathcal{L}; S_{k:n}}(s|j_1, \ldots,j_{k-1}) \,\prod_{i=1}^{k-1}\frac{\lambda_{j_i}(\{\,j_1, j_2, \ldots,j_{i-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{i-1}\})}\nonumber \\[3pt]&\quad \times \frac{\lambda_{i}(\{\,j_1, j_2, \ldots, j_{k-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{k-1}\})}.\end{align}

Moreover,

(12) \begin{align}\mathbb{E}S_i & = \sum_{k=1}^n \sum_{(j_1, \ldots, j_{k-1}) \in \Pi([n]\setminus \{i\})}\left( \sum_{i=1}^k \frac{1}{\Lambda(\{\,j_1, \ldots,j_{i-1}\})}\right)\prod_{i=1}^{k-1}\frac{\lambda_{j_i}(\{\,j_1, j_2, \ldots, j_{i-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{i-1}\})}\nonumber \\[3pt]&\quad \times \frac{\lambda_{i}(\{\,j_1, j_2, \ldots, j_{k-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{k-1}\})}.\end{align}

The difference between the expressions (9)–(10) and (11)–(12) lies in that the latter contain one summation more; the second summations in (11)–(12) are performed over the sequences of length $k-1$ of elements of [n] other than i, and each mixing coefficient in (11) and (12) has one more factor

\begin{equation*}\frac{\lambda_{i}(\{\,j_1, j_2, \ldots, j_{k-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{k-1}\})}\end{equation*}

than the corresponding terms in the first sums of (9)–(10). To justify (11) we note that the density function of the ith component lifetime is identical with that of the kth consecutive failure under the condition that $(j_1,\ldots,j_{k-1},i)$ is the sequence of consecutively failed components for arbitrary $(j_1, \ldots, j_{k-1}) \in \Pi([n]\setminus \{i\})$ and for some $k=1,\ldots,n$. The probability of the above sequence of failures is

\begin{equation*}\prod_{i=1}^{k-1}\frac{\lambda_{j_i}(\{\,j_1, j_2, \ldots,j_{i-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{i-1}\})}\,\frac{\lambda_{i}(\{\,j_1, j_2, \ldots, j_{k-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{k-1}\})}.\end{equation*}

Hence the representation (11) follows from the law of total probability, and (12) is its consequence.

A similar construction can be accomplished for the system lifetime. For this purpose we first introduce the function

\begin{equation*}\chi (j_1, \ldots, j_n) = \sum_{k=1}^n k\, [ \varphi (\{\,j_1, \ldots, j_k\}) - \varphi (\{\,j_1, \ldots, j_{k-1}\})]\end{equation*}

acting on all the permutations of the set [n]. If $(j_1, \ldots, j_n)$ represents the sequence of subsequently failing components, then $\chi (j_1, \ldots, j_n)$ is just the number of consecutively failed components that causes the failure of the system.

Proposition 3. The lifetime S of the system with structure $\varphi$ and component lifetimes satisfying the assumptions of the time-homogeneous load-sharing model with parameters (5) has the density function

(13) \begin{equation}f_{\mathcal{L};S}(s)= \sum_{k=1}^n \sum_{(j_1, \ldots, j_{n}) \in \chi^{-1}(k)}f_{\mathcal{L}; S_{k:n}}(s|j_1, \ldots,j_{k-1})\prod_{i=1}^{n-1}\frac{\lambda_{j_i}(\{\,j_1, j_2, \ldots, j_{i-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{i-1}\})}\end{equation}

and expectation

(14) \begin{equation}\mathbb{E}S = \sum_{k=1}^n \sum_{(j_1, \ldots, j_{n}) \in \chi^{-1}(k)}\left( \sum_{i=1}^k \frac{1}{\Lambda(\{\,j_1, \ldots,j_{i-1}\})}\right)\prod_{i=1}^{n-1}\frac{\lambda_{j_i}(\{\,j_1, j_2, \ldots, j_{i-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{i-1}\})}.\end{equation}

The system lifetime density function coincides with that of the kth consecutive failure in the sequence of all consecutively failed components $(j_1,\ldots,j_n)$ if and only if $\chi(j_1,\ldots,j_n)=k$, i.e., for this sequence of component failures the system fails at the kth consecutive failure. The probability of the particular failure sequence $(j_1,\ldots,j_n)$ is represented by the product in (13), and (13) itself results from the total probability rule.

Remark 3. In many cases it is natural in practice to assume that

(15) \begin{equation}\lambda_j(\{\,j_1, \ldots, j_{k-1} \} ) \leq \lambda_j(\{\,j_1, \ldots, j_{k-1},j_k \}), \quad j \in \{\,j_1, \ldots, j_{k-1},j_k \}^c, \;k=1,\ldots, n-1,\end{equation}

which means that a failure of any component $j_k\in \{\,j_1, \ldots, j_{k-1} \}^c$ implies that the conditional failure rates of all living components $j \in \{\,j_1, \ldots, j_{k-1},j_k \}^c$ increase (or at least do not decrease) with the failure of any component $j_k \in \{\,j_1, \ldots, j_{k-1} \}^c$, $ j_k \neq j$. Moreover, it is practically justified to have similar relations for the cumulative hazard rates

(16) \begin{equation}\Lambda(\emptyset) \leq \Lambda(\{\,j_1 \}) \leq \ldots \leq \Lambda (\{\,j_1, \ldots,j_{n-1} \}).\end{equation}

We get the classic load-sharing model if $\Lambda(\emptyset) = \ldots = \Lambda (\{\,j_1, \ldots,j_{n-1} \})$, which means that there is a constant load acting on the system, and this is shared by its components. Then the conditional distribution of each $S_{k:n}$ is the Erlang (gamma) distribution with shape parameter k. However, it is more realistic to assume that the aggregate hazard rate does not depend linearly on the burden that the system undergoes, but rather increases as the number of failed components increases. The conditions $\Lambda(\emptyset) < \ldots < \Lambda (\{\,j_1, \ldots,j_{n-1} \})$ imply that the conditional distribution of each $S_{k:n}$ described in (8) is a linear combination of k exponential distributions with different scales and some possibly negative coefficients summing to 1. In the most general case this is a linear combination of Erlang distributions with possibly different shapes (see, e.g., Smaili et al. [Reference Smaili, Kadri and Kadry37, Theorem 1]).

3. Time-heterogeneous load-sharing model

The model presented above has a number of convincing motivations. On one hand, it realistically depicts the situation that the system components have different strengths, and these also depend on the circumstances under which they are working: the failures of some components may not necessarily imply the failure of the system, yet may make the still living components work harder, because of the increased burden on the whole system. On the other hand, it is assumed that the still working components have stable aging tendencies, expressed by constant failure rates. Clearly, the latter assumption is sometimes violated in practice. Below we present a generalization of the model given in Section 2 which admits variations of the inter-failure hazard rates in time, but still inherits some natural and useful properties of the time-homogeneous load-sharing model.

Let F be an absolutely continuous lifetime distribution function strictly increasing on its support interval $[a,b] \subset [0,+\infty)$, with density function f. We say that the component lifetimes $T_1,\ldots,T_n$ satisfy the assumptions of the time-heterogeneous load-sharing model, with baseline distribution function F and parameters (5), if the respective order statistics satisfy

(17) \begin{eqnarray}&& (T_{k:n}| (J_1, \ldots, J_{k-1})= (j_1, \ldots, j_{k-1}) ) \nonumber \\[3pt]& \stackrel{\mathrm{d}}{=} &(F^{-1}( 1-\exp (\!- S_{k:n}))| (J_1, \ldots, J_{k-1})= (j_1, \ldots, j_{k-1}) ), \quad k=1,\ldots,n, \end{eqnarray}

with $S_{k:n}$, $k=1, \ldots,n$, being the order statistics coming from the time-homogeneous load-sharing model with parameters (5), while

(18) \begin{equation}\mathbb{P}(T_i= T_{k:n}| (J_1, \ldots, J_{k-1})= (j_1, \ldots, j_{k-1}) )= \frac{\lambda_i(\{\,j_1, \ldots, j_{k-1}\})}{\Lambda(\{\,j_1, \ldots, j_{k-1}\})},\;\: i \in \{\,j_1, \ldots, j_{k-1}\}^c.\end{equation}

Actually, we could replace $T_{k:n}$ and $S_{k:n}$ in (17) by $T_i$ and $S_i$, respectively, but the formulae with order statistics are more convenient for our further calculations.

Proposition 4. The sequence $(T_{k:n},I_k)$, $k=1,\ldots ,n$, is a Markov chain with initial and transition probabilities

(19) \begin{equation}\mathbb{P}(T_{1:n}> t, \; I_1= \{\,j_1\}) = [1-F(t)]^{\Lambda(\emptyset)}\,\frac{\lambda_{j_1}(\emptyset)}{\Lambda(\emptyset)},\qquad a<t<b, \;j_1 \in [n],\end{equation}

and

(20) \begin{align}& \mathbb{P}(T_{k:n}> t, \; I_k= A_{k-1} \cup \{\,j_k\}|T_{k-1:n}= t_{k-1}, \; I_{k-1}= A_{k-1} ) \nonumber \\[3pt]&\quad = \left[ \frac{1-F(t)}{1-F(t_{k-1})} \right]^ {\Lambda(A_{k-1})}\,\frac{\lambda_{j_k}(A_{k-1})}{\Lambda(A_{k-1})},\qquad t_{k-1}< t <b, \;j_k \in A_{k-1}^c, \nonumber \\[3pt] & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\,\,\,\, |A_{k-1}|=k-1, \;k=2,\ldots,n,\end{align}

respectively.

Proof. Since $t \mapsto F^{-1}(1-\exp\!(\!-t))$ is a one-to-one continuous strictly increasing transformation of $S_{k:n} > S_{k-1:n}$ onto $T_{k:n} \in (T_{k-1:n}, b)$, in view of the conditional independence of $S_{k:n}$ and $I_k$, it suffices to calculate

\begin{align*}\mathbb{P}(T_{1:n} >t)& = \mathbb{P}(F^{-1}( 1-\exp\!(\!- S_{1:n}) >t)= \mathbb{P}(S_{1:n} > -\ln[1-F(t)]) \\[3pt]& = \exp \left( \ln[1-F(t)]\, \Lambda(\emptyset) \right) = [1-F(t)]^{\Lambda(\emptyset)}\end{align*}

and

\begin{align*}\hspace*{17pt}& \mathbb{P}(T_{k:n} >t |T_{k-1:n}= t_{k-1}, \; I_{k-1}= A_{k-1} ) \\[3pt] &\quad = \mathbb{P}(S_{k:n} > -\ln[1-F(t)] |S_{k-1:n}= -\ln[1-F(t_{k-1})], \; I_{k-1}= A_{k-1} ) \\[3pt] &\quad = \exp \left(\!-\{ -\ln[1-F(t)]+\ln[1-F(t_{k-1})] \}\, \Lambda(A_{k-1}) \right) = \left[ \frac{1-F(t)}{1-F(t_{k-1})} \right]^ {\Lambda(A_{k-1})}.\qquad\qed\end{align*}

Corollary 5. The joint density function of $(T_{1:n},\ldots, T_{k:n},I_1, \ldots,I_k)$, $k=1,\ldots,n$, with respect to the product of k-dimensional Lebesgue measure on $(a,b)^k$ and k-dimensional counting measure on $[n]^k$ has the form

(21) \begin{align}& f_{\mathcal{L}F;T_{1:n},\ldots, T_{k:n},I_1, \ldots,I_k}(t_1,t_2, \ldots,t_k, \{\,j_1\},\{\,j_1, j_2\} , \ldots, \{\,j_1, \ldots, j_k\}) \nonumber \\[3pt] &\quad = \prod_{i=1}^{k} \lambda_{j_i} (\{\,j_1, \ldots, j_{i-1}\})\, f(t_i)\, \prod_{i=1}^{k-1} [1-F(t_i)]^{\Lambda (\{\,j_1, \ldots, j_{i-1}\})-\Lambda (\{\,j_1, \ldots, j_{i}\})-1}\nonumber\\ &\qquad \times [1-F(t_k)]^{\Lambda (\{\,j_1, \ldots, j_{k-1}\})-1}\end{align}

for $a < t_1 < \ldots < t_k <b$ and $\{\,j_1,\ldots,j_{k}\} \subset [n]$.

Proposition 5. Let $V_1,\ldots, V_k$ for some $1 \leq k \leq n$ denote independent standard exponential random variables with the common mean 1. Let $U_1,\ldots, U_k$ stand for independent standard uniform random variables supported on (0,1). Then the conditional distribution of $T_{k:n}$ with respect to $I_1, \ldots ,I_{k-1}$ has the representations

(22) \begin{align}& (T_{k:n}|I_1=\{\,j_1\}, I_2=\{\,j_1,j_2\} \ldots , I_{k-1}=\{\,j_1,j_2, \ldots, j_{k-1}\}) \nonumber \\[3pt]&\quad \stackrel{\mathrm{d}}{=} (T_{k:n}|J_1=j_1, J_2=j_2, \ldots , J_{k-1}= j_{k-1})\nonumber \\[3pt]&\quad \stackrel{\mathrm{d}}{=} F^{-1} \!\left ( 1- \exp \left( -\sum_{i=1}^k \frac{V_i}{\Lambda(\{\,j_1, \ldots,j_{i-1}\})}\right)\! \right) \nonumber\\[3pt] & \stackrel{\mathrm{d}}{=} F^{-1}\! \left( \!1- \prod_{i=1}^k U_i^{\frac 1{\Lambda(\{\,j_1, \ldots,j_{i-1}\})}}\right) \!.\end{align}

Similarly, the joint conditional distribution of $(T_{i:n})_{i=1}^k$ can be written as

\begin{align}& ((T_{i:n})_{i=1}^k|I_1=\{\,j_1\}, I_2=\{\,j_1,j_2\} \ldots , I_{k-1}=\{\,j_1,j_2, \ldots, j_{k-1}\}) \nonumber \\[3pt] &\quad \stackrel{\mathrm{d}}{=} ((T_{i:n})_{i=1}^k|J_1=j_1, J_2=j_2, \ldots , J_{k-1}= j_{k-1})\nonumber \\[3pt]&\quad \stackrel{\mathrm{d}}{=} \left(F^{-1} \left ( 1- \exp \left( - \sum_{m=1}^i \frac{V_m}{\Lambda(\{\,j_1, \ldots,j_{m-1}\})}\right) \right)\right)_{i=1}^k \nonumber \\[3pt] &\quad \stackrel{\mathrm{d}}{=} \left(F^{-1} \left( 1- \prod_{m=1}^i U_m^{\frac 1{\Lambda(\{\,j_1, \ldots,j_{m-1}\})}}\right)\right)_{i=1}^k .\nonumber \end{align}

Proof. The first representation in (22) immediately follows from the definition and Corollary 2 (see (8)). In order to get the last one, we first rewrite the exponential function in (22) as

\begin{equation*}\exp \left( - \sum_{i=1}^k \frac{V_i}{\Lambda(\{\,j_1, \ldots,j_{i-1}\})}\right)= \prod_{i=1}^k [\!\exp\!(\!-\!V_i)]^{\frac 1{\Lambda(\{\,j_1, \ldots,j_{k-1}\})}}.\end{equation*}

Note that random variables

\begin{equation*}\tilde{U}_i= 1- \exp\!(\!-V_i), \qquad i=1, \ldots,k,\end{equation*}

are independent uniformly distributed on (0,1), and so are $U_i=1-\tilde{U}_i$, $i=1,\ldots,k$. Therefore the two representations in (22) are equivalent. Similarly one obtains the representation of the joint conditional distribution of the first k components failures.

The relations between $(T_{k:n}|J_1,\ldots,J_{k-1})$ and $(T_{k-1:n}|J_1,\ldots,J_{k-2})$ are nicely illustrated by the formulae

\begin{align*}& (T_{k:n}|J_1,\ldots,J_{k-1}) \\[3pt] &\quad \stackrel{\mathrm{d}}{=} \left( F^{-1}\left( 1-[1-F(T_{k-1:n})] \left. \exp \left( \frac{-V_k}{\Lambda(\{J_1, \ldots,J_{k-1}\})}\right)\right)\right| J_1,\ldots,J_{k-2}\right) \\[3pt] &\quad \stackrel{\mathrm{d}}{=} \left( F^{-1}\left( 1-[1-F(T_{k-1:n})]\left. U_k^{\frac{1}{\Lambda(\{J_1, \ldots,J_{k-1}\})}}\right) \right| J_1,\ldots,J_{k-2}\right),\end{align*}

easily deduced from (22).

Proposition 5 shows that the joint distribution of $T_{1:n}, \ldots, T_{n:n}$ under the condition that $J_1=j_1, J_2=j_2, \ldots , J_{n-1}= j_{n-1}$ is just the distribution of generalized order statistics with parent distribution function F and parameters $(\gamma_1, \gamma_2, \ldots , \gamma_n ) = (\Lambda(\emptyset), \Lambda(\{\,j_1\}), \ldots, \Lambda(\{\,j_1, j_2, \ldots, j_{n-1} \})$. The generalized order statistic is a distributional concept introduced by Kamps [Reference Kamps18] and applied for a unified description of distributions of various ordered random variables. It is defined using a single distribution function F and a set of positive parameters $\gamma_i$. The joint density function of the first r generalized order statistics, with distribution function F and density f, and parameters $\gamma_1,\ldots,\gamma_r$, has the form

\begin{equation*}f^*_{\gamma_1,\ldots,\gamma_r,F}(x_1,\ldots,x_r)= \left( \prod_{j=1}^r \gamma_j \right)\! \left( \prod_{j=1}^{r-1} [1-F(x_j)]^{\gamma_j-\gamma_{j+1}-1} f(x_j) \right)\! [1-F(x_{r})]^{\gamma_r-1} f(x_r)\end{equation*}

for $F^{-1}(0+) < x_1 < \ldots <x_r < F^{-1}(1-\!)$. For various choices of parameters, this describes the joint density function of, e.g., order statistics from independent and identically distributed (i.i.d.) samples with parent F, upper records in i.i.d. sequences with marginal F, their generalizations to kth records (which are new kth maxima in the sequences), Type-II censored samples, and progressively Type-II censored samples, where after consecutive failures fixed numbers of randomly selected living items are removed from the experiment for ethical or economic reasons. Since the publication of the book [Reference Kamps18], the theory and applications of generalized order statistics have been developed in hundreds of papers. In particular, there have been attempts to employ the notion in reliability theory (see, e.g., [Reference Burkschat8], [Reference Burkschat and Rychlik9], and [Reference Hollander and Peña17]).

It follows that the time-heterogeneous load-sharing model generalizes the model of generalized order statistics. Its weak point, however, is that it cannot be sequentially extended to an arbitrary dimension as the standard generalized order statistics can. In order to get marginal density functions of a single or several order statistics in the model, it suffices to integrate properly the joint density (21). However, we cannot expect closed and nice formulae in general. Cramer and Kamps [Reference Cramer and Kamps10] proved that in the case of the standard uniform distribution function U, the kth generalized order statistic with parameters $(\gamma_1, \gamma_2, \ldots , \gamma_k )$ has the density function

\begin{equation*}g^*_{k;\gamma_1,\ldots,\gamma_k, U}(t) = \left (\prod_{i=1}^k \gamma_i \right)\sum_{j=1}^l \sum_{m=0}^{d_j-1} \frac{K_{jm}}{ m!} (1-t)^{\delta_j-1}\frac{[\!- \ln (1-t)]^{d_j - m-1}}{(d_j-m-1)!}, \quad 0<t<1,\end{equation*}

where $ \delta_1 < \ldots < \delta _l$ represent the different values in the sequence $(\gamma_1, \ldots, \gamma_k)$, and $d_1, \ldots, d_j$ are their respective multiplicities, while

\begin{align*}K_{j0} & = \prod_{\stackrel{\scriptstyle q=1}{ q\neq j}}^l \frac{1}{(\delta_q-\delta_j)^{d_q}}, \\[3pt] K_{jm} & = \sum_{p=0}^{m-1} (\!-1)^{p+1} \frac {(m-1)!}{(m-1-p)!} K_{j,m-1-p}\sum_{\stackrel{\scriptstyle q=1}{ q\neq j}}^l \frac{d_q}{(\delta_q-\delta_j)^{p+1}}, \quad j \geq 1.\end{align*}

This density function does not depend on n and $\gamma_i$ with $i>k$. If the generalized order statistics have the baseline distribution F, then the marginal density function of the kth generalized order statistic $T_{k:n}$ takes the form

\begin{equation*}g^*_{k;\gamma_1,\ldots,\gamma_k,F}(t) = f(t)\,g^*_{k;\gamma_1,\ldots,\gamma_k,U}(F(t)), \qquad a<t<b.\end{equation*}

The kth order statistic in the time-heterogeneous load-sharing model with parent distribution function F and parameter set (5) has the conditional density function

\begin{equation*}f_{\mathcal{L},F;T_{k:n}}(t|j_1, \ldots,j_{k-1})= g^*_{k;\Lambda(\emptyset), \Lambda(\{\,j_1\}), \ldots, \Lambda(\{\,j_1, j_2, \ldots, j_{k-1} \}),F}(t),\qquad a<t<b,\end{equation*}

with respect to the random event $(J_1,J_2,\ldots,J_{k-1})=(j_1,j_2, \ldots,j_{k-1})$. The unconditional distribution of the kth subsequent component failure in our model is a convex combination of $\frac{n!}{(n-k)!}$ distributions of kth generalized order statistics with common F and various parameters.

Corollary 6. The unconditional density of $T_{k:n}$ is given by

\begin{equation*}f_{\mathcal{L},F;T_{k:n}}(t)=\! \sum_{(j_1, \ldots, j_{k-1}) \in \Pi([n])}g^*_{k;\Lambda(\emptyset), \Lambda(\{\,j_1\}), \ldots, \Lambda(\{\,j_1, j_2, \ldots, j_{k-1} \}),F}(t)\! \prod_{i=1}^{k-1}\!\frac{\lambda_{j_i}(\{\,j_1, j_2, \ldots, j_{i-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{i-1}\})}.\end{equation*}

Similarly, the density functions of every component and of the system are also convex combinations of the density functions of generalized order statistics.

Corollary 7. The density function of the ith component lifetime $T_i$ has the form

\begin{align}f_{\mathcal{L},F; T_i}(t)& = \sum_{k=1}^n \sum_{(j_1, \ldots, j_{k-1}) \in \Pi([n]\setminus \{i\})}g^*_{k;\Lambda(\emptyset), \Lambda(\{\,j_1\}), \ldots, \Lambda(\{\,j_1, j_2, \ldots, j_{k-1} \}),F}(t)\nonumber \\[3pt]&\quad \times \prod_{i=1}^{k-1}\frac{\lambda_{j_i}(\{\,j_1, j_2, \ldots, j_{i-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{i-1}\})}\frac{\lambda_{i}(\{\,j_1, j_2, \ldots, j_{k-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{k-1}\})}.\nonumber \end{align}

Corollary 8. The lifetime T of the system with structure $\varphi$ and component lifetimes $T_1,\ldots,T_n$ satisfying the assumptions of the time-heterogeneous load-sharing model with parent distribution function F and parameters (5) has the density function

\begin{equation*}f_{\mathcal{L},F;T}(t)= \sum_{k=1}^n \!\sum_{(j_1, \ldots, j_{n}) \in \chi^{-1}(k)}\!g^*_{k;\Lambda(\emptyset), \Lambda(\{\,j_1\}), \ldots, \Lambda(\{\,j_1, j_2, \ldots, j_{k-1} \}),F}(t)\!\prod_{i=1}^{n-1}\!\frac{\lambda_{j_i}(\{\,j_1, j_2, \ldots, j_{i-1}\})}{\Lambda(\{\,j_1, j_2, \ldots, j_{i-1}\})}.\end{equation*}

Proposition 6. In the time-heterogeneous load-sharing model based on the distribution function F and parameters $\mathcal{L}$, the multivariate conditional hazard rates of component lifetimes do not depend on the particular failure times of the components that have failed previously, and can be written as

(23) \begin{equation}\lambda_j(t| A_{k-1}, t_1,\ldots, t_{k-1})= \lambda_j( A_{k-1})\frac{f(t)}{1-F(t)}, \quad j \in A_{k-1}^c, \; |A_{k-1}|=k-1, \; a<t<b.\end{equation}

Formally, the formula (23) holds for $a<t_1< \ldots <t_{k-1} <t$, but $t_1,\ldots,t_{k-1}$ could be located arbitrarily close to a, and so (23) has to be defined for all $t>a$.

Proof. We first determine the multivariate conditional failure rates at subsequent failure times. By (19) and (20), the conditional survival function of the kth order statistic

\begin{equation*}\mathbb{P}(T_{k:n} >t| T_{k-1:n}=t_{k-1}, I_{k-1}=A_{k-1}) = \left[ \frac{1-F(t)}{1-F(t_{k-1})}\right]^{\Lambda(A_{k-1})}, \quad k=1, \ldots ,n,\end{equation*}

depends only on the labels of the already failed components $A_{k-1}$ and the last preceding failure time $t_{k-1}$. It is independent of the moments of other earlier failures and of the order of consecutively failing components. The corresponding conditional density is

\begin{equation*}f_{\mathcal{L},F; T_{k:n}}(t| T_{k-1:n}=t_{k-1}, I_{k-1}=A_{k-1}) = \frac{\Lambda(A_{k-1})[1-F(t)]^{\Lambda(A_{k-1})-1}f(t)}{[1-F(t_{k-1})]^{\Lambda(A_{k-1})}}.\end{equation*}

Writing the respective hazard rate

\begin{equation*}\lambda_{\mathcal{L},F; T_{k:n}}(t| T_{k-1:n}=t_{k-1}, I_{k-1}=A_{k-1}) =\Lambda(A_{k-1})\,\frac{f(t)}{1-F(t)}= \lambda_{\mathcal{L},F; T_{k:n}}(t| I_{k-1}=A_{k-1}),\end{equation*}

we also eliminate the dependence on $t_{k-1}$ and get a linear transformation of the failure rate $\lambda_F(t)=\frac{f(t)}{1-F(t)}$ of the model distribution function F. For the component lifetime, we have

\begin{align*}& \lambda_{\mathcal{L},F; T_{j}}(t|I_{k-1}=A_{k-1}) = \lim_{\triangle\, t \searrow 0} \frac 1{\triangle\, t} \mathbb{P}(T_j\leq t+ \triangle\, t|I_{k-1}=A_{k-1},T_{k:n}>t )\\[3pt] &\quad = \lim_{\triangle\, t \searrow 0} \frac 1{\triangle\, t} \mathbb{P}(T_j=T_{k:n}, T_{k:n} \leq t+ \triangle\, t|I_{k-1}=A_{k-1},T_{k:n}>t ) .\end{align*}

In view of (18), under the condition $I_{k-1}=A_{k-1}$, the probability that $T_j=T_{k:n}$ is independent of the distribution of $T_{k:n}$. Therefore

\begin{align*}\lambda_{\mathcal{L},F; T_{j}}(t|I_{k-1}=A_{k-1}) & = \mathbb{P}(T_j= T_{k:n}|I_{k-1}=A_{k-1}) \\[3pt] &\quad \times \lim_{\triangle\, t \searrow 0} \frac 1{\triangle\, t}\mathbb{P}(T_{k:n} \leq t+ \triangle\, t|I_{k-1}=A_{k-1},T_{k:n}>t ) \\[3pt] & = \frac{\lambda_j(A_{k-1})}{\Lambda(A_{k-1})}\,\Lambda(A_{k-1})\,\frac{f(t)}{1-F(t)}= \lambda_j(A_{k-1}) \lambda_F(t),\end{align*}

as desired.

Proposition 6 shows that our time-heterogeneous load-sharing model based on some fixed distribution function F and parameters (5) follows the linear breakdown rule (4). The form of the multivariate conditional hazard rates (23) refers to some intuitive arguments. We assume that the components have some immanent durability properties, expressed by means of the failure rate $\lambda_F$ of the parent model distribution, which has fixed shape and monotonicity properties. After the system begins operation, and then after the working status of other components changes, the level of aging intensity of each component changes abruptly without violating its further variability. The degree of change depends on the location of the component in the system and the working status of the remaining system components.

We claim that the only joint distributions of the component lifetimes that follow the linear breakdown rule are those satisfying the assumptions of the time-heterogeneous load-sharing model with some baseline distribution function F and parameters $\lambda_j(A)$, $j \not \in A \subsetneqq [n]$.

In the proof we apply the following auxiliary result.

Proposition 7. Suppose that $T_1,\ldots,T_n$ have a joint distribution determined by the multivariate conditional hazard rates (2) and (3). Then for every $k=0,\ldots,n-1$, $j_1,\ldots,j_k \in [n]$ and $0<t_1<\ldots<t_k$, the random variables $T_i-t_k$, $i \in \{\,j_1,\ldots,j_k\}^c$, and independent positive random variables $Z_i$, $i \in \{\,j_1,\ldots,j_k\}^c$, with respective failure rates

\begin{equation*}r_i(t) = \lambda_i(t+t_k|j_1,\ldots,j_k,t_1,\ldots,t_k), \qquad t>0,\end{equation*}

satisfy

\begin{align*}& \mathbb{P}(T_i-t_k= \min\{ T_j-t_k\,{:}\, j \in \{\,j_1,\ldots,j_k\}^c\} \in B |T_{j_1}=t_1, \ldots ,T_{j_k}=t_k<T_{k+1:n})\\[1pt] & \quad =\, \mathbb{P}(Z_i= \min\{ Z_j\,{:}\, j \in \{\,j_1,\ldots,j_k\}^c \}\in B)\end{align*}

for every $ i \in \{\,j_1,\ldots,j_k\}^c$ and B an arbitrary Borel subset of $\mathbb{R}_+$.

This result immediately follows from De Santis et al. [Reference De Santis, Malinovsky and Spizzichino12, Proposition 1]. The special case with $k=0$ was formulated in Proposition 2 of [Reference De Santis, Malinovsky and Spizzichino12]. In our more general version, we consider conditional distributions of the residual lifetimes $T_i-t_k$, $i \in \{\,j_1,\ldots,j_k\}^c$, given $\{ T_{j_1}=t_1,\ldots,T_{j_k}=t_k,T_i > t_k, i \in \{\,j_1,\ldots,j_k \}^c\}$. The assumption of absolute continuity of $T_1,\ldots,T_n$ ensures the possibility of constructing the multivariate conditional hazard rates $\lambda_\cdot(\!\cdot|\cdot\!)$ on one hand, and the existence of the joint conditional density function of the aforementioned conditional distribution on the other. The latter implies that there exist multivariate conditional hazard rates $\tilde{\lambda}_\cdot(\!\cdot|\cdot\!)$ of the conditional distribution. In view of the definitions, we immediately notice that the relations

\begin{equation*}\tilde{\lambda}_j(t|\emptyset) = \lambda_j(t+t_k|j_1,\ldots,j_k,t_1,\ldots ,t_k),\end{equation*}

with $j \in \{\,j_1,\ldots,j_k\}^c$ and $t>0$, and

\begin{equation*}\tilde{\lambda}_j(t|i_1,\ldots,i_\ell,s_1,\ldots,s_\ell) = \lambda_j(t+t_k|j_1,\ldots,j_k,i_1,\ldots, i_\ell,t_1,\ldots ,t_k,t_k+s_1,\ldots,t_k+s_\ell),\end{equation*}

with $ \ell= 0,\ldots,n-k-1$, $\{i_1,\ldots,i_\ell,j\} \subset \{\,j_1,\ldots,j_k\}^c$, and $0 <s_1<\ldots s_\ell <t$, hold.

Proof of Theorem 2. We first show that the function L(t) is the failure rate of a lifetime distribution function with a positive density on (a, b). The function is certainly positive on (a, b), because so is $\lambda_j(A)$ for every j and A. Consequently, it suffices to show that $\int_a^b L(t)dt =+\infty$.

The multivariate conditional failure rates of the jth component lifetime have the forms

(24) \begin{align}& \lambda_j(t|J_1=j_1, \ldots, J_k=j_k, T_{j_1}=t_1 < \ldots < T_{j_k}=t_k <t< \min_{i \in \{\,j_1, \ldots, j_k \}^c} T_i ) \nonumber \\[1pt]&\quad = \lambda_j(\{\,j_1, \ldots, j_k \}) L(t), \qquad \qquad a<t<b, \quad j \in \{\,j_1, \ldots, j_k \}^c. \end{align}

In view of the fact that such conditional failure rates do not depend on the particular times of the previous failures, we use the total probability law to calculate the unconditional version of the failure rate,

\begin{equation*}\lambda_j(t) =\sum_{k=0}^{n-1} \sum_{(j_1,\ldots,j_k) \in \Pi([n]\setminus\{\,j\})}\!\! \lambda_j(\{\,j_1, \ldots, j_k \})\mathbb{P}(T_{j_1} < \ldots < T_{j_k} <t < \!\min_{i \in \{\,j_1, \ldots, j_k \}^c} T_i ) L(t),\end{equation*}

which satisfies

\begin{align*}+\infty= \int_a^b \lambda_j(t)\, dt & = \sum_{k=0}^{n-1} \sum_{(j_1,\ldots,j_k) \in \Pi([n]\setminus\{\,j\})} \lambda_j(\{\,j_1, \ldots, j_k \})\\[1pt]&\quad \times \int_a^b \mathbb{P}(T_{j_1}< \ldots < T_{j_k} <t < \min_{i \in \{\,j_1, \ldots, j_k \}^c} T_i )\, L(t)\, dt\end{align*}

if and only if $\int_a^b L(t)dt =+\infty$.

Now we prove the relations (18). We fix $0 \leq k \leq n-1$, $j_1,\ldots,j_k \in [n]$, and $0<t_1< \ldots\,{<}\, t_k$. Under the notation of Proposition 7, the independent random variables $Z_i$, $i \in \{\,j_1,\ldots,j_k \}^c$, have failure rates

\begin{equation*}r_i(t) = \lambda_i(\{\,j_1,\ldots,j_k\}) L(t+t_k), \qquad 0 < t < b-t_k.\end{equation*}

Their survival and density functions are

\begin{align*}\bar{G}_i(t) & = \exp \left( - \lambda_i(\{\,j_1,\ldots,j_k\}) \int_0^t L(u+t_k)\,du \right),\\[3pt] g_i (t) & = \lambda_i(\{\,j_1,\ldots,j_k\}) L(t+t_k)\,\exp \left( - \lambda_i(\{\,j_1,\ldots,j_k\}) \int_0^t L(u+t_k)\,du \right),\end{align*}

respectively. Then

\begin{align*}& \mathbb{P}(Z_i= \min\{ Z_j\,{:}\, j \in \{\,j_1,\ldots,j_k\}^c \}) =\mathbb{P}(Z_i\leq \min\{ Z_j\,{:}\, j \in \{\,j_1,\ldots,j_k,j\}^c \})\\[3pt]&\quad = \int_0^{b-t_k} \left[ \prod_{j \in \{\,j_1,\ldots,j_k,j\}^c} \bar{G}_j(t) \right] g_i(t) \,dt= \int_0^{b-t_k} \!\lambda_i(\{\,j_1,\ldots,j_k\}) L(t+t_k) \\& \qquad \times \exp \left( - \sum_{j \in \{\,j_1,\ldots,j_k\}^c} \lambda_j(\{\,j_1,\ldots,j_k\}) \int_0^t L(u+t_k)\,du \right) dt\\[3pt] &\quad = \frac{\lambda_i(\{\,j_1,\ldots,j_k\})}{\sum_{j \in \{\,j_1,\ldots,j_k\}^c} \lambda_j(\{\,j_1,\ldots,j_k\})}.\end{align*}

By Proposition 7, independently of the particular values of $T_{j_1}=t_1<\ldots<T_{j_k}=t_k<T_{k+1:n}$, we obtain

\begin{equation*}\mathbb{P}(T_i= T_{k+1:n}|J_1=j_1\ldots,J_k=j_k)\\= \frac{\lambda_i(\{\,j_1,\ldots,j_k\})}{\sum_{j \in \{\,j_1,\ldots,j_k\}^c} \lambda_j(\{\,j_1,\ldots,j_k\})}.\end{equation*}

Summing up, under the condition (4) the function L(t), $a<t<b$, is the failure rate of a unique distribution function F, say, supported on (a, b). The representation (4) implies (18) as well. It follows that the assumptions of the time-heterogeneous load-sharing model with baseline distribution function F and parameters (5) are satisfied. Since the family of multivariate conditional hazard rate functions uniquely determines the joint distribution of $(T_1,\ldots,T_n)$, the assumption (4) describes a time-heterogeneous load-sharing model as defined at the beginning of this section.

The time-heterogeneous load-sharing model distribution uniquely determines the respective set of parameters $\mathcal{L}$ and F up to the transformations $\mathcal{L}_\theta = \{ \theta \lambda_j(A)\,{:}\,A \subsetneqq [n],\; j \in A^c \}$ and $F_\theta(t)= 1-[1-F(t)]^{1/\theta}$ for $\theta>0$. Note that the multivariate conditional hazard rate functions have the same form (4) for all $\theta>0$.

4. Special cases of the model parameters

In this section we analyze the time-heterogeneous load-sharing model in the cases when the set of model parameters (5) takes on simplified forms. The first one is equivalent to the exchangeability of component lifetimes, and the other two provide some generalizations of the exchangeable case.

Exchangeable components. Here we assume that the behavior of any system component is indistinguishable from that of the others; i.e. the joint density $f_{T_1, \ldots ,T_n}(t_1, \ldots,t_n)$ of the component lifetimes is invariant under permutations of its arguments $t_1,\ldots,t_n$. In terms of the multivariate conditional hazard rates, this assumption can be equivalently expressed by the condition that for any pair (j, A) with $A \varsubsetneq [n]$ and $j \in A^c$, the load-sharing parameter $\lambda_j(A)$ is independent of j, and depends on A only through its cardinality $|A|$. In other words, we assume existence of positive numbers $\alpha(i)$, $i=1,\ldots, n-1$, such that

(25) \begin{equation}\lambda_j(A)= \alpha(|A|), \qquad A \varsubsetneq [n], \quad j \in A^c.\end{equation}

This implies that $\Lambda(A)= (n-|A|)\alpha(|A|)$, $A \varsubsetneq [n]$. Note that in the exchangeable submodel, the number of parameters is reduced from $n2^{n-1}$ to n. The equivalence between exchangeability and (25) can be proven by resorting to the formulae which express the multivariate conditional hazard rate functions in terms of the joint density and vice versa. See also, e.g., the arguments in [Reference Spizzichino38], and [Reference Spizzichino40]. By the exchangeability assumption and Corollary 2, the random sequence of consecutive subsets of failed components $I_1,\ldots, I_n$ ($I_k \subset I_{k+1}$, $|I_k|=k$) forms a Markov chain with initial probability $\mathbb{P} (I_1=\{\,j_1\}) = \frac 1n$ and transition probability law

\begin{equation*}\mathbb{P}(I_{k+1}= A_k\cup \{\,j_{k+1}\}|I_k=A_k) = \frac 1{n+1-k}, \qquad k=1,\ldots,n,\end{equation*}

for $A_k$ denoting a value of the random variable $I_k$, i.e., the set of the first k failed components. This implies that

\begin{equation*}\mathbb{P}(I_1=A_1,\ldots,I_k=A_k)= \frac{(n-k)!}{n!}\end{equation*}

and

\begin{equation*}\mathbb{P}(I_k=A_k) = \frac 1{\left(\begin{array}{l}n \\ k\end{array}\right)}\end{equation*}

for $k=1,\ldots,n$. In particular, all the sequences of labels of consecutively failed components have the same probability $\mathbb{P}(I_1=A_1,\ldots,I_n=A_n)= \frac 1{n!}$.

By Proposition 4, the sequence of pairs $(T_{k:n},I_k)$, $k=1,\ldots,n$, is a two-dimensional Markov chain with probability measure determined by

\begin{equation*}\mathbb{P}(T_{1:n}>t, I_1=\{\,j_1\}) =[1-F(t)]^{n\alpha(0)} \, \frac 1n, \qquad t>0,\end{equation*}

and

\begin{align*}& \mathbb{P}(T_{k:n}>t, I_k=A_{k-1}\cup \{\,j_k\}|T_{k-1:n}=t_{k-1}, I_{k-1}=A_{k-1}) \\&\quad = \left[ \frac{1-F(t)}{1-F(t_{k-1})}\right]^{(n+1-k)\alpha(k-1)} \, \frac 1{n+1-k}, \qquad t>t_{k-1},\;k=2,\ldots,n.\end{align*}

It follows that the sequence of order statistics $T_{k:n}$, $k=1,\ldots,n$, itself is a Markov chain with initial and transition probability laws

\begin{align*}\mathbb{P}(T_{1:n}>t) & = [1-F(t)]^{n\alpha(0)}, \qquad \quad \qquad \quad \;\; t>0, \\[3pt] \mathbb{P}(T_{k:n}>t|T_{k-1:n}=t_{k-1}) & = \left[ \frac{1-F(t)}{1-F(t_{k-1})}\right]^{(n+1-k)\alpha(k-1)}, \quad t>t_{k-1},\;k=2,\ldots,n,\end{align*}

respectively. This implies that the distribution of consecutive failures $T_{1:n},\ldots,T_{n:n}$ is identical with the distribution of n generalized order statistics with baseline distribution function F and parameters $\gamma_k= (n+1-k)\alpha(k-1)$, $k=1,\ldots,n$. Therefore the unconditional density function of the kth system component failure can be written as

\begin{equation*}f_{\mathcal{L},F,T_{k:n}}(t) = g^*_{k; n\alpha(0), \ldots,(n+1-k)\alpha(k-1),F}(t).\end{equation*}

By Corollary 7, we also have

\begin{align*}f_{\mathcal{L},F,T_i} (t) & = \sum_{k=1}^n \frac{(n-1)!}{(n-k)!}\,g^*_{k; n\alpha(0), \ldots,(n+1-k)\alpha(k-1),F}(t)\, \frac{(n-k)!}{n!} \\[3pt]& = \frac 1n \sum_{k=1}^n g^*_{k; n\alpha(0), \ldots,(n+1-k)\alpha(k-1),F}(t) = \frac 1n \sum_{k=1}^n f_{\mathcal{L},F,T_{k:n}}(t).\end{align*}

This illustrates the classic property of exchangeable random variables that the marginal distribution of any single variable is the uniform mixture of the distributions of order statistics. By Corollary 8, we get

(26) \begin{equation}f_{\mathcal{L},F,T} (t) = \sum_{k=1}^n \frac{|\chi^{-1}(k)|}{n!} g^*_{k; n\alpha(0), \ldots,(n+1-k)\alpha(k-1),F}(t)= \sum_{k=1}^n \frac{|\chi^{-1}(k)|}{n!} f_{\mathcal{L},F,T_{k:n}}(t),\end{equation}

which is the mixture of density functions of order statistics with the coefficient vector

\begin{equation*}\textbf{s}= (s_1, \ldots,s_n)= \left( \frac{|\chi^{-1}(1)|}{n!}, \ldots, \frac{|\chi^{-1}(n)|}{n!} \right),\end{equation*}

dependent merely on the structure function of the system. The vector $\textbf{s}$ is called the structural Samaniego signature of the system. The representation (26) was proven in Samaniego [Reference Samaniego30] in the case of i.i.d. component lifetimes, and it was extended to the exchangeable case by Navarro et al. [Reference Navarro, Samaniego, Balakrishnan and Bhattacharya24] (see also [Reference Navarro and Rychlik23]). Note that the Samaniego signature has also the probabilistic meaning $s_k=p_k=\mathbb{P}(T=T_{k:n})$, $k=1,\ldots,n$. Hence (26) can be rewritten as

\begin{equation*}f_{\mathcal{L},F,T} (t) = \sum_{k=1}^n \mathbb{P}(T=T_{k:n}) f_{\mathcal{L},F,T_{k:n}}(t),\end{equation*}

which means that under the assumption of exchangeability of $T_1,\ldots,T_n$ the component failure times are independent of the system structure. The vector $\textbf{p}=(p_1,\ldots,p_n)$ is called the probabilistic signature of the system; under violation of the exchangeability assumption it usually differs from the structural one. The time-heterogeneous load-sharing model with exchangeable components is also called the failure-dependent proportional hazard model, because the multivariate conditional hazard rates here,

\begin{equation*}\lambda_j(t|A_{k-1},t_1, \ldots,t_{k-1}) = \alpha(k-1) \,\frac{f(t)}{1-F(t)}, \qquad k=1,\ldots,n, \; t>0\end{equation*}

(cf. (23)), are proportional to the baseline hazard rate $f(t)/(1-F(t))$, and change only at the failure times of the components. The model was studied by Hollander and Pna [Reference Hollander and Peña17], Aki and Hirano [Reference Aki and Hirano1], Burkschat [Reference Burkschat8], and Burkschat and Rychlik [Reference Burkschat and Rychlik9], among others.

Components with uniform supporting ability. Assume now that the model parameters $\lambda_j(A)$, $j \not \in A \varsubsetneq [n]$, depend on the number of failed components, but do not depend on their particular labels. The submodel is thus characterized by $n^2$ parameters

\begin{equation*}\beta_j(k) = \lambda_j(A) \qquad \mathrm{iff}\qquad |A|=k, \; j=1,\ldots,n,\;k=0,\ldots,n-1.\end{equation*}

It is natural, but not necessary, to assume that for fixed j all the sequences $\beta_j(k)$ are nondecreasing.

The quantity

\begin{equation*}\varsigma_{i,j}(A)=\mathcal{\lambda}_{j}(A\cup\{i\})-\mathcal{\lambda}_{j}(A), \qquad i \neq j \in A^c, \;A \varsubsetneq [n],\end{equation*}

which can be regarded as a measure of the ability of component i to reduce the stress suffered by component j when all the components in A have failed, does not depend on the label of the supporting component, but only on the label of the supported one. This can be interpreted as the uniformity of the ability of operating components to reduce the stress on other operating components.

In contrast to the exchangeable case, it appears here that despite a significant reduction of the number of parameters, the distributional properties of component and system lifetimes and respective formulae do not simplify much compared to those in the general model. We have

\begin{align*}f_{\mathcal{L},F,T_{k:n}}(t) & = \sum_{(j_1,\ldots,j_{k-1}) \in \Pi([n])} g^*_{k;B(\emptyset),\ldots,B(\{\,j_1, \ldots,j_{k-1}\}),F}(t)\prod_{i=1}^k \frac{\beta_{j_i}(i-1)}{B(\{\,j_1, \ldots,j_{i-1}\})}, \\[3pt] & k=1,\ldots,n,\\[3pt] f_{\mathcal{L},F,T_i}(t) & = \sum_{k=1}^n\!\beta_i(k-1)\!\! \sum_{(j_1,\ldots,j_{k-1}) \in \Pi([n]\setminus \{i\})}\!\frac{g^*_{k;B(\emptyset),\ldots,B(\{\,j_1, \ldots,j_{k-1}\}),F}(t)\!\prod_{i=1}^{k-1} \beta_{j_i}(i-1)}{\prod_{i=1}^k \!B(\{\,j_1, \ldots,j_{i-\!1}\})}, \\[3pt] &\quad i=1,\ldots,n,\\f_{\mathcal{L},F,T}(t) & = \sum_{k=1}^n \sum_{(j_1,\ldots,j_{n}) \in \chi^{-1}(k)} g^*_{k;B(\emptyset),\ldots,B(\{\,j_1, \ldots,j_{n}\}),F}(t)\prod_{i=1}^n \frac{ \beta_{j_i}(i-1)}{ B(\{\,j_1, \ldots,j_{i-1}\})}\end{align*}

for $B(A)=\Lambda(A)= \sum_{j \in A^c} \beta_j(|A|)$, $A \varsubsetneq [n]$, denoting the cumulative hazard rate of living components when those of A have failed.

As in the general model, $(T_{k:n},I_k)$ and $(I_k)$, $k=1,\ldots,n$, are Markov chains, but $(T_{k:n})$, $k=1,\ldots,n$, is not. We finally observe that

\begin{equation*}p_k= \mathbb{P}(T=T_{k:n}) = \sum_{(j_1,\ldots,j_{n}) \in \chi^{-1}(k)}\prod_{i=1}^n \frac{ \beta_{j_i}(i-1)}{ B(\{\,j_1, \ldots,j_{i-1}\})}, \quad k=1,\ldots,n,\end{equation*}

are different from the structural signature coefficients, and the conditional density functions

\begin{equation*}f_{\mathcal{L},F,T|T=T_{k:n}}(t) = \frac{\sum_{(j_1,\ldots,j_{n}) \in \chi^{-1}(k)} g^*_{k;B(\emptyset),\ldots,B(\{\,j_1, \ldots,j_{n}\}),F}(t)\prod_{i=1}^{n-1} \frac{ \beta_{j_i}(i-1)}{ B(\{\,j_1, \ldots,j_{i-1}\})}}{\sum_{(j_1,\ldots,j_{n}) \in \chi^{-1}(k)}\prod_{i=1}^{n-1} \frac{ \beta_{j_i}(i-1)}{ B(\{\,j_1, \ldots,j_{i-1}\})}},\end{equation*}

$k=1,\ldots,n$, strongly depend on the structure of the system, unlike in the exchangeable case.

Components with uniform frailty. In this subsection, we consider the case when the load-sharing parameters $\mathcal{\lambda}_{j}(A)$ are independent of j for any $A\varsubsetneq\left[n\right] $ such that $j\in A^{c}$. That is, we assume the existence of $2^n-1$ positive numbers $\delta(A)$ satisfying

\begin{equation*}\mathcal{\lambda}_{j}(A)=\delta(A), \qquad j\in A^c,\; A\varsubsetneq\left[n\right].\end{equation*}

The quantities $\delta(A)$ do not depend on the specific identity j of each still surviving component, but they may depend on the set A of already failed components. Unlike in the exchangeable setup we do not assume here the relation $|A^{\prime}|=|A^{\prime\prime}|\Rightarrow\delta(A^{\prime})=\delta(A^{\prime\prime})$. The special form of the assumptions can be interpreted as uniform frailty or uniform sensitivity of all still operating components to the load acting on the system.

We first notice that

\begin{align*}\Lambda(A)=\sum_{j\in A^{c}}\mathcal{\lambda}_{j}(A) & = \left(n-|A|\right)\delta(A), \\[3pt] \frac{\mathcal{\lambda}_{j}(A)}{\Lambda\left( A\right) }& = \frac{1}{n-|A|}, \qquad \qquad A \varsubsetneq [n].\end{align*}

This implies that the distributions of the sequences $(I_1,\ldots,I_k)$, $k=1,\ldots,n$, are

\begin{equation*}\mathbb{P}(I_1=\{\,j_1\}, \ldots, I_k=\{\,j_1,\ldots,j_k\}) = \frac{(n-k)!}{n!},\end{equation*}

as in the exchangeable case. Applying Corollary 6, we conclude that

\begin{equation*}f_{\mathcal{L},F,T_{k:n}}(t) = \frac{(n+1-k)!}{n!} \sum_{(j_1,\ldots,j_{k-1})\in \Pi([n])} g^*_{k;n\delta(\emptyset), \ldots, (n+1-k)\delta(\{\,j_1,\ldots,j_{k-1}\}),F}(t),\end{equation*}

which means that the distribution of the kth consecutive component failure is the uniform mixture of kth generalized order statistics with parent distribution function F and parameters $\gamma_i= (n+1-i) \delta(\{\,j_1,\ldots,j_{i-1}\})$, $i=1,\ldots,k$, over all subsequences of different elements of [n] of length $k-1$. Furthermore, we have

\begin{align*}f_{\mathcal{L},F,T_i}(t) & = \frac 1{n} \sum_{k=1}^n \frac{(n-k)!}{(n-1)!}\sum_{(j_1,\ldots,j_{k-1})\in \Pi([n]\setminus \{i\})} g^*_{k;n\delta(\emptyset), \ldots, (n+1-k)\delta(\{\,j_1,\ldots,j_{k-1}\}),F}(t), \\[3pt]f_{\mathcal{L},F,T}(t) & = \frac 1{n!} \sum_{k=1}^n\sum_{(j_1,\ldots,j_{n})\in \chi^{-1}(k)} g^*_{k;n\delta(\emptyset), \ldots, (n+1-k)\delta(\{\,j_1,\ldots,j_{k-1}\}),F}(t).\end{align*}

The first is the mean over all $k=1,\ldots,n$ of uniform mixtures of kth generalized order statistic densities with parameters $n\delta(\emptyset), \ldots, (n+1-k)\delta(\{\,j_1,\ldots,j_{k-1}\})$ and baseline F, with $(j_1,\ldots,j_{k-1})$ being the sequences of different elements of [n] other than i. The latter is the average of all conditional density functions of system failure under all possible orders of consecutively failing components. Note that although the uniform frailty model has many more parameters than the uniform supporting-ability model, the distributions of component and system lifetimes have much simpler forms. Moreover, under the uniform frailty assumption, the structural and probabilistic signatures coincide. On the other hand, the conditional density functions

\begin{equation*}f_{\mathcal{L},F,T|T=T_{k:n}}(t) = \frac 1{|\chi^{-1}(k)|} \sum_{(j_1,\ldots,j_{n})\in \chi^{-1}(k)} g^*_{k;n\delta(\emptyset), \ldots, (n+1-k)\delta(\{\,j_1,\ldots,j_{k-1}\}),F}(t)\end{equation*}

evidently depend on the system structure, in contrast to their independence in the exchangeable model. Also, the sequence of consecutive failures does not form a Markov chain.

5. Expectations of system lifetimes for special baseline distributions

For some parent distribution functions F, one can establish analytic formulae for the expectations of component lifetimes, their order statistics, and system lifetimes. The most natural choice is the standard exponential distribution function $F(t)=1-\exp\!(\!-t)$, $t>0$, with the quantile function $F^{-1}(u) = - \ln (1-u)$, $0<u<1$. Plugging this into the model of Section 3 (see (17)), we obtain exactly the time-homogeneous load-sharing model with parameters $\mathcal{L}$. In consequence, for the standard exponential model distribution function F we can simply recall all the establishments of Section 2. Location-scale transformations of the parent distribution result in analogous transformations of the random variables in the model.

Another example with frequent applications in the lifetime analysis is the Lomax distribution function $F_\alpha(t) = 1- (1+t)^{-\alpha}$, $t>0$, $\alpha>0$. It has the inverse $F^{-1}_\alpha(u) = (1-u)^{-1/\alpha}-1$. By Proposition 5, we have the representation

\begin{equation*}(T_{k:n}|J_1=j_1, J_2=j_2 \ldots , J_{k-1}= j_{k-1}) \stackrel{\mathrm{d}}{=}\prod_{i=1}^k U_i^{\frac{- 1}{\alpha \Lambda(\{\,j_1, \ldots,j_{i-1}\})}} -1\end{equation*}

with the conditional expectation

\begin{equation*}\mathbb{E} (T_{k:n}|J_1=j_1, J_2=j_2 \ldots , J_{k-1}= j_{k-1}) =\prod_{i=1}^k \frac{ \alpha \Lambda (\{\,j_1,\ldots, j_{i-1}\})}{\alpha \Lambda (\{\,j_1,\ldots, j_{i-1}\})-1}-1,\end{equation*}

which is finite under the assumption that all $\Lambda (\{\,j_1,\ldots, j_{i-1}\}$, $i=1,\ldots,k$, are greater than $\frac 1\alpha$. In consequence, we have

(27) \begin{align}\mathbb{E}T_{k:n} & = \sum_{(j_1, \ldots, j_{k-1}) \in \Pi([n])}\frac{ \alpha^k \Lambda (\{\,j_1,\ldots,j_{k-1}\})\prod_{i=1}^{k-1} \lambda_{j_i} (\{\,j_1,\ldots, j_{i-1}\})}{\prod_{i=1}^{k}[\alpha \Lambda (\{\,j_1,\ldots, j_{i-1}\})-1]}-1,\nonumber \\[3pt] \mathbb{E}T_i & = \sum_{k=1}^n \sum_{(j_1, \ldots, j_{k-1}) \in \Pi([n]\setminus \{i\})}\frac{ \alpha^k \lambda_{i} (\{\,j_1,\ldots, j_{k-1}\})\prod_{i=1}^{k-1}\lambda_{j_i} (\{\,j_1,\ldots, j_{i-1}\})}{\prod_{i=1}^k[\alpha \Lambda (\{\,j_1,\ldots, j_{i-1}\})-1]}-1,\nonumber \\[3pt] \mathbb{E}T & = \sum_{k=1}^n \! \sum_{(j_1, \ldots, j_{n}) \in \chi^{-1}(k)}\!\prod_{i=1}^k \! \frac{ \alpha \lambda_{j_i} (\{\,j_1,\ldots, j_{i-1}\})}{\alpha \Lambda (\{\,j_1,\ldots, j_{i-1}\})-1}\!\prod_{i=k+1}^n \!\!\frac{ \lambda_{j_i} (\{\,j_1,\ldots, j_{i-1}\})}{\Lambda (\{\,j_1,\ldots, j_{i-1}\})}-1,\end{align}

under the condition that all the denominators are positive.

The last example of this type is provided by the reflected power distribution function $F_\alpha(t) = 1-(1-t)^\alpha$, $0<t<1$, $\alpha>0$, with the quantile function $F^{-1}_\alpha(u)= 1-(1-u)^{1/\alpha}$, $0<u<1$. By (22), this yields

\begin{equation*}(T_{k:n}|J_1=j_1, J_2=j_2 \ldots , J_{k-1}= j_{k-1}) \stackrel{\mathrm{d}}{=} 1-\prod_{i=1}^k U_i^{\frac 1{\alpha \Lambda(\{\,j_1, \ldots,j_{i-1}\})}}.\end{equation*}

Since the respective expectation is

\begin{equation*}\mathbb{E} (T_{k:n}|J_1=j_1, J_2=j_2 \ldots , J_{k-1}= j_{k-1}) = 1-\prod_{i=1}^k \frac{ \alpha \Lambda (\{\,j_1,\ldots, j_{i-1}\})}{\alpha \Lambda (\{\,j_1,\ldots, j_{i-1}\})+1},\end{equation*}

we conclude the following:

(28) \begin{align}\mathbb{E}T_{k:n} & = 1- \sum_{(j_1, \ldots, j_{k-1}) \in \Pi([n])}\frac{ \alpha^k \Lambda (\{\,j_1,\ldots,j_{k-1}\})\prod_{i=1}^{k-1} \lambda_{j_i} (\{\,j_1,\ldots, j_{i-1}\})}{\prod_{i=1}^{k}[\alpha \Lambda (\{\,j_1,\ldots, j_{i-1}\})+1]}, \nonumber \\[3pt] \mathbb{E}T_i & = 1- \sum_{k=1}^n \sum_{(j_1, \ldots, j_{k-1}) \in \Pi([n]\setminus \{i\})}\frac{ \alpha^k \lambda_{i} (\{\,j_1,\ldots, j_{k-1}\})\prod_{i=1}^{k-1}\lambda_{j_i} (\{\,j_1,\ldots, j_{i-1}\})}{\prod_{i=1}^k[\alpha \Lambda (\{\,j_1,\ldots, j_{i-1}\})+1]},\nonumber \\[3pt] \mathbb{E}T & = 1 - \sum_{k=1}^n \sum_{(j_1, \ldots, j_{n}) \in \chi^{-1}\!(k)}\!\prod_{i=1}^k \! \frac{ \alpha \lambda_{j_i} (\{\,j_1,\ldots, j_{i-1}\})}{\alpha \Lambda (\{\,j_1,\ldots, j_{i-1}\})\!+\!1}\!\prod_{i=k+1}^n \!\! \frac{ \lambda_{j_i} (\{\,j_1,\ldots, j_{i-1}\})}{\Lambda (\{\,j_1,\ldots, j_{i-1}\})}. \end{align}

In the special case $\alpha=1$ we obtain the formulae for F being the standard uniform distribution function. The Lomax, exponential, and reflected power distributions together form the family of generalized Pareto distributions, which have multiple applications in the theory and practice of lifetime data analysis.

We complete the paper by presenting a numerical example.

It has minimal path sets $\{1,2\}$, $\{4,5\}$, $\{1,3,5\}$, $\{2,3,4\}$, and minimal cut sets $\{1,4\}$, $\{2,5\}$, $\{1,3,5\}$, $\{2,3,4\}$ (for the definitions of these notions, we refer the reader to, e.g., [5, p. 9]). Its Samaniego signature is $\big( 0, \frac 15, \frac 35,\frac 15, 0 \big)$, which means in particular that the system cannot fail because of one component failure, and cannot survive four component failures. This is also easily verified by a cursory look at the figure.

Another immediate conclusion of the system diagram analysis is the fact that the components in positions 1, 2, 4, and 5 are equally important to the functioning of the system, while the component in position 3 affects it less. We may support this claim more formally using component importance measures. For example, the Barlow–Proschan importance measure $I_{BP}(i)$ of the ith component, in the case of identical exchangeable components, is the proportion of the number of sequences of labels of consecutively failing components such that the failure of component i implies the failure of the system to the total number of sequences (see [Reference Barlow and Proschan4]). For the bridge system we have $I_{BP}(3)= \frac 1{15}$ and $I_{BP}(j) = \frac 7{30}$, $j \neq 3$.

Figure 1: Bridge system.

If we have four equally strong components and one weaker one, we certainly locate the weakest component at the least important position 3. Assume that this is reflected by the following assumption on the parameters of the load-sharing model: for different $j_1,j_2,j_3, j_4 \in \{1,2,4,5\}$, we set

\begin{align*}& \lambda_{j_1}(\emptyset) = 1 < \lambda_{3}( \emptyset) = \frac{4}3 < \lambda_{j_1}(\{ 3\}) = \frac{5}3 <\lambda_{j_1}(\{\,j_2\}) = 2 < \lambda_{3}(\{\,j_1\}) = \frac{7}3 \\[3pt] &\quad < \lambda_{j_1}(\{ 3,j_2\}) = \frac{8}3 <\lambda_{j_1}(\{\,j_2,j_3\}) = 3 < \lambda_{3}(\{\,j_1,j_2\}) = \frac{10}3 < \lambda_{j_1}(\{ 3, j_2,j_3\}) = \frac{17}3 \\[3pt] &\quad < \lambda_{j_1}(\{\,j_2,j_3, j_4\}) = 6 < \lambda_{3}(\{\,j_1,j_2, j_3\}) = \frac{19}3.\end{align*}

Since for system lifetime analysis we do not need to consider the last failure, we do not determine $\lambda_{j_1}(\{ 3, j_2,j_3, j_4\})$ nor $\lambda_{3}(\{\,j_1, j_2,j_3, j_4\})$. The above definition can be concisely written as

\begin{equation*}\lambda_j(A)= \left\{\begin{array}{l@{\quad}l}|A| +2 \lfloor \frac{|A|}3\rfloor +1, & j \in A^c \not \ni 3, \\[3pt] |A| +2 \lfloor \frac{|A|}3\rfloor +\frac 43, & j =3 \in A^c , \\[3pt] |A| +2 \lfloor \frac{|A|}3\rfloor +\frac 23, & j \neq 3 \in A^c.\end{array}\right.\end{equation*}

In particular, the natural condition $\lambda_i(A) < \lambda_j(B)$ holds for $|A|<|B|$ and $i \not \in A$, $j \not \in B$. Moreover, for the sets with identical cardinalities we have

\begin{equation*}\lambda_{j_1}(A \cup\{3\}) < \lambda_{j_1}(A \cup\{\,j_2\}) < \lambda_{3}(A \cup\{\,j_2\}), \qquad \{\,j_1,j_2,3\} \cap A = \emptyset.\end{equation*}

The latter inequality means that in any system state, component 3 is more liable to fail than any other component. The former asserts that a component has a possibility of living longer when the still operating components are different from component 3. Note that, in view of the system structure and the homogeneity of four of the components, the number of parameters is reduced from 80 to 11.

Using the definition, we calculate the cumulative hazard rates:

\begin{align*}& \Lambda(\emptyset)= \frac{16}3 < \Lambda(\{3\})= \frac{20}3 < \Lambda(\{\,j_1\})= \frac{23}3 < \Lambda(\{3, j_1\})= 8 \\[3pt] & < \Lambda(\{\,j_1, j_2\})= \frac{28}3 < \Lambda(\{3,j_1, j_2\})= \frac{34}3 <\Lambda(\{\,j_1, j_2,j_3\})=\frac{37}3.\end{align*}

It follows that $\Lambda$ increases with the increase of cardinality of its argument, and moreover $\Lambda(A \cup \{3\}) < \Lambda(A \cup \{\,j\})$, $3\neq j \in A^c$. We determine the following probabilities for labels of consecutively failing components:

\begin{align*}\mathbb{P}(J_1=j_1,J_2=j_2) & = \frac{\lambda_{j_1}(\emptyset)}{\Lambda(\emptyset)} \frac{\lambda_{j_2}(\{\,j_1\})}{\Lambda(\{\,j_1\})}= \frac{9}{184}, \\[3pt]\mathbb{P}(J_1=3,J_2=j_2,J_3=j_3) & = \frac{\lambda_{3}(\emptyset)}{\Lambda(\emptyset)} \frac{\lambda_{j_2}(\{3\})}{\Lambda(\{3\})}\frac{\lambda_{j_3}(\{3,j_2\})}{\Lambda(\{3,j_2\})}= \frac{1}{48}, \\[3pt] \mathbb{P}(J_1=j_1,J_2=3,J_3=j_3) & = \frac{\lambda_{j_1}(\emptyset)}{\Lambda(\emptyset)} \frac{\lambda_{3}(\{\,j_1\})}{\Lambda(\{\,j_1\})}\frac{\lambda_{j_3}(\{\,j_1,3\})}{\Lambda(\{\,j_1,3\})}= \frac{7}{320}, \\[3pt] \mathbb{P}(J_1=j_1,J_2=j_2,J_3=3) & = \frac{\lambda_{j_1}(\emptyset)}{\Lambda(\emptyset)} \frac{\lambda_{j_2}(\{\,j_1\})}{\Lambda(\{\,j_1\})}\frac{\lambda_{3}(\{\,j_1,j_2\})}{\Lambda(\{\,j_1,j_2\})}= \frac{45}{2576}, \\[3pt] \mathbb{P}(J_1=j_1,J_2=j_2,J_3=j_3) & = \frac{\lambda_{j_1}(\emptyset)}{\Lambda(\emptyset)} \frac{\lambda_{j_2}(\{\,j_1\})}{\Lambda(\{\,j_1\})}\frac{\lambda_{j_3}(\{\,j_1,j_2\})}{\Lambda(\{\,j_1,j_2\})}= \frac{81}{6808}.\end{align*}

These can further be used to evaluate the system lifetime. For simplicity of calculation, we assume that the baseline model distribution is standard exponential, and apply (14). Analogous results can be established for the Lomax and reflected power distributions using (27) and (28), respectively. In the exponential case, we determine that

\begin{align*} \mathbb{E}T &= 4 \left( \frac 1{\Lambda(\emptyset)}+\frac 1{\Lambda(\{\,j_1\})}\right) \mathbb{P}(J_1=j_1,J_2=j_2) \\&\quad + 8 \left( \frac 1{\Lambda(\emptyset)} + \frac 1{\Lambda(\{3\})} + \frac 1{\Lambda(\{3,j_2\})}\right) \mathbb{P}(J_1=3,J_2= j_2,J_3=j_3)\\[3pt]&\quad + 8 \left( \frac 1{\Lambda(\emptyset)} + \frac 1{\Lambda(\{\,j_1\})} + \frac 1{\Lambda(\{\,j_1,3\})} \right)\mathbb{P}(J_1=j_1,J_2=3,J_3=j_3)\\[3pt] &\quad + 4 \left( \frac 1{\Lambda(\emptyset)} + \frac 1{\Lambda(\{\,j_1\})} + \frac 1{\Lambda(\{\,j_1,j_2\})}\right) \mathbb{P}(J_1=j_1,J_2=j_2,J_3=3)\\[3pt] &\quad + 16 \left( \frac 1{\Lambda(\emptyset)} + \frac 1{\Lambda(\{\,j_1\})} + \frac 1{\Lambda(\{\,j_1,j_2\})}\right) \mathbb{P}(J_1=j_1,J_2=j_2,J_3=j_3)\\[3pt] &\quad + 4 \left( \frac 1{\Lambda(\emptyset)} + \frac 1{\Lambda(\{3\})} + \frac 1{\Lambda(\{3,j_2\})}+ \frac 1{\Lambda(\{3,j_2,j_3\})}\right) \mathbb{P}(J_1=3,J_2=j_2,J_3=j_3)\\[3pt] &\quad + 4 \left( \frac 1{\Lambda(\emptyset)} + \frac 1{\Lambda(\{\,j_1\})} + \frac 1{\Lambda(\{\,j_1,3\})}+ \frac 1{\Lambda(\{\,j_1,3,j_3\})}\right) \mathbb{P}(J_1=j_1,J_2=3,J_3=j_3)\\[3pt] &\quad + 4 \left( \frac 1{\Lambda(\emptyset)} + \frac 1{\Lambda(\{\,j_1\})} + \frac 1{\Lambda(\{\,j_1,j_2\})}+ \frac 1{\Lambda(\{\,j_1,j_2,3\})}\right)\! \mathbb{P}(J_1=j_1,J_2=j_2,J_3=3).\end{align*}

The first term represents system failure caused by two component failures, which happens in the four cases when $I_2$ is equal to either $\{1,4\}$ or $\{2,5\}$, and the failure order is arbitrary. The last three expressions correspond to the case $T=T_{4:5}$ if $I_3=\{1,2,3\}$ or $I_3=\{3,4,5\}$, and component 3 is respectively the first, second, or third component to fail. The remaining summands are connected with T coinciding with the third failure. The consecutive terms express the contribution of $\mathbb{E}T$ in the cases that component 3 fails respectively first, second, third, or later. Performing numerical calculations yields $\mathbb{E}T \approx 0.45588$.

In order to check whether the proper location of components yields any benefit, we compare these results with the performance of identical system components with an exchangeable joint distribution. To make the comparison fair, we assume that the inter-failure cumulative hazard rates $\Lambda_i$, $i=0,1,2,3$, in the exchangeable case are identical with their expected counterparts in the non-exchangeable case; i.e., $\Lambda_0=\Lambda(\emptyset)$, and

\begin{equation*}\Lambda_i= \Lambda(A_{i-1} \cup \{3\})\mathbb{P}(I_i=A_{i-1} \cup \{3\}) + \Lambda(A_{i-1} \cup \{\,j\})[1-\mathbb{P}(I_i=A_{i-1} \cup \{3\})]\end{equation*}

for $j\neq 3$, $|A_{i-1}|=i-1$, $|I_i|=i$, and $i=1,2,3$. By elementary calculations we obtain $\Lambda_0= 5 \frac 13$, $\Lambda_1= 7\frac 5{12}$, $\Lambda_2=9 \frac 4{23}$, and $\Lambda_3= 12 \frac{14073}{51520}$. By the Samaniego formula, the corresponding expectation of the system lifetime,

\begin{equation*}\mathbb{E}\tilde{T}= \frac{1}{\Lambda_0} + \frac{1}{\Lambda_1} + \frac{4}{5\Lambda_2} + \frac{1}{5\Lambda_3} \approx 0.42581,\end{equation*}

is $6.528\%$ lower than $\mathbb{E}T$.