2. Ordered system signature for independent and non-identical systems

Based on a life test of several independent and identical coherent systems, [Reference Balakrishnan and Volterman5] introduced the concept of ordered system signature as follows.

Definition 2.1. (Ordered system signature.) For a life test of n independent and identical coherent systems, each of which has m i.i.d. components and a common system signature $\boldsymbol{s}$ , the ith $(i=1,\ldots,n)$ ordered system signature is defined as the vector ${\boldsymbol{s}^{(i:n)}} = \big(s_1^{(i:n)}, \ldots ,s_m^{(i:n)}\big)$ , where, for $j = 1, \ldots ,m$ ,

\[s_j^{(i:n)} = \sum\limits_{k = 1}^n {\mathbb{P}\Big\{ {T_{i:n}} = X_{j:m}^{(k)}\Big\} } = \sum\limits_{k = 1}^n {\mathbb{P}\Big\{ {T_{i:n}} = X_{j:m}^{(k)}\mid {{T_{i:n}} = {T_k}} \Big\} } \]

is the probability that the ith ordered system failure corresponds to a system that failed due to its jth ordered component failure. Here, for system k $(k=1,\ldots,n)$ , component lifetimes $X_1^{(k)}, \ldots ,X_m^{(k)}$ are assumed to have a common continuous distribution function F, and $X_{1:m}^{(k)}, \ldots ,X_{m:m}^{(k)}$ are the corresponding ordered (in ascending order) lifetimes; similarly, the system lifetimes ${T_1}, \ldots ,{T_n}$ are ordered (in ascending order) as ${T_{1:n}}, \ldots ,{T_{n:n}}$ .

Even though this definition was given in [Reference Balakrishnan and Volterman5] for the case of independent and identical coherent systems, it can easily be directly extended to the case of independent and non-identical coherent systems of the same size, and then subsequently to independent and non-identical coherent systems of different sizes by using the idea of equivalent systems and related transformation formulas developed, for example, in [Reference Navarro, Samaniego, Balakrishnan and Bhattacharya21]. The use of equivalent systems, of course, hides information about components in the systems, such as the expected number of failed components and possible maintenance policies in the original system. However, since equivalent systems of the same size share the same system lifetime distribution and component lifetime distribution, the generalization can be quite useful for describing system structures and in developing statistical inferences. Here, we first discuss some important properties of ordered system signature for the case of independent and non-identical coherent systems.

Let us consider a life test of n independent coherent systems with all their components i.i.d., and assume that they can be divided into N groups according to their equivalent systems of size m, namely, the $k_i$ systems in each group $i=1,\ldots,N$ have the same equivalent system with m components and a system signature ${\boldsymbol{s}^{(i)}} = \big(s_1^{(i)}, \ldots ,s_m^{(i)}\big)$ . Note that m is often chosen to be the largest number of components in the n independent coherent systems. Also, ${k_i} \in \{ 1, \ldots ,n\} $ for all $i = 1, \ldots ,N$ , with $\sum\nolimits_{i = 1}^N {{k_i}} = n$ , and ${\boldsymbol{s}^{(i)}}$ are different for different i $(i=1,\ldots,N)$ . The component lifetimes and the system lifetimes are defined and ordered exactly as explained in Definition 2.1. Then, along the lines of [Reference Balakrishnan and Volterman5], associated properties of their ordered system signatures can be presented. The first of these is the distribution-free property, as established in the following proposition.

Proof. In each group $i = 1, \ldots ,N$ , assume that there are ${l_{i,j}}$ $(j = 1, \ldots ,m)$ of the ${k_i}$ i.i.d. coherent systems that failed due to the jth ordered component failure. Evidently, all possible combinations of those ${l_{i,j}}$ can be given as

\[{\mathcal{L}_\boldsymbol{k}} = \{ \boldsymbol{l} = ({l_{i,j}},\,1 \le i \le N,\,1 \le j \le m) \,:\, {l_{i,1}} + \cdots + {l_{i,m}} = {k_i}{\rm{\,for\,all\, }}i\} .\]

Then, as in the discussions of [Reference Balakrishnan and Volterman5], $({l_{i,1}}, \ldots ,{l_{i,m}})$ are distributed as multinomial with parameters $\Big( {{k_i},s_1^{(i)}, \ldots ,s_m^{(i)}} \Big)$ , $i = 1, \ldots ,N$ , so that, for $i = 1, \ldots ,N$ and $j = 1, \ldots ,m$ we have

\[s_j^{(i:n)} = \sum\limits_{\boldsymbol{l} \in {\mathcal{L}_\boldsymbol{k}}} {p_{j\,|\, \boldsymbol{l}}^{(i:n)} \cdot \prod\limits_{w = 1}^N {\Bigg\{ {\Bigg( \begin{matrix} {{k_w}} \\ {{l_{w,1}}, \ldots ,{l_{w,m}}} \end{matrix}\Bigg) \prod\limits_{r = 1}^m {{{\big[ {s_r^{(w)}} \big]}^{{l_{w,r}}}}} } \Bigg\}}} ,\]

where $p_{j\,|\, \boldsymbol{l}}^{(i:n)}$ is the conditional probability that the ith system under test failed due to the jth ordered component failure, given a fixed value of $\boldsymbol{l}$ , that is, given that ${l_j} = {l_{1,j}} + \cdots + {l_{N,j}}$ $(j = 1, \ldots ,m)$ systems failed due to the jth ordered component failure. Note that $p_{j\,|\, \boldsymbol{l}}^{(i:n)}$ depends on $\boldsymbol{l}$ only through ${l_1}, \ldots ,{l_m}$ , which means that it can also be denoted as $p_{j\,|\, {{l_1}, \ldots ,{l_m}}}^{(i:n)}$ . Clearly, $p_{j\,|\, \boldsymbol{l}}^{(i:n)}$ can be expressed as probabilities of orderings of $X_{{j_k}:m}^{(k)}$ , which are independent of the component lifetime distribution F. Hence, the proposition.

According to Proposition 2.1, the ordered system signature ${\boldsymbol{s}^{(i:n)}}$ can be computed directly from system signatures ${\boldsymbol{s}^{(1)}}, \ldots ,{\boldsymbol{s}^{(N)}}$ . The required computation can be simplified by utilizing some properties of the conditional probabilities $p_{j\,|\, \boldsymbol{l}}^{(i:n)}$ presented in the following lemma.

Based on Proposition 2.1 and Lemma 2.1, we can simplify the computation of ordered system signature in the following manner.

Proof. For $s_j^{(1)} = \cdots = s_j^{(N)} = 0$ , the terms in the expression

\[s_j^{(i:n)} = \sum\limits_{\boldsymbol{l} \in {\mathcal{L}_\boldsymbol{k}}} {p_{j\,|\, \boldsymbol{l}}^{(i:n)} \cdot \prod\limits_{w = 1}^N {\Bigg\{ {\Bigg( \begin{matrix} {{k_w}} \\ {{l_{w,1}}, \ldots ,{l_{w,m}}} \end{matrix} \Bigg)\prod\limits_{r = 1}^m {{{\big[ {s_r^{(w)}} \big]}^{{l_{w,r}}}}} } \Bigg\}} } \]

can be classified into two classes: for $\boldsymbol{l} \in {\mathcal{L}_\boldsymbol{k}}$ such that ${l_{1,j}} = \cdots = {l_{N,j}} = 0$ , corresponding terms will all be 0 with $p_{j\,|\, \boldsymbol{l}}^{(i:n)} = 0$ according to Lemma 2.1(ii), and for $\boldsymbol{l} \in {\mathcal{L}_\boldsymbol{k}}$ such that $\sum\nolimits_{w = 1}^N {{l_{w,j}}} \ne 0$ , the corresponding terms will all be 0 with

$$\prod\limits_{w = 1}^N {{{\big[ {s_j^{(w)}} \big]}^{{l_{w,j}}}}} = {\big[ {s_j^{(1)}} \big]^{\sum\nolimits_{w = 1}^N {{l_{w,j}}} }} = 0.$$

Then, it is clear that we have $s_j^{(i:n)} = 0$ for any i, i.e. $s_j^{(1:n)} = \cdots = s_j^{(n:n)} = 0$ .

For $s_j^{(1:n)} = \cdots = s_j^{(n:n)} = 0$ , we have

\begin{align*} 0 = \sum\limits_{i = 1}^n {s_j^{(i:n)}} & = \sum\limits_{\boldsymbol{l} \in {\mathcal{L}_\boldsymbol{k}}} {\Bigg[ {\sum\limits_{i = 1}^n {p_{j\,|\, \boldsymbol{l}}^{(i:n)}} } \Bigg] \cdot \prod\limits_{w = 1}^N {\Bigg\{ {\Bigg( \begin{matrix} {{k_w}} \\ {{l_{w,1}}, \ldots ,{l_{w,m}}} \end{matrix} \Bigg)\prod\limits_{r = 1}^m {{{\big[ {s_r^{(w)}} \big]}^{{l_{w,r}}}}} } \Bigg\}} } \\ & = \sum\limits_{\boldsymbol{l} \in {\mathcal{L}_\boldsymbol{k}}} {\Bigg( {\sum\limits_{s = 1}^N {{l_{s,j}}} } \Bigg) \cdot \prod\limits_{w = 1}^N {\Bigg\{ {\Bigg( \begin{matrix} {{k_w}} \\ {{l_{w,1}}, \ldots ,{l_{w,m}}} \end{matrix} \Bigg)\prod\limits_{r = 1}^m {{{\big[ {s_r^{(w)}} \big]}^{{l_{w,r}}}}} } \Bigg\}} } \\ & = \sum\limits_{s = 1}^N \sum\limits_{{\boldsymbol{l}_s} \in {{\mathcal{\tilde L}}_s}} {\Bigg\{ {{l_{s,j}} \cdot \Bigg( \begin{matrix} {{k_s}} \\ {{l_{s,1}}, \ldots ,{l_{s,m}}} \end{matrix} \Bigg)\prod\limits_{r = 1}^m {{{\big[ {s_r^{(s)}} \big]}^{{l_{s,r}}}}} } \Bigg\}} = \sum\limits_{s = 1}^N {{k_s}s_j^{(s)}} , \end{align*}

since vectors ${\boldsymbol{l}_s} \in {\mathcal{\tilde L}_s} = \{ ({l_{s,1}}, \ldots , {l_{s,m}})\,:\,{l_{s,1}} + \cdots + {l_{s,m}} = {k_s}\} $ are distributed as multinomial with parameters $\big( {{k_s},s_1^{(s)}, \ldots ,s_m^{(s)}} \big)$ for $s = 1, \ldots ,N$ . Then, we clearly have $s_j^{(1)} = \cdots = s_j^{(N)}=0$ .

In addition to the above-stated properties, there are also some other interesting symmetry properties for ordered system signatures that could be used to further simplify the computational process.

Proposition 2.2. The ordered system signatures ${\boldsymbol{s}^{(1:n)}}, \ldots ,{\boldsymbol{s}^{(n:n)}}$ satisfy

\[\sum\limits_{i = 1}^n {{\boldsymbol{s}^{(i:n)}}} = \sum\limits_{s = 1}^N {{k_i}{\boldsymbol{s}^{(s)}}}, \qquad {\mathrm{rev}}\,{\boldsymbol{s}^{(i:n)}} = {(\mathrm{rev}\,\boldsymbol{s})^{(n - i + 1:n)}}, \qquad i = 1, \ldots ,n,\]

where $\boldsymbol{s} = \big({\boldsymbol{s}^{(1)}}, \ldots ,{\boldsymbol{s}^{(N)}}\big)$ and $\mathrm{rev}\,\boldsymbol{s} = ({\mathrm{rev}}\,{\boldsymbol{s}^{(1)}}, \ldots ,{\mathrm{rev}}\,{\boldsymbol{s}^{(N)}})$ , with ${\mathrm{rev}}\,{\boldsymbol{s}^{(s)}} = \big(s_m^{(s)}, \ldots,$ $s_1^{(s)}\big)$ $(s = 1, \ldots ,N)$ given by the reverse ordering of ${\boldsymbol{s}^{(s)}} = \big(s_1^{(s)}, \ldots ,s_m^{(s)}\big)$ .

Proof. From the proof of Corollary 2.1, we have

\[\sum\limits_{i = 1}^n {s_j^{(i:n)}} = \sum\limits_{s = 1}^N {{k_s}s_j^{(s)}} ,\qquad j = 1, \ldots ,m,\]

which clearly leads to $\sum\nolimits_{i = 1}^n {{\boldsymbol{s}^{(i:n)}}} = \sum\nolimits_{s = 1}^N {{k_s}{\boldsymbol{s}^{(s)}}} $ . This means that the summation of all ordered system signatures equals the summation of all system signatures of the original equivalent systems of size m.

From the formula of $s_j^{(i:n)}$ $(i = 1, \ldots ,n,\,j = 1, \ldots ,m)$ in Proposition 2.1, we have

\begin{align*} s_{m - j + 1}^{(n - i + 1:n)} & = \sum\limits_{\boldsymbol{l} \in {\mathcal{L}_\boldsymbol{k}}} {p_{m - j + 1\,|\, \boldsymbol{l}}^{(n - i + 1:n)} \cdot \prod\limits_{w = 1}^N {\Bigg\{ {\Bigg( \begin{matrix} {{k_w}} \\ {{l_{w,1}}, \ldots ,{l_{w,m}}} \end{matrix} \Bigg)\prod\limits_{r = 1}^m {{{\big[ {s_r^{(w)}} \big]}^{{l_{w,r}}}}} } \Bigg\}} } \\ & = \sum\limits_{\boldsymbol{l} \in {\mathcal{L}_\boldsymbol{k}}} {p_{j\,|\, {\mathrm{rev}\,\boldsymbol{l}}}^{(i:n)} \cdot \prod\limits_{w = 1}^N {\Bigg\{ {\Bigg( \begin{matrix} {{k_w}} \\ {{l_{w,1}}, \ldots ,{l_{w,m}}} \end{matrix} \Bigg)\prod\limits_{r = 1}^m {{{\big[ {s_r^{(w)}} \big]}^{{l_{w,r}}}}} } \Bigg\}} } \\ & = \sum\limits_{\boldsymbol{l} \in {\mathcal{L}_\boldsymbol{k}}} {p_{j\,|\, \boldsymbol{l}}^{(i:n)} \cdot \prod\limits_{w = 1}^N {\Bigg\{ {\Bigg( \begin{matrix} {{k_w}} \\ {{l_{w,1}}, \ldots ,{l_{w,m}}} \end{matrix} \Bigg)\prod\limits_{r = 1}^m {{{\big[ {s_{m - r + 1}^{(w)}} \big]}^{{l_{w,r}}}}} } \Bigg\}} } = ({\mathrm{rev}}\,s)_j^{(i:n)}, \end{align*}

which leads to ${\mathrm{rev}}\,{\boldsymbol{s}^{(i:n)}} = {({\mathrm{rev}}\,\boldsymbol{s})^{(n - i + 1:n)}}$ $(i = 1, \ldots ,n)$ .

The ordered system signatures also possess another important property: stochastic orderings between them, as presented in the following proposition.

Proof. For $1 \le {i_1} < {i_2} \le n$ , the property that ${\boldsymbol{s}^{({i_1}:n)}} \le _{{\rm{st}}} {\boldsymbol{s}^{({i_2}:n)}}$ can be proved similarly to [Reference Balakrishnan and Volterman5, Proposition 3], and the only change caused by the difference in system signatures ${\boldsymbol{s}^{(1)}}, \ldots ,{\boldsymbol{s}^{(N)}}$ is in the formula for probability $C(S,u,v,{j_U},{j_L})$ . For this reason, the corresponding proof is omitted here for brevity.

If ${\boldsymbol{s}^{({i_1}:n)}} \ge _{{\rm{st}}} {\boldsymbol{s}^{({i_2}:n)}}$ for any $1 \le {i_1} < {i_2} \le n$ , then clearly we have ${\boldsymbol{s}^{({i_1}:n)}} = {\boldsymbol{s}^{({i_2}:n)}}$ . Let ${\kappa _s}$ , $s = 1, \ldots ,N$ , be the smallest k such that $s_k^{(s)} > 0$ , and ${\kappa}$ be the smallest among the ${\kappa _s}$ , $s = 1, \ldots ,N$ . We then have

\begin{equation*} 0 = s_\kappa ^{({i_1}:n)} - s_\kappa ^{({i_2}:n)} = \sum\limits_{\boldsymbol{l} \in {\mathcal{L}_\boldsymbol{k}}} {\Big( {p_{\kappa \,|\, \boldsymbol{l}}^{({i_1}:n)} - p_{\kappa \,|\, \boldsymbol{l}}^{({i_2}:n)}} \Big) \cdot \prod\limits_{w = 1}^N {\Bigg\{ {\Bigg( \begin{matrix} {{k_w}} \\ {{l_{w,1}}, \ldots ,{l_{w,m}}} \end{matrix} \Bigg)\prod\limits_{r = 1}^m {{{\big[ {s_r^{(w)}} \big]}^{{l_{w,r}}}}} } \Bigg\}} } . \end{equation*}

To prove ${\kappa _1} = \cdots = {\kappa _N} = \kappa $ , we assume that there exists at least one ${\kappa _s}$ such that ${\kappa _s} > \kappa $ . Let $\boldsymbol{l}$ be such that ${l_{i,{\kappa _i}}} = {k_i}$ $(i = 1, \ldots ,N)$ . Then, by an argument similar to that in [Reference Balakrishnan and Volterman5], we have $p_{\kappa \,|\, \boldsymbol{l}}^{({i_1}:n)} - p_{\kappa \,|\, \boldsymbol{l}}^{({i_2}:n)} > 0$ , which implies an impossible result that $\prod\nolimits_{w = 1}^N {{{\big( {s_{{\kappa _w}}^{(w)}} \big)}^{{k_w}}}} = 0$ , i.e. $s_{{\kappa _1}}^{(1)} = \cdots = s_{{\kappa _N}}^{(N)} = 0$ . Thus, we conclude that ${\kappa _1} = \cdots = {\kappa _N} = \kappa$ . As discussed above, for any $s = 1, \ldots ,N$ and any ${\kappa} < {\kappa^{\prime}_{s}} \le m$ , let $\boldsymbol{l}$ be such that ${l_{s,\kappa^{\prime}_{s}}} = {k_s}$ and ${l_{i,\kappa }} = {k_i}$ $(i \ne s)$ , and we then have

$${\big[ {s^{(s)}_{\kappa^{\prime}_{s}}} \big]^{{k_s}}}\prod\limits_{w = 1}^N {{{\big( {s_\kappa ^{(w)}} \big)}^{{k_w}{{\textbf{1}}_{\{ w \ne s\} }}}}} = 0,$$

i.e. $ {s^{(s)}_{\kappa^{\prime}_{s}}} =0$ . This implies that ${\boldsymbol{s}^{(s)}} = (\underbrace {0, \ldots ,0}_{\kappa{-}1},1,\underbrace {0, \cdots ,0}_{m - \kappa })$ for all $s = 1, \ldots ,N$ .

3. Dynamic ordered system signature

As mentioned in Section 2, the concept of ordered system signature is applicable for independent and non-identical coherent systems of any sizes. In this section we introduce a new concept, called dynamic ordered system signature, which will be useful in examining dynamic properties of used coherent systems.

Consider a life test of n independent coherent systems with all their components being i.i.d., and assume that they can be divided into N groups such that the ${k_i}$ systems in each group, $i = 1, \ldots ,N$ , have the same system size ${m_i}$ and the same system signature ${\boldsymbol{s}^{(i)}} = \Big(s_1^{(i)}, \ldots ,s_{{m_i}}^{(i)}\Big)$ . Note that ${k_i} \in \{ 1, \ldots ,n\} $ for all $i = 1, \ldots ,N$ , with $\sum\nolimits_{i = 1}^N {{k_i}} = n$ , and the ${\boldsymbol{s}^{(i)}}$ $(i = 1, \ldots ,N)$ are different for different i. The component lifetimes and the system lifetimes are defined and ordered exactly as explained in Definition 2.1. Let us further use ${E_\boldsymbol{k}}(t)$ , $\boldsymbol{k} = ({k_{i,j}},\,1 \le i \le N,\,1 \le j \le {m_i})$ , to denote the event that there are ${k_{i,j}}$ $(i = 1, \ldots ,N,\, j = 1, \ldots ,{m_i})$ working systems in group i with exactly j working components at time t, i.e. there are exactly ${k_0} = n - \sum\nolimits_{i = 1}^N {\sum\nolimits_{j = 1}^{{m_i}} {{k_{i,j}}} } $ failed systems at time t with $0 \le \sum\nolimits_{j = 1}^{{m_i}} {{k_{i,j}}} \le {k_i}$ for $i = 1, \ldots ,N$ . Let $\tilde n = \sum\nolimits_{i = 1}^N {\sum\nolimits_{j = 1}^{{m_i}} {{k_{i,j}}} } $ be the total number of working systems at time t, $m = \sup \{ j\,:\,{k_{i,j}} \ne 0,\,i = 1, \ldots ,N,\,j = 1, \ldots ,{m_i}\} $ denote the largest number of working components in each of the $\tilde n$ working systems at time t, and ${\tilde m_i} = \min \{ {m_i},m\} $ denote the smaller number between ${m_i}$ and m. Then, the concept of dynamic ordered system signature can be introduced as follows.

Definition 3.1. (Dynamic ordered system signature.) The ith $(i = 1, \ldots ,\tilde n)$ dynamic ordered system signature is given by ${\boldsymbol{s}^{(i\,|\, \boldsymbol{k})}} = \big(s_1^{(i\,|\, \boldsymbol{k})}, \ldots ,s_m^{(i\,|\, \boldsymbol{k})}\big)$ , where, for $j = 1, \ldots ,m$ ,

\[s_j^{(i\,|\, \boldsymbol{k})} = \sum\limits_{k = 1}^n \mathbb{P}\Big\{{T_{i + {k_0}:n}} = \tilde X_{j:m}^{(k)}\mid {{E_\boldsymbol{k}}(t)}\Big\} = \sum\limits_{k = 1}^n \mathbb{P}\Big\{ {T_{i + {k_0}:n}} = \tilde X_{j:m}^{(k)}\mid {{E_\boldsymbol{k}}(t),{T_{i + {k_0}:n}} = {T_k}}\Big\} \]

is the conditional probability that the failure of the $(i + {k_0})$ th system corresponds to the failure of a system whose equivalent system (of size m) at time t fails due to the jth ordered component failure, given that there are ${k_{i,j}}$ $(i = 1, \ldots ,N,\,j = 1, \ldots ,{m_i})$ working systems in group i with exactly j working components at time t. Here, for system k $(k = 1, \ldots ,n)$ , the component lifetimes $\tilde X_1^{(k)}, \ldots ,\tilde X_m^{(k)}$ of its equivalent system (of size m) at time t also have a common continuous distribution function F and are ordered (in ascending order) as $\tilde X_{1:m}^{(k)}, \ldots ,\tilde X_{m:m}^{(k)}$ , and similarly the (equivalent) system lifetimes ${T_1}, \ldots ,{T_n}$ are ordered as ${T_{1:n}}, \ldots ,{T_{n:n}}$ .

Then, along the lines of Section 2, some properties of dynamic ordered system signature can be presented as follows.

Proof. Given event ${E_\boldsymbol{k}}(t)$ , there are $\tilde n = \sum\nolimits_{i = 1}^N {\sum\nolimits_{j = 1}^{{m_i}} {{k_{i,j}}} } = \sum\nolimits_{i = 1}^N {\sum\nolimits_{j = 1}^{{{\tilde m}_i}} {{k_{i,j}}} } $ working systems in the life test at time t. For the ${k_{i,j}}$ $(i = 1, \ldots ,N,\,j = 1, \ldots ,{\tilde m_i})$ working systems in group i with exactly j working components and ${m_i} - j$ failed components, according to [Reference Samaniego, Balakrishnan and Navarro24], their dynamic signature can be given from the system signature ${\boldsymbol{s}^{(i)}} = \Big(s_1^{(i)}, \ldots ,s_{{m_i}}^{(i)}\Big)$ as

\[{\boldsymbol{s}^{(i,j)}} = \frac{1}{{\sum\nolimits_{w = {m_i} - j + 1}^{{m_i}} {s_w^{(i)}} }} \cdot \Big(s_{{m_i} - j + 1}^{(i)}, \ldots ,s_{{m_i}}^{(i)}\Big),\]

with the corresponding component lifetime distribution $\tilde F$ being the left-truncated form of F given by $\tilde F(x\mid t) = [F(x) - F(t)]/[1 - F(t)]$ , $x > t$ . According to [Reference Balakrishnan, Beutner and Cramer2], the signature of their equivalent system (of size m) at time t is given by , where, for $r = 1, \ldots ,m$ ,

\[\tilde s_r^{(i,j)} = \sum\limits_{l = \max \{ 1,j + r - m\} }^{\min \{ j,r\} } {\Bigg[ {{\frac{s_{{m_i} - j + l}^{(i)}}{\sum\nolimits_{w = {m_i} - j + 1}^{{m_i}} {s_w^{(i)}} }} \cdot {{\Bigg( \begin{matrix} m \\ j \end{matrix} \Bigg)}^{ - 1}}\Bigg( \begin{matrix} {r - 1} \\ {l - 1} \end{matrix} \Bigg)\Bigg( \begin{matrix} {m - r} \\ {j - l} \end{matrix} \Bigg)} \Bigg]} .\]

Then, the distribution-free property of ${\boldsymbol{s}^{(i\,|\, \boldsymbol{k})}} = \Big(s_1^{(i\,|\, \boldsymbol{k})}, \ldots ,s_m^{(i\,|\, \boldsymbol{k})}\Big)$ follows directly as $s_j^{(i\,|\, \boldsymbol{k})}$ $(j = 1, \ldots ,m)$ can be expressed as

\[s_j^{(i\,|\, \boldsymbol{k})}{\rm{ = }}\sum\limits_{\boldsymbol{l} \in {\mathcal{L}_\boldsymbol{k}}} {p_{j\,|\, \boldsymbol{l}}^{(i\,|\, \boldsymbol{k})} \cdot \prod\limits_{u = 1}^N {\prod\limits_{v = 1}^{{{\tilde m}_u}} {\Bigg\{ {\Bigg( \begin{matrix} {{k_{u,v}}} \\ {{l_{u,v,1}}, \ldots ,{l_{u,v,m}}} \end{matrix} \Bigg)\prod\limits_{r = 1}^m {{{\big[ {\tilde s_r^{(u,v)}} \big]}^{{l_{u,v,r}}}}} } \Bigg\}} } } ,\]

with $p_{j\,|\, \boldsymbol{l}}^{(i\,|\, \boldsymbol{k})}$ , $\boldsymbol{l} \in {\mathcal{L}_\boldsymbol{k}}$ , as defined in Proposition 2.1,

\begin{align*} {\mathcal{L}_\boldsymbol{k}} = \{ \boldsymbol{l} = ({l_{u,v,r}},\,1 \le u \le N,\,1 \le v \le {\tilde m_u}, \,1 \le r \le m)\,:\, {l_{u,v,1}} + \cdots + {l_{u,v,m}} = {k_{u,v}}{\rm{\ for\ all\ }}u,v\} , \end{align*}

and $p_{j\,|\, \boldsymbol{l}}^{(i\,|\, \boldsymbol{k})}$ is the conditional probability that the failure of the $(i + {k_0})$ th system corresponds to the failure of a system whose equivalent system (of size m) at time t is due to the jth ordered component failure, given a fixed value of $\boldsymbol{l}$ , i.e. given that ${l_j} = \sum\nolimits_{u = 1}^N {\sum\nolimits_{v = 1}^{{{\tilde m}_u}} {{l_{u,v,j}}} } $ $(j = 1, \ldots ,m)$ systems failed due to the jth ordered component failure. Note that $p_{j\,|\, \boldsymbol{l}}^{(i\,|\, \boldsymbol{k})}$ depends on $\boldsymbol{l}$ only through ${l_1}, \ldots ,{l_m}$ and is independent of $\boldsymbol{k}$ , which implies that it can also be denoted as $p_{j\,|\, {{l_1}, \ldots ,{l_m}}}^{(i:n)}$ . Clearly, similar to Proposition 2.1, $p_{j\,|\, \boldsymbol{l}}^{(i\,|\, \boldsymbol{k})}$ is independent of the component lifetime distribution F. Hence, the proposition.

As in Corollary 2.2, the elements in the dynamic ordered system signatures ${\boldsymbol{s}^{(1\,|\, \boldsymbol{k})}}, $ $\ldots ,{\boldsymbol{s}^{(\tilde n\,|\, \boldsymbol{k})}}$ are zero if and only if the corresponding elements in the system signatures ${\boldsymbol{s}^{(1)}}, \ldots ,{\boldsymbol{s}^{(N)}}$ are zero.

Similarly, as in Proposition 2.3, the dynamic ordered system signatures ${\boldsymbol{s}^{(1\,|\, \boldsymbol{k})}}, \ldots$ , ${\boldsymbol{s}^{(\tilde n\,|\, \boldsymbol{k})}}$ are also stochastically ordered, and any two of them are the same only when the corresponding elements in the system signatures ${\boldsymbol{s}^{(1)}}, \ldots ,{\boldsymbol{s}^{(N)}}$ are zero.

4. Some illustrative examples

In this section, for a clear understanding of the theoretical results established in the preceding sections, we discuss the computation of the dynamic ordered system signatures for two independent coherent systems (System 1 and System 2) with three components, when they are both series-parallel systems with a common signature $\boldsymbol{s} = \big( {\frac{1}{3},\frac{2}{3},0} \big)$ , parallel-series systems with a common signature $\boldsymbol{s} = \big( {0,\frac{2}{3},\frac{1}{3}} \big)$ , and parallel systems with a common signature $\boldsymbol{s} = (0, 0, 1)$ ; and when they are a series-parallel system and a parallel-series system, a series-parallel system and a parallel system, and a parallel-series system and a parallel system. Assume that both systems are working at time t. Then, for example, we shall consider the number of failed components in each system to be as follows:

1. Case 1: There is no failed component in any of the two systems at time t, i.e. ${k_{11}} = 0$ , ${k_{12}} = 0$ , ${k_{13}} = 1$ and ${k_{21}} = 0$ , ${k_{22}} = 0$ , ${k_{23}} = 1$ , for which case the pertinent calculations are presented in Example 4.1.

2. Case 2: There is no failed component in System 1 but one failed component in System 2, i.e. ${k_{11}} = 0$ , ${k_{12}} = 0$ , ${k_{13}} = 1$ and ${k_{21}} = 0$ , ${k_{22}} = 1$ , ${k_{23}} = 0$ , for which case the pertinent calculations are presented in Example 4.2.

3. Case 3: There is no failed component in System 1 but two failed components in System 2, i.e. ${k_{11}} = 0$ , ${k_{12}} = 0$ , ${k_{13}} = 1$ and ${k_{21}} = 1$ , ${k_{22}} = 0$ , ${k_{23}} = 0$ , for which case the pertinent calculations are presented in Example 4.3.

4. Case 4: There is one failed component in each of the two systems, i.e. ${k_{11}} = 0$ , ${k_{12}} = 1$ , ${k_{13}} = 0$ and ${k_{21}} = 0$ , ${k_{22}} = 1$ , ${k_{23}} = 0$ , for which case the pertinent calculations are presented in Example 4.4.

5. Case 5: There is one failed component in System 1 but two failed components in System 2, i.e. ${k_{11}} = 0$ , ${k_{12}} = 1$ , ${k_{13}} = 0$ and ${k_{21}} = 1$ , ${k_{22}} = 0$ , ${k_{23}} = 0$ , for which case the pertinent calculations are presented in Example 4.5.

Before presenting these five examples, we need to provide some basic discussions first. For System i with signature ${\boldsymbol{s}^{(i)}} = \big(s_1^{(i)},s_2^{(i)},s_3^{(i)}\big)$ $(i = 1,2)$ , according to [Reference Samaniego, Balakrishnan and Navarro24], its dynamic signature with two failed components and one failed component are given by ${\boldsymbol{s}^{(i,1)}} = 1$ if $s_3^{(i)} \ne 0$ and

$${\boldsymbol{s}^{(i,2)}} = \Bigg( {{\frac{s_2^{(i)}}{s_2^{(i)} + s_3^{(i)}}},{\frac{s_3^{(i)}}{s_2^{(i)} + s_3^{(i)}}}} \Bigg)$$

if $s_1^{(i)} \ne 1$ , respectively. Also, from [Reference Navarro, Samaniego, Balakrishnan and Bhattacharya21], system signatures of their equivalent systems with three components are given by

and

respectively. Moreover, as presented in [Reference Balakrishnan and Volterman5], Systems 1 and 2 have their ordered system signatures as ${\boldsymbol{s}^{(1:2)}} = \big(s_1^{(1:2)},s_2^{(1:2)},s_3^{(1:2)}\big)$ and ${\boldsymbol{s}^{(2:2)}} = \big(s_1^{(2:2)},s_2^{(2:2)},s_3^{(2:2)}\big)$ , where

\begin{align*} s_1^{(1:2)} & = s_1^{(1)}s_1^{(2)} + \frac{4}{5}\Big[s_1^{(1)}s_2^{(2)} + s_2^{(1)}s_1^{(2)}\Big] + {\frac{19}{20}}\Big[s_1^{(1)}s_3^{(2)} + s_3^{(1)}s_1^{(2)}\Big], \\ s_2^{(1:2)} & = \frac{1}{5}\Big[s_1^{(1)}s_2^{(2)} + s_2^{(1)}s_1^{(2)}\Big] + s_2^{(1)}s_2^{(2)} + \frac{4}{5}\Big[s_2^{(1)}s_3^{(2)} + s_3^{(1)}s_2^{(2)}\Big], \\ s_3^{(1:2)} & = \frac{1}{20}\Big[s_1^{(1)}s_3^{(2)} + s_3^{(1)}s_1^{(2)}\Big] + \frac{1}{5}\Big[s_2^{(1)}s_3^{(2)} + s_3^{(1)}s_2^{(2)}\Big] + s_3^{(1)}s_3^{(2)}, \\ s_1^{(2:2)} & = s_1^{(1)}s_1^{(2)} + \frac{1}{5}\Big[s_1^{(1)}s_2^{(2)} + s_2^{(1)}s_1^{(2)}\Big] + \frac{1}{20}\Big[s_1^{(1)}s_3^{(2)} + s_3^{(1)}s_1^{(2)}\Big], \\ s_2^{(2:2)} & = \frac{4}{5}\Big[s_1^{(1)}s_2^{(2)} + s_2^{(1)}s_1^{(2)}\Big] + s_2^{(1)}s_2^{(2)} + \frac{1}{5}\Big[s_2^{(1)}s_3^{(2)} + s_3^{(1)}s_2^{(2)}\Big], \\ s_3^{(2:2)} & = {\frac{19}{20}}\Big[s_1^{(1)}s_3^{(2)} + s_3^{(1)}s_1^{(2)}\Big] + \frac{4}{5}\Big[s_2^{(1)}s_3^{(2)} + s_3^{(1)}s_2^{(2)}\Big] + s_3^{(1)}s_3^{(2)}. \end{align*}

For the computation of dynamic ordered system signatures, the probability that an order statistic ${X_{i:m}}$ is less than another order statistic ${\tilde X_{j:m}}$ when they arise from two independent groups of m i.i.d. component lifetimes with the same distribution F can be given, for $i = 1, \ldots ,m$ and $j = 1, \ldots ,m$ , with $m = 2,3$ , as follows:

\begin{align*} \mathbb{P}\{ {X_{i:m}} < {{\tilde X}_{j:m}}\} & = \int\limits_0^\infty {j\Bigg( \begin{matrix} m \\ j \end{matrix} \Bigg){F^{j - 1}}(x){{\bar F}^{m - j}}(x)\mathbb{P}\{ {X_{i:m}} < x\} \, {\mathrm{d}} F(x)} \\ & = \int\limits_0^\infty {j\Bigg( \begin{matrix} m \\ j \end{matrix} \Bigg){F^{j - 1}}(x){{\bar F}^{m - j}}(x)\sum\limits_{k = i}^m {\Bigg( \begin{matrix} m \\ k \end{matrix} \Bigg){F^k}(x){{\bar F}^{m - k}}(x)} \, {\mathrm{d}} F(x)} \\[-5pt] & = \sum\limits_{k = i}^m {j\Bigg( \begin{matrix} m \\ j \end{matrix} \Bigg)\Bigg( \begin{matrix} m \\ k \end{matrix} \Bigg)\int\limits_0^1 {{u^{k + j - 1}}{{(1 - u)}^{2m - k - j}}\,{\mathrm{d}} u} } \\[-5pt] & = {\sum\limits_{k = i}^m {\frac{j}{k + j}\Bigg( \begin{matrix} m \\ j \end{matrix} \Bigg)\Bigg( \begin{matrix} m \\ k \end{matrix} \Bigg)\Bigg( \begin{matrix} {2m} \\ {k + j} \end{matrix} \Bigg)} ^{ - 1}}, \end{align*}

which is clearly free of F, i.e. the probability remains the same if F is replaced by $\tilde F$ . (See [Reference Balakrishnan, Beutner and Cramer2, Lemma 1] for an alternate proof.) Table 1 presents values of these probabilities (with NA meaning ‘not applicable’).

Table 1. Probabilities $\mathbb{P}\{ {X_{i:m}} < {\tilde X_{j:m}}\} $ for $i = 1, \ldots ,m$ , $j = 1, \ldots ,m$ , $m = 2,3$ .

Table 2. Values of $p_{j\,|\, {{l_1},{l_2},{l_3}}}^{(i:2)}$ for different i, j and ${l_1},{l_2},{l_3}$ .

Then, according to the discussions in Proposition 3.1, for $m=3$ (Cases 1–3), we have

\begin{align*} p_{1\,|\, {(2,0,0)}}^{(1:2)} & = p_{1\,|\, {(2,0,0)}}^{(2:2)} = p_{2\,|\, {(0,2,0)}}^{(1:2)} = p_{2\,|\, {(0,2,0)}}^{(2:2)} = p_{3\,|\, {(0,0,2)}}^{(1:2)} = p_{3\,|\, {(0,0,2)}}^{(2:2)} = 1, \\ p_{1\,|\, {(1,1,0)}}^{(1:2)} & = p_{2\,|\, {(1,1,0)}}^{(2:2)} = {\mathbb P}\{ {X_{1:3}} < {{\tilde X}_{2:3}}\} = \tfrac{4}{5}, \\ p_{1\,|\, {(1,1,0)}}^{(2:2)} & = p_{2\,|\, {(1,1,0)}}^{(1:2)} = {\mathbb P}\{ {X_{1:3}} > {{\tilde X}_{2:3}}\} = \tfrac{1}{5}, \\ p_{1\,|\, {(1,0,1)}}^{(1:2)} & = p_{3\,|\, {(1,0,1)}}^{(2:2)} = {\mathbb P}\{ {X_{1:3}} < {{\tilde X}_{3:3}}\} = \tfrac{19}{20}, \\ p_{1\,|\, {(1,0,1)}}^{(2:2)} & = p_{3\,|\, {(1,0,1)}}^{(1:2)} = {\mathbb P}\{ {X_{1:3}} > {{\tilde X}_{3:3}}\} = \tfrac{1}{20}, \\ p_{2\,|\, {(0,1,1)}}^{(1:2)} & = p_{3\,|\, {(0,1,1)}}^{(2:2)} = {\mathbb P}\{ {X_{2:3}} < {{\tilde X}_{3:3}}\} = \tfrac{4}{5}, \\ p_{2\,|\, {(0,1,1)}}^{(2:2)} & = p_{3\,|\, {(0,1,1)}}^{(1:2)} = {\mathbb P}\{ {X_{2:3}} > {{\tilde X}_{3:3}}\} = \tfrac{1}{5}, \end{align*}

and $p_{j\,|\, {{l_1},{l_2},{l_3}}}^{(i:2)} = 0$ for other i, j, ${l_1}$ , ${l_2}$ , ${l_3}$ . With these, we find the values of $p_{j\,|\, {{l_1},{l_2},{l_3}}}^{(i:2)}$ as presented in Table 2.

Similarly, for $m = 2$ (Cases 4 and 5), we have

\begin{align*} p_{1\,|\, {(2,0)}}^{(1:2)} & = p_{1\,|\, {(2,0)}}^{(2:2)} = p_{2\,|\, {(0,2)}}^{(1:2)} = p_{2\,|\, {(0,2)}}^{(2:2)} = 1, \\ p_{1\,|\, {(1,1)}}^{(1:2)} & = p_{2\,|\, {(1,1)}}^{(2:2)} = {\mathbb P}\{ {X_{1:2}} < {{\tilde X}_{2:2}}\} = \tfrac{5}{6}, \\ p_{1\,|\, {(1,1)}}^{(2:2)} & = p_{2\,|\, {(1,1)}}^{(1:2)} = {\mathbb P}\{ {X_{1:2}} > {{\tilde X}_{2:2}}\} = \tfrac{1}{6},\end{align*}

and $p_{j\,|\, {{l_1},{l_2}}}^{(i:2)} = 0$ for other i, j, ${l_1}$ , ${l_2}$ . With these, we find the values of $p_{j\,|\, {{l_1},{l_2}}}^{(i:2)}$ as presented in Table 3.

Table 3. Values of $p_{j\,|\, {{l_1},{l_2}}}^{(i:2)}$ for different i, j and ${l_1},{l_2}$ .

We now proceed to the computation of dynamic order system signatures for the five cases listed at the start of this section.

Example 4.1. To present the dynamic ordered system signatures of the two coherent systems at time t in Case 1, values of $p_{j\,|\, \boldsymbol{l}}^{(i\,|\, {0,0,1;0,0,1})}$ $(i = 1,2,\,j = 1,2,3,\, \boldsymbol{l} \in {\mathcal{L}_{0,0,1;0,0,1}})$ need to be presented first by

$$p_{j\,|\, \boldsymbol{l}}^{(i\,|\, {0,0,1;\,0,0,1})} = p_{j\,|\, {{l_1},{l_2},{l_3}}}^{(i:2)} = p_{j\,|\, {{l_{1,3,1}} + {l_{2,3,1}},{l_{1,3,2}} + {l_{2,3,2}},{l_{1,3,3}} + {l_{2,3,3}}}}^{(i:2)},$$

with the values of $p_{j\,|\, {{l_1},{l_2},{l_3}}}^{(i:2)}$ in Table 2 and

\begin{align*} \mathcal{L} _{0,0,1;\,0,0,1} & = \{ \boldsymbol{l} = ({l_{1,3,1}},{l_{1,3,2}},{l_{1,3,3}};\,{l_{2,3,1}},{l_{2,3,2}},{l_{2,3,3}})\,:\, \\ & \qquad {l_{1,3,1}} + {l_{1,3,2}} + {l_{1,3,3}} = 1, {l_{2,3,1}} + {l_{2,3,2}} + {l_{2,3,3}} = 1\} \\ & = \{(1,0,0;\,1,0,0),(1,0,0;\,0,1,0),(1,0,0;\,0,0,1),(0,1,0;\,1,0,0), \\ & \qquad\!\! (0,1,0;\,0,1,0), (0,1,0;\,0,0,1),(0,0,1;\,1,0,0),(0,0,1;\,0,1,0),(0,0,1;\,0,0,1)\} . \end{align*}

Then, for $\boldsymbol{k} = (0,0,1;\,0,0,1)$ , the dynamic ordered system signatures ${\boldsymbol{s}^{(1\,|\, \boldsymbol{k})}} = \Big(s_1^{(1\,|\, \boldsymbol{k})}$ , $s_2^{(1\,|\, \boldsymbol{k})},s_3^{(1\,|\, \boldsymbol{k})}\Big)$ and ${\boldsymbol{s}^{(2\,|\, \boldsymbol{k})}} = \Big(s_1^{(2\,|\, \boldsymbol{k})},s_2^{(2\,|\, \boldsymbol{k})},s_3^{(2\,|\, \boldsymbol{k})}\Big)$ are given by

that is,

From these expressions, we have, for different ${\boldsymbol{s}^{(1)}}$ and ${\boldsymbol{s}^{(2)}}$ , the dynamic ordered system signatures in Case 1, as presented in Table 4. From the results in Table 4, theoretical results like Propositions 3.2 and 3.3 can be readily verified.

Table 4. Dynamic ordered system signatures ${\boldsymbol{s}^{(1\,|\, \boldsymbol{k})}}$ , ${\boldsymbol{s}^{(2\,|\, \boldsymbol{k})}}$ in Cases 1–5 for different ${\boldsymbol{s}^{(1)}}$ , ${\boldsymbol{s}^{(2)}}$ .

Example 4.2. To present the dynamic ordered system signatures of the two coherent systems at time t in Case 2, values of $p_{j\,|\, \boldsymbol{l}}^{(i\,|\, 0,0,1;0,1,0)}$ $(i = 1,2,\,j = 1,2,3,\,\boldsymbol{l} \in {\mathcal{L}_{0,0,1;0,1,0}})$ need to be presented first by

$$p_{j\,|\, \boldsymbol{l}}^{(i\,|\, {0,0,1;0,1,0})} = p_{j\,|\, {{l_1},{l_2},{l_3}}}^{(i:2)} = p_{j\,|\, {{l_{1,3,1}} + {l_{2,2,1}},{l_{1,3,2}} + {l_{2,2,2}},{l_{1,3,3}} + {l_{2,2,3}}}}^{(i:2)},$$

with the values of $p_{j\,|\, {{l_1},{l_2},{l_3}}}^{(i:2)}$ in Table 2 and

\begin{align*} {\mathcal{L}_{0,0,1;\,0,1,0}} & = \{ \boldsymbol{l} = ({l_{1,3,1}},{l_{1,3,2}},{l_{1,3,3}};\,{l_{2,2,1}},{l_{2,2,2}},{l_{2,2,3}})\,:\, \\ & \qquad {l_{1,3,1}} + {l_{1,3,2}} + {l_{1,3,3}} = 1,\, {l_{2,2,1}} + {l_{2,2,2}} + {l_{2,2,3}} = 1\} \\ & = \{ (1,0,0;\,1,0,0),(1,0,0;\,0,1,0),(1,0,0;\,0,0,1),(0,1,0;\,1,0,0), \\ & \qquad\!\! (0,1,0;\,0,1,0),(0,1,0;\,0,0,1),(0,0,1;\,1,0,0),(0,0,1;\,0,1,0),(0,0,1;\,0,0,1)\} . \end{align*}

Then, the dynamic system ordered system signatures ${\boldsymbol{s}^{(1\,|\, {0,0,1;0,1,0})}}$ , ${\boldsymbol{s}^{(2\,|\, {0,0,1;0,1,0})}}$ can be given by replacing the signature ${\boldsymbol{s}^{(2)}} = \Big(s_1^{(2)},s_2^{(2)},s_3^{(2)}\Big)$ of System 2 in Example 4.1 by

namely,

\begin{align*} {\boldsymbol{s}^{(1\,|\, {0,0,1};0,1,0)}} & = \Big(s_1^{(1\,|\, {0,0,1};0,1,0)},s_2^{(1\,|\, {0,0,1};0,1,0)},s_3^{(1\,|\, {0,0,1};0,1,0)}\Big) \\ & = {\frac{1}{30\Big[s_2^{(2)} + s_3^{(2)}\Big]}} \cdot \Big( {28s_1^{(1)}s_2^{(2)} + 27s_1^{(1)}s_3^{(2)} + 16s_2^{(1)}s_2^{(2)} + 19s_3^{(1)}s_2^{(2)},} \\ & \qquad \quad 14s_2^{(1)}s_2^{(2)} + 26s_2^{(1)}s_3^{(2)} + 2s_1^{(1)}s_2^{(2)} + 2s_1^{(1)}s_3^{(2)} + 8s_3^{(1)}s_2^{(2)} + 8s_3^{(1)}s_3^{(2)}, \\ & \qquad \quad {22s_3^{(1)}s_3^{(2)} + s_1^{(1)}s_3^{(2)} + 3s_3^{(1)}s_2^{(2)} + 4s_2^{(1)}s_3^{(2)}} \Big), \\ {\boldsymbol{s}^{(2\,|\, {0,0,1};0,1,0)}} & = \Big(s_1^{(2\,|\, {0,0,1};0,1,0)},s_2^{(2\,|\, {0,0,1};0,1,0)},s_3^{(2\,|\, {0,0,1};0,1,0)}\Big) \\ & = \frac{1}{{30\Big[s_2^{(2)} + s_3^{(2)}\Big]}} \cdot \Big( {22s_1^{(1)}s_2^{(2)} + 3s_1^{(1)}s_3^{(2)} + 4s_2^{(1)}s_2^{(2)} + s_3^{(1)}s_2^{(2)},} \\ & \qquad \quad 26s_2^{(1)}s_2^{(2)} + 14s_2^{(1)}s_3^{(2)} + 8s_1^{(1)}s_2^{(2)} + 8s_1^{(1)}s_3^{(2)} + 2s_3^{(1)}s_2^{(2)} + 2s_3^{(1)}s_3^{(2)}, \\ & \qquad \quad {28s_3^{(1)}s_3^{(2)} + 19s_1^{(1)}s_3^{(2)} + 27s_3^{(1)}s_2^{(2)} + 16s_2^{(1)}s_3^{(2)}} \Big). \end{align*}

From these expressions, we have, for different ${\boldsymbol{s}^{(1)}}$ and ${\boldsymbol{s}^{(2)}}$ , the dynamic ordered system signatures in Case 2 as presented in Table 4.

Example 4.3. To present the dynamic ordered system signatures of the two coherent systems at time t in Case 3, values of $p_{j\,|\, \boldsymbol{l}}^{(i\,|\, {0,0,1;1,0,0})}$ $(i = 1,2,\,j = 1,2,3,\,\boldsymbol{l} \in {\mathcal{L}_{0,0,1;1,0,0}})$ need to be presented first by

$$p_{j\,|\, \boldsymbol{l}}^{(i\,|\, {0,0,1;\,1,0,0})} = p_{j\,|\, {{l_1},{l_2},{l_3}}}^{(i:2)} = p_{j\,|\, {{l_{1,3,1}} + {l_{2,1,1}},{l_{1,3,2}} + {l_{2,1,2}},{l_{1,3,3}} + {l_{2,1,3}}}}^{(i:2)},$$

with the values of $p_{j\,|\, {{l_1},{l_2},{l_3}}}^{(i:2)}$ in Table 2 and

\begin{align*} {\mathcal{L}_{0,0,1;\,1,0,0}} & = \{ \boldsymbol{l} = ({l_{1,3,1}},{l_{1,3,2}},{l_{1,3,3}};\,{l_{2,1,1}},{l_{2,1,2}},{l_{2,1,3}})\,:\, \\ & \qquad {l_{1,3,1}} + {l_{1,3,2}} + {l_{1,3,3}} = 1, {l_{2,1,1}} + {l_{2,1,2}} + {l_{2,1,3}} = 1\} \\ & = \{ (1,0,0;\,1,0,0),(1,0,0;\,0,1,0),(1,0,0;\,0,0,1),(0,1,0;\,1,0,0), \\ & \qquad\!\! (0,1,0;\, 0,1,0), (0,1,0;\,0,0,1),(0,0,1;\,1,0,0),(0,0,1;\,0,1,0),(0,0,1;\,0,0,1)\} . \end{align*}

Then, the dynamic ordered system signatures ${\boldsymbol{s}^{(1\,|\, {0,0,1;1,0,0})}}$ , ${\boldsymbol{s}^{(2\,|\, {0,0,1;1,0,0})}}$ can be given by replacing the signature ${\boldsymbol{s}^{(2)}} = \big(s_1^{(2)},s_2^{(2)},s_3^{(2)}\big)$ of System 2 in Example 4.1 by :

\begin{align*} {\boldsymbol{s}^{(1\,|\, {0,0,1;1,0,0})}} & = \Big(s_1^{(1\,|\, {0,0,1;1,0,0})},s_2^{(1\,|\, {0,0,1;1,0,0})},s_3^{(1\,|\, {0,0,1;1,0,0})}\Big) \\ & = \frac{1}{60} \cdot \Big( {55s_1^{(1)} + 16s_2^{(1)} + 19s_3^{(1)},4s_1^{(1)} + 40s_2^{(1)} + 16s_3^{(1)},} {s_1^{(1)} + 4s_2^{(1)} + 25s_3^{(1)}}\Big), \\ {\boldsymbol{s}^{(2\,|\, {0,0,1;1,0,0})}} & = \Big(s_1^{(2\,|\, {0,0,1;1,0,0})},s_2^{(2\,|\, {0,0,1;1,0,0})},s_3^{(2\,|\, {0,0,1;1,0,0})}\Big) \\ & = \frac{1}{60} \cdot \Big( {25s_1^{(1)} + 4s_2^{(1)} + s_3^{(1)},16s_1^{(1)} + 40s_2^{(1)} + 4s_3^{(1)},} {19s_1^{(1)} + 16s_2^{(1)} + 55s_3^{(1)}}\Big). \end{align*}

From these expressions, we have, for different ${\boldsymbol{s}^{(1)}}$ and ${\boldsymbol{s}^{(2)}}$ , the dynamic ordered system signatures in Case 3 as presented in Table 4.

Example 4.4. To present the dynamic ordered system signatures of the two coherent systems at time t in Case 4, values of $p_{j\,|\, \boldsymbol{l}}^{(i\,|\, {0,1,0;0,1,0})}$ $(i = 1,2,\,j = 1,2,\,\boldsymbol{l} \in {\mathcal{L}_{0,1,0;0,1,0}})$ need to be presented first by

$$p_{j\,|\, \boldsymbol{l}}^{(i\,|\, {0,1,0;\,0,1,0})} = p_{j\,|\, {{l_1},{l_2}}}^{(i:2)} = p_{j\,|\, {{l_{1,2,1}} + {l_{2,2,1}},{l_{1,2,2}} + {l_{2,2,2}}}}^{(i:2)},$$

with the values of $p_{j\,|\, {{l_1},{l_2}}}^{(i:2)}$ in Table 3 and

\begin{align*} {\mathcal{L}_{0,1,0;\,0,1,0}} & = \{ \boldsymbol{l} = ({l_{1,2,1}},{l_{1,2,2}};\,{l_{2,2,1}},{l_{2,2,2}})\,:\, {l_{1,2,1}} + {l_{1,2,2}} = 1,{l_{2,2,1}} + {l_{2,2,2}} = 1\} \\ & = \{ (1,0;\,1,0),(1,0;\,0,1),(0,1;\,1,0),(0,1;\,0,1)\} . \end{align*}

Then, the dynamic ordered system signatures ${\boldsymbol{s}^{(1\,|\, {0,1,0;0,1,0})}} $ and ${\boldsymbol{s}^{(2\,|\, {0,1,0;0,1,0})}}$ are given by replacing ${\boldsymbol{s}^{(1)}} = \big(s_1^{(1)},s_2^{(1)}\big)$ and ${\boldsymbol{s}^{(2)}} = \big(s_1^{(2)},s_2^{(2)}\big)$ in

\begin{align*} s_1^{(1\,|\, {0,1,0;\,0,1,0})} & = p_{1\,|\, {(2,0)}}^{(1:2)} \cdot s_1^{(1)}s_1^{(2)} + p_{1\,|\, {(1,1)}}^{(1:2)} \cdot \Big[ {s_1^{(1)}s_2^{(2)} + s_2^{(1)}s_1^{(2)}} \Big], \\ s_1^{(2\,|\, {0,1,0;\,0,1,0})} & = p_{1\,|\, {(2,0)}}^{(2:2)} \cdot s_1^{(1)}s_1^{(2)} + p_{1\,|\, {(1,1)}}^{(2:2)} \cdot \Big[ {s_1^{(1)}s_2^{(2)} + s_2^{(1)}s_1^{(2)}} \Big], \\ s_2^{(1\,|\, {0,1,0;\,0,1,0})} & = p_{2\,|\, {(0,2)}}^{(1:2)} \cdot s_2^{(1)}s_2^{(2)} + p_{2\,|\, {(1,1)}}^{(1:2)} \cdot \Big[ {s_1^{(1)}s_2^{(2)} + s_2^{(1)}s_1^{(2)}} \Big], \\ s_2^{(2\,|\, {0,1,0;\,0,1,0})} & = p_{2\,|\, {(0,2)}}^{(2:2)} \cdot s_2^{(1)}s_2^{(2)} + p_{2\,|\, {(1,1)}}^{(2:2)} \cdot \Big[ {s_1^{(1)}s_2^{(2)} + s_2^{(1)}s_1^{(2)}} \Big] \end{align*}

with

$${\boldsymbol{s}^{(1)}} = \Bigg( {{\frac{s_2^{(1)}}{s_2^{(1)} + s_3^{(1)}}},{\frac{s_3^{(1)}}{s_2^{(1)} + s_3^{(1)}}}} \Bigg), \qquad {\boldsymbol{s}^{(2)}} = \Bigg( {{\frac{s_2^{(2)}}{s_2^{(2)} + s_3^{(2)}}},{\frac{s_3^{(2)}}{s_2^{(2)} + s_3^{(2)}}}} \Bigg),$$

that is,

\begin{align*} {\boldsymbol{s}^{(1\,|\, {0,1,0;\,0,1,0})}} & = \big(s_1^{(1\,|\, {0,1,0;\,0,1,0})},s_2^{(1\,|\, {0,1,0;\,0,1,0})}\big) \\ & = \frac{1}{{6\big[s_2^{(1)} + s_3^{(1)}\big]\big[s_2^{(2)} + s_3^{(2)}\big]}} \cdot \Big( {6s_2^{(1)}s_2^{(2)} + 5s_2^{(1)}s_3^{(2)} + 5s_3^{(1)}s_2^{(2)},} \\ & \qquad \qquad {6s_3^{(1)}s_3^{(2)} + s_2^{(1)}s_3^{(2)} + s_3^{(1)}s_2^{(2)}} \Big), \\ {\boldsymbol{s}^{(2\,|\, {0,1,0;\,0,1,0})}} & = (s_1^{(2\,|\, {0,1,0;\,0,1,0})},s_2^{(2\,|\, {0,1,0;\,0,1,0})}) \\ & = \frac{1}{{6\big[s_2^{(1)} + s_3^{(1)}\big]\big[s_2^{(2)} + s_3^{(2)}\big]}} \cdot \Big( {6s_2^{(1)}s_2^{(2)} + s_2^{(1)}s_3^{(2)} + s_3^{(1)}s_2^{(2)},} \\ & \qquad \qquad {6s_3^{(1)}s_3^{(2)} + 5s_2^{(1)}s_3^{(2)} + 5s_3^{(1)}s_2^{(2)}} \Big). \end{align*}

Then, their equivalent systems with three components have the following signatures:

From these expressions, we have, for different ${\boldsymbol{s}^{(1)}}$ and ${\boldsymbol{s}^{(2)}}$ , the dynamic ordered system signatures in Case 4 as presented in Table 4.

Example 4.5. To present the dynamic ordered system signatures of the two coherent systems at time t in Case 5, values of $p_{j\,|\, \boldsymbol{l}}^{(i\,|\, {0,1,0;1,0,0})}$ $(i = 1,2,\,j = 1,2,\,\boldsymbol{l} \in {\mathcal{L}_{0,1,0;1,0,0}})$ need to be presented first, where

\begin{align*} {\mathcal{L}_{0,1,0;\,1,0,0}} & = \{ \boldsymbol{l} = ({l_{1,2,1}},{l_{1,2,2}};\,{l_{2,1,1}},{l_{2,1,2}})\,:\, {l_{1,2,1}} + {l_{1,2,2}} = 1,\,{l_{2,1,1}} + {l_{2,1,2}} = 1\} \\ & = \{ (1,0;\,1,0),(1,0;\,0,1),(0,1;\,1,0),(0,1;\,0,1)\} . \end{align*}

Then, the dynamic ordered system signatures ${\boldsymbol{s}^{(1\,|\, {0,1,0;1,0,0})}} $ and ${\boldsymbol{s}^{(2\,|\, {0,1,0;1,0,0})}}$ are given by replacing ${\boldsymbol{s}^{(1)}} = \big(s_1^{(1)},s_2^{(1)}\big)$ and ${\boldsymbol{s}^{(2)}} = \big(s_1^{(2)},s_2^{(2)}\big)$ in Example 4.4 with ${\boldsymbol{s}^{(1)}} = \big( {\frac{1}{2},\frac{1}{2}} \big)$ and ${\boldsymbol{s}^{(2)}} = \big( {\frac{1}{2},\frac{1}{2}} \big)$ , respectively, that is,

\begin{align*} {\boldsymbol{s}^{(1\,|\, {0,1,0;\,1,0,0})}} & = \frac{1}{{12\Big[s_2^{(1)} + s_3^{(1)}\Big]}} \cdot \Big( {11s_2^{(1)} + 5s_3^{(1)},s_2^{(1)} + 7s_3^{(1)}} \Big), \\ {\boldsymbol{s}^{(2\,|\, {0,1,0;\,1,0,0})}} & = \frac{1}{{12\Big[s_2^{(1)} + s_3^{(1)}\Big]}} \cdot \Big( {7s_2^{(1)} + s_3^{(1)},5s_2^{(1)} + 11s_3^{(1)}} \Big). \end{align*}

Then, their equivalent systems with three components have the following signatures:

From these expressions, we have, for different ${\boldsymbol{s}^{(1)}}$ and ${\boldsymbol{s}^{(2)}}$ , the dynamic ordered system signatures in Case 5 as presented in Table 4.

5. Applications to evaluation of aging properties of used systems

The computation of dynamic ordered system signatures discussed in the preceding sections facilitates comparison of used systems at time t in different cases (see Table 4 for their dynamic ordered system signatures). Suppose the component lifetimes in each system are all i.i.d. from an exponential distribution F with $F(x) = 1 - {{\mathrm{e}}^{ - x}}$ , $x \ge 0$ , which leads to the corresponding residual lifetime distribution as $\tilde F(x\mid t) = 1 - {{\mathrm{e}}^{ - (x - t)}}$ , $x > t$ . In this case, the respective conditional survival probabilities in Cases 1–5 that the first/second failed system among the two used systems fails after time $x+t$ , given ${E_\boldsymbol{k}}(t)$ , are plotted in Figs. 1–6 for different ${\boldsymbol{s}^{(1)}}$ and ${\boldsymbol{s}^{(2)}}$ , along with the survival probability that the first/second failed system among the two original systems, with signatures ${\boldsymbol{s}^{(1)}}$ and ${\boldsymbol{s}^{(2)}}$ and component lifetime distribution $\tilde F(x\mid t) = 1 - {{\mathrm{e}}^{ - (x - t)}}$ , $x > t$ , fails after time $x+t$ . As seen in Figs. 1–6, for any ${\boldsymbol{s}^{(1)}}$ and ${\boldsymbol{s}^{(2)}}$ in Table 4, the best systems are the original systems or the systems in Case 1.

Figure 1. Conditional survival probabilities of the first (left) and second (right) failed coherent systems in Cases 1–5 for ${\boldsymbol{s}^{(1)}} = {\boldsymbol{s}^{(2)}} = \big(\frac{1}{3},\frac{2}{3},0\big)$ .

Figure 2. Conditional survival probabilities of the first (left) and second (right) failed coherent systems in Cases 1–5 for ${\boldsymbol{s}^{(1)}} = {\boldsymbol{s}^{(2)}} = \big(0,\frac{2}{3},\frac{1}{3}\big)$ .

Figure 3. Conditional survival probabilities of the first (left) and second (right) failed coherent systems in Cases 1–5 for ${\boldsymbol{s}^{(1)}} = {\boldsymbol{s}^{(2)}} = (0, 0, 1)$ .

Figure 4. Conditional survival probabilities of the first (left) and second (right) failed coherent systems in Cases 1–5 for ${\boldsymbol{s}^{(1)}} = \big(\frac{1}{3},\frac{2}{3},0\big)$ , ${\boldsymbol{s}^{(2)}} = \big(0,\frac{2}{3},\frac{1}{3}\big)$ .

Figure 5. Conditional survival probabilities of the first (left) and second (right) failed coherent systems in Cases 1–5 for ${\boldsymbol{s}^{(1)}} = \big(\frac{1}{3},\frac{2}{3},0\big)$ , ${\boldsymbol{s}^{(2)}} = (0, 0, 1)$ .

Figure 6. Conditional survival probabilities of the first (left) and second (right) failed coherent systems in Cases 1–5 for ${\boldsymbol{s}^{(1)}} = \big(0,\frac{2}{3},\frac{1}{3}\big)$ , ${\boldsymbol{s}^{(2)}} = (0, 0, 1)$ .

6. Concluding remarks

In this paper, we first generalized the notion of ordered system signature from independent and identical coherent systems to the case of independent and non-identical coherent systems, and then established some related properties for the purpose of simplifying its computation. Based on such a general ordered system signature, a new concept, called dynamic ordered system signature, was then proposed for several coherent systems under a life-testing experiment. Then, several examples were presented to illustrate the established results. The usefulness of these results in the evaluation of aging properties of used systems was also demonstrated.

It is important to mention here that the notions introduced and their properties would be quite useful in developing parametric/non-parametric inferential methods for component lifetimes in coherent systems along the lines of [Reference Balakrishnan, Ng and Navarro3, Reference Balakrishnan, Ng and Navarro4, Reference Yang, Ng and Balakrishnan26, Reference Yang, Ng and Balakrishnan27]. As shown in [Reference Yang, Ng and Balakrishnan26], the ordered system signature leads to a more efficient method than the system signature for inferences on component lifetimes based on system lifetime data in a life test of several i.i.d. coherent systems. With the use of ordered system signature generalized in this paper, inferential methods can be developed for a life test of several independent and non-identical coherent systems, which would be a less restrictive life test in practice. Furthermore, the concept of dynamic ordered system signature can be applied to study dynamic properties of used systems in a life test, which would assist in studying maintenance policy, for example. We are currently working on these problems and hope to report the findings in a future paper.