2. Preliminaries

If the random variable $\xi$ has a matrix-geometric distribution, then it has a rational p.g.f. which is given by

(2.1) \begin{equation}\psi(z)=\mathbb{E}(z^{\xi})=\dfrac{c_{1}z+\cdots +c_{m}z^{m}}{1+d_{1}z+\cdots +d_{m}z^{m}}.\end{equation}

Using (2.1), the probability mass function (p.m.f.) and survival function of $\xi$ can be represented as

\begin{equation*}\mathbb{P}\{ \xi=l\} =\textbf{aQ}^{l-1}\textbf{u}^{\prime}\end{equation*}

and

\begin{equation*}\mathbb{P}\{ \xi>l\} =\textbf{aQ}^{l}(\textbf{I}-\textbf{Q})^{-1}\textbf{u}^{\prime}, \end{equation*}

where $\textbf{a=(}1,0,\ldots,0\textbf{)}$ and

\begin{equation*}\textbf{Q}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}-d_{1} & 0 & 0 & \cdots & 0 & 1\\ \\[-7pt]-d_{m} & 0 & 0 & \cdots & 0 & 0\\ \\[-7pt]-d_{m-1} & 1 & 0 & \cdots & 0 & 0\\ \\[-7pt]-d_{m-2} & 0 & 1 & & 0 & 0\\ \\[-7pt]\vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ \\[-7pt]-d_{2} & 0 & 0 & & 1 & 0\end{array}\right]\! ,\quad \textbf{u}^{\prime}=\left[\begin{array}{c}c_{1}\\ \\[-7pt]c_{m}\\ \\[-7pt]c_{m-1}\\ \\[-7pt]c_{m-2}\\ \\[-7pt]\vdots\\ \\[-7pt]c_{2}\end{array}\right]\end{equation*}

(see e.g. [Reference Bladt and Nielsen2]). Consider a coherent system with structure function $\phi(x_{1},\ldots,x_{n})$ such that $\phi(x_{1},\ldots,x_{n})=1$ if the system works and $\phi(x_{1},\ldots,x_{n})=0$ if the system has failed, where $x_{i}$ denotes the state of the ith component with $x_{i}=1$ if the ith component works and $x_{i}=0$ if it has failed. Let $r=\mathbb{P}\{ X_{i}=1\} $ be the common reliability of its components, $i=1,2,\ldots,n$ . Assume that the reliability of the system can be represented by the probability

(2.2) \begin{equation}\mathbb{P}\{ \phi(X_{1},\ldots,X_{n})=1\} =\mathbb{P}\{ \xi>n\} ,\end{equation}

where $\xi\sim MG(\textbf{a,Q}_{r}\textbf{,u}_{r})$ . Currently we do not know if a random variable $\xi$ satisfying (2.2) and having the distribution $MG(\textbf{a,Q}_{r}\textbf{,u}_{r})$ exists for any coherent system. However, as will be illustrated throughout the paper, there are many well-known and less-known coherent system models following the assumption.

The random variable $\xi$ actually represents the waiting time for the occurrence of the system failure. More explicitly, it is the total number of trials until the occurrence of a specific event in a sequence of binary trials with possible outcomes as either 0 (failure) or 1 (working). Note that all or some of the coefficients $c_{1},\ldots,c_{m}$ and $d_{1},\ldots,d_{m}$ depend on component reliability r. That is, the probability (2.2) depends on r through the matrix $\textbf{Q}_{r}$ and the vector $\textbf{u}_{r}$ . Therefore $\textbf{Q}_{r}$ and $\textbf{u}_{r}$ depend on r.

For an illustration, below we give examples for well-known coherent structures.

Example 2.1. Let $\phi(x_{1},\ldots,x_{n})=\min(x_{1},\ldots,x_{n})$ , i.e. the system has a series structure. Then $\psi(z)$ corresponds to the p.g.f. of the random variable $\xi$ that denotes the number of trials until the first failure in a sequence of independent and identical binary trials. Clearly

\begin{equation*}\psi(z)=\dfrac{(1-r)z}{1-zr}.\end{equation*}

Therefore $\xi\sim MG(1\textbf{,}r\textbf{,}1-r)$ .

Example 2.2. Let the system have k-out-of-n:F structure. Because the system fails as soon as k failures occur, the random variable $\xi$ denotes the waiting time for a total of k failures in a binary sequence of two possible outcomes as either 0 (failure) or 1 (working). Thus the random variable $\xi$ is the sum of k independent geometric variables, and

\begin{equation*}\psi(z)=\biggl[ \dfrac{(1-r)z}{1-zr}\biggr] ^{k}=\dfrac{(1-r)^{k}z^{k}}{1+\sum_{i=1}^{k}(-1)^{i}\binom{k}{i}r^{i}z^{i}}.\end{equation*}

Therefore $\xi\sim MG(\textbf{a,Q}_{r}\textbf{,u}_{r})$ with $\textbf{a=(}1,0,\ldots,0\textbf{)}$ and

\begin{equation*}\textbf{Q}_{r}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}kr & 0 & 0 & \cdots & 0 & 1\\ \\[-7pt](-1)^{k+1}r^{k} & 0 & 0 & \cdots & 0 & 0\\ \\[-7pt](-1)^{k}\binom{k}{k-1}r^{k-1} & 1 & 0 & \cdots & 0 & 0\\ \\[-7pt] (-1)^{k-1}\binom{k}{k-2}r^{k-2} & 0 & 1 & & 0 & 0\\ \\[-7pt]\vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ \\[-7pt]-\binom{k}{2}r^{2} & 0 & 0 & & 1 & 0\end{array}\right]\! ,\quad \textbf{u}_{r}^{\prime}=\left[\begin{array}{c}0\\ \\[-7pt](1-r)^{k}\\ \\[-7pt]0\\ \\[-7pt]0\\ \\[-7pt]\vdots\\ \\[-7pt]0\end{array}\right]\! .\end{equation*}

3. Computing the minimal signature

In this section we develop a method for computing the minimal signature of a coherent system by utilizing matrix-geometric distributions that were mentioned in the previous section.

Theorem 3.1. Let $\phi(x_{1},\ldots,x_{n})$ be the structure function of a binary coherent system and let r be the common reliability of its components. Suppose that the reliability of the system is represented by $P\{ \xi>n\} $ , where $\xi$ is a discrete random variable having a matrix-geometric distribution with p.g.f.

\begin{equation*}\psi(z)=\dfrac{c_{1}z+\cdots +c_{m}z^{m}}{1+d_{1}z+\cdots +d_{m}z^{m}}.\end{equation*}

Then the minimal signature of the system with structure function $\phi$ can be computed from

\begin{equation*}\boldsymbol{\omega}=\textbf{A}^{-1}\textbf{b,} \end{equation*}

where

\begin{equation*}\textbf{A}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}1 & 1 & \cdots & 1\\ \dfrac{1}{2} & \dfrac{1}{2^{2}} & \cdots & \dfrac{1}{2^{n}}\\ \vdots & & \ddots & \\ \dfrac{1}{2^{n-1}} & \dfrac{1}{2^{2(n-1)}} & \cdots & \dfrac{1}{2^{n(n-1)}}\end{array}\right]\end{equation*}

and

\begin{equation*}\textbf{b}=\begin{bmatrix}1\\ \\[-7pt] {a\Delta}_{1}^{n}(\textbf{I}-\boldsymbol{\Delta}_{1})^{-1}\textbf{d}_{1}^{\prime}\\ \\[-7pt] \vdots\\ \\[-7pt] {a\Delta}_{n-1}^{n}(\textbf{I}-\boldsymbol{\Delta}_{n-1})^{-1}\textbf{d}_{n-1}^{\prime}\end{bmatrix}\! ,\end{equation*}

where $\boldsymbol{\Delta}_{t}$ and $\textbf{d}_{t}$ are obtained by replacing r in $\textbf{Q}_{r}$ and $\textbf{u}_{r}$ with ${1}/{2^{t}}$ , $t=1,2,\ldots,n-1$ .

Proof. Because the minimal signature only depends on the system structure, without loss of generality, assume that the system with lifetime $T=\phi(T_{1},\ldots,T_{n})$ consists of independent and identically distributed components with common lifetime distribution $F(t)=1-(1-p)^{t}$ , $t=1,2,\ldots,n$ . That is, the lifetimes of components follow a geometric distribution with mean ${1}/{p}$ . Then

\begin{equation*} p(t) =\mathbb{P}\{ T>t\} =\sum_{i=1}^{n}\omega_{i}[ (1-p)^{t}] ^{i} =\omega_{1}(1-p)^{t}+\omega_{2}(1-p)^{2t}+\cdots +\omega_{n}(1-p)^{nt}.\end{equation*}

Clearly

\begin{align*}p(0) & =1,\\ p(1) & =\omega_{1}(1-p)+\omega_{2}(1-p)^{2}+\cdots +\omega_{n}(1-p)^{n}\\& \vdots\\ p(n-1) & =\omega_{1}(1-p)^{n-1}+\omega_{2}(1-p)^{2(n-1)}+\cdots +\omega_{n}(1-p)^{n(n-1)}.\end{align*}

Thus the minimal signature $(\omega_{1},\omega_{2},\ldots,\omega_{n})$ is obtained from

\begin{equation*}\begin{bmatrix}\omega_{1}\\ \\[-7pt] \omega_{2}\\ \\[-7pt] \vdots\\ \\[-7pt] \omega_{n}\end{bmatrix} =\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}1 & 1 & \cdots & 1\\ \\[-7pt](1-p) & (1-p)^{2} & \cdots & (1-p)^{n}\\ \\[-7pt]\vdots & & \ddots & \\ \\[-7pt](1-p)^{n-1} & (1-p)^{2(n-1)} & \cdots & (1-p)^{n(n-1)}\end{array}\right]^{-1}\begin{bmatrix}1\\ \\[-7pt]p(1)\\ \\[-7pt]\vdots\\ \\[-7pt]p(n-1)\end{bmatrix}\! .\end{equation*}

Because $r=\bar{F}(t)$ , from equation (2.2) we have

\begin{equation*}p(t) =\mathbb{P}\{ T>t\} =\textbf{aQ}_{\bar{F}(t)}^{n}(\textbf{I}-\textbf{Q}_{\bar{F}(t)})^{-1}\textbf{u}_{\bar{F}(t)}^{\prime} ={a\boldsymbol\Delta}_{t}^{n}(\textbf{I}-\boldsymbol{\Delta}_{t})^{-1}\textbf{d}_{t}^{\prime},\end{equation*}

where $\boldsymbol{\Delta}_{t}$ and $\textbf{d}_{t}$ are obtained by replacing r in $\textbf{Q}_{r}$ and $\textbf{u}_{r}$ with $\bar{F}(t)=(1-p)^{t}$ , $t=1,2,\ldots,n-1$ . The result now follows upon choosing $p=\frac{1}{2}$ .

As is clear from Theorem 3.1, the computation of the minimal signature vector requires the p.g.f. of the random variable $\xi$ corresponding to the system structure. For some systems, the derivation of the p.g.f. for the occurrence of the system failure is simpler than obtaining an expression for the system reliability. Consider the following examples.

Example 3.1. The (n,f,k) system consists of n components ordered in a line, and it fails if and only if there exist at least f failed components or at least k consecutive failed components. The p.g.f. of the waiting time for the occurrence of the (n,f,2) system failure can be obtained from Proposition 1 of Triantafyllou and Koutras [Reference Triantafyllou and Koutras20]. Note that [20, Proposition 1] includes the p.g.f. for the reliability function defined by

\begin{equation*}R(z;r)=\sum_{n=0}^{\infty}z^{n} \,\mathbb{P}\{ \xi>n\} .\end{equation*}

The p.g.f. for the waiting time for the occurrence of the system failure can be obtained from $\psi(z)=1-R(z;r)(1-z)$ (see Appendix A) and is given by

\begin{equation*}\psi(z)=1-\dfrac{(1-rz)^{f}-((1-r)z)^{2} [\sum_{i=0}^{f-3} (r(1-r)z^{2}) ^{i}(1-rz)^{f-i-1}+(r(1-r)z^{2})^{f-2}]}{(1-rz)^{f}}.\end{equation*}

If in particular $f=3$ , then

\begin{equation*}\psi(z)=\dfrac{(1-r)^{2}z^{2}-2r(1-r)^{2}z^{3}+r(1-r)^{2}z^{4}}{1-3rz+3r^{2}z^{2}-r^{3}z^{3}}.\end{equation*}

Thus $\xi\sim MG(\textbf{a,Q}_{r}\textbf{,u}_{r})$ with $\textbf{a=(}1,0,0,0\textbf{)}$ and

\begin{equation*}\textbf{Q}_{r}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}3r & 0 & 0 & 1\\ \\[-7pt]0 & 0 & 0 & 0\\ \\[-7pt]r^{3} & 1 & 0 & 0\\ \\[-7pt]-3r^{2} & 0 & 1 & 0\end{array}\right],\quad \textbf{u}_{r}^{\prime}=\begin{bmatrix}0\\ \\[-7pt]r(1-r)^{2}\\ \\[-7pt]-2r(1-r)^{2}\\ \\[-7pt](1-r)^{2}\end{bmatrix}\! .\end{equation*}

Using Theorem 3.1, the minimal signature of the (n,3,2) system can be computed from

\begin{equation*}\begin{bmatrix}\omega_{1}\\ \\[-7pt]\omega_{2}\\ \\[-7pt]\vdots\\ \\[-7pt]\omega_{n}\end{bmatrix} =\begin{bmatrix}1 & 1 & \cdots & 1\\ \\[-7pt] \dfrac{1}{2} & \dfrac{1}{2^{2}} & \cdots & \dfrac{1}{2^{n}}\\ \\[-7pt]\vdots & & \ddots & \\ \\[-7pt] \dfrac{1}{2^{n-1}} & \dfrac{1}{2^{2(n-1)}} & \cdots & \dfrac{1}{2^{n(n-1)}}\end{bmatrix}^{-1}\begin{bmatrix}1\\ \\[-7pt]p(1)\\ \\[-7pt]\vdots\\ \\[-7pt]p(n-1)\end{bmatrix}\! ,\end{equation*}

where

\begin{align*}p(t) & =\textbf{(}1,0,0,0\textbf{)}\begin{bmatrix}3\dfrac{1}{2^{t}} &\quad 0 &\quad 0 &\quad 1\\ \\[-7pt] 0 &\quad 0 &\quad 0 &\quad 0\\ \\[-7pt]\biggl( \dfrac{1}{2^{t}}\biggr) ^{3} &\quad 1 &\quad 0 &\quad 0\\ \\[-7pt] -3\biggl( \dfrac{1}{2^{t}}\biggr) ^{2} &\quad 0 &\quad 1 &\quad 0\end{bmatrix}^{n}\\ \\[-7pt]&\quad \quad\, \times\left(\begin{bmatrix}1 &\quad 0 &\quad 0 &\quad 0\\ \\[-7pt]0 &\quad 1 &\quad 0 &\quad 0\\ \\[-7pt]0 &\quad 0 &\quad 1 &\quad 0\\ \\[-7pt]0 &\quad 0 &\quad 0 &\quad 1\end{bmatrix} -\begin{bmatrix}3\dfrac{1}{2^{t}} &\quad 0 &\quad 0 &\quad 1\\ \\[-7pt] 0 &\quad 0 &\quad 0 &\quad 0\\ \\[-7pt]\biggl( \dfrac{1}{2^{t}}\biggr) ^{3} &\quad 1 &\quad 0 &\quad 0\\ \\[-7pt] -3\biggl( \dfrac{1}{2^{t}}\biggr) ^{2} &\quad 0 &\quad 1 &\quad 0\end{bmatrix} \right) ^{-1}\begin{bmatrix}0\\ \\[-7pt]\dfrac{1}{2^{t}}\biggl(1-\dfrac{1}{2^{t}}\biggr)^{2}\\ \\[-7pt] -2\dfrac{1}{2^{t}}\biggl(1-\dfrac{1}{2^{t}}\biggr)^{2}\\ \\[-7pt] \biggl(1-\dfrac{1}{2^{t}}\biggr)^{2}\end{bmatrix}\! ,\end{align*}

for $t=1,2,\ldots,n-1$ . In Table 1 we present $\boldsymbol{\omega}=(\omega_{1},\ldots,\omega_{n})$ for the (n,3,2) system for selected values of n.

Table 1.

Minimal signature of the (n,3,2) system.

Example 3.2. A consecutive-k-out-of-n:F system with sparse d is a system that consists of n linearly ordered components such that the system fails if and only if there exist at least k consecutive failed components with sparse d, i.e. there must be k or more failures and sparse d or less if the system fails [Reference Zhao, Cui and Kuo22]. The p.g.f. of the waiting time for the occurrence of the system failure has been obtained as [Reference Mohan, Agarwal and Sen14]

\begin{equation*}\psi(z)=\dfrac{((1-r)z)^{k} ( 1-(rz)^{d+1}) ^{k-1}(1-z+(1-r)z ( rz ) ^{d+1} ) }{(1-rz) ( (1-z)(1-rz)^{k-1}+((1-r)z)^{k}(rz)^{d+1} ( 1-(rz)^{d+1} ) ^{k-1}) }.\end{equation*}

Let $k=2$ and $d=1$ . Then

\begin{equation*}\psi(z)=\dfrac{c_{1}z+\cdots +c_{7}z^{7}}{1+d_{1}z+\cdots +d_{7}z^{7}},\end{equation*}

where

\begin{alignat*}{4}c_{1}&=0, & c_{2}&=(1-r)^{2}, \quad & c_{3}&=-(1-r)^{2}, \qquad & c_{4}&=-r^{2}(1-r)^{2}, \\ c_{5}&=r^{2}(1-r)^{2}(2-r), \quad & c_{6}&=0, & c_{7}&=-r^{4}(1-r)^{3},\\d_{1}&=-(1+2r), \quad & d_{2}&=r(r+2), \quad & d_{3}&=-r^{2},\quad & d_{4}&=r^{2}(1-r)^{2}, \\ d_{5}&=-r^{3}(1-r)^{2}, & d_{6}&=-r^{4}(1-r)^{2}, \quad & d_{7}&=r^{5}(1-r)^{2}.\end{alignat*}

If, for example, $n=5$ , then using Theorem 3.1, the minimal signature of the consecutive-2-out-of-5:F system with sparse $d=1$ is obtained as $(0,0,3,-1,-1)$ .

4. Minimal signature of a series system with two modules

Consider a coherent system consisting of $n+m$ components with the index set of components $C=\{ 1,\ldots,n+m\} $ . Suppose that the system with the component index set C consists of two disjoint modules with respective component index sets $\{ 1,\ldots,n\} $ and $\{n+1,\ldots,n+m\} $ and structure functions $\phi_{1}$ and $\phi_{2}$ . If the overall system has a series structure, i.e. the disjoint modules are serially connected, then the system’s reliability is represented as

\begin{equation*}\mathbb{P}\{ \phi_{1}(X_{1},\ldots,X_{n})=1,\phi_{2}(X_{n+1},\ldots,X_{n+m})=1\}=\mathbb{P}\{ \xi_{1}>n\} \,\mathbb{P}\{ \xi_{2}>m\} ,\end{equation*}

where $\xi_{i}\sim MG(\textbf{a}^{(i)}\textbf{,Q}_{r}^{(i)}\textbf{,u}_{r}^{(i)}),i=1,2$ . The problem is to compute the minimal signature of the series system with structure function $\min(\phi_{1}(x_{1},\ldots,x_{n}),\phi_{2}(x_{n+1},\ldots,x_{n+m}))$ based on the minimal signatures of the systems with structure functions $\phi_{1}(x_{1},\ldots,x_{n})$ and $\phi_{2}(x_{n+1},\ldots,x_{n+m})$ . This problem has been considered in the literature by using a different approach that is based on the number of path sets of the structures (see e.g. [Reference Da, Xia and Hu4,Reference Eryilmaz6]).

Let $\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})$ and $\boldsymbol{\beta}=(\beta_{1},\beta_{2},\ldots,\beta_{m})$ , respectively, denote the minimal signatures of the systems with structure functions $\phi_{1}$ and $\phi_{2}$ . Then the reliability of the system with structure function $\min(\phi_{1}(x_{1},\ldots,x_{n}),\phi_{2}(x_{n+1},\ldots,x_{n+m}))$ can be written as

\begin{equation*}\mathbb{P}\{ \xi_{s}>n+m\} =\sum_{i=1}^{n}\alpha_{i}r^{i}\sum_{j=1}^{m}\beta_{j}r^{j} =( \boldsymbol{\alpha\otimes\beta}) ( \textbf{r}_{1}^{\prime}\otimes\textbf{r}_{2}^{\prime}) =\sum_{i=1}^{n+m}\omega_{i}r^{i}, \end{equation*}

where $\boldsymbol{\otimes}$ denotes the Kronecker product and $\textbf{r}_{1}=[r,r^{2},\ldots,r^{n}] $ , $\textbf{r}_{2}=[ r,r^{2},\ldots,r^{m}]$ .

For given $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ , the minimal signature $(\omega_{1},\omega_{2},\ldots,\omega_{n+m})$ can be computed by collecting the coefficients of the terms $r^{i}$ in the expansion of $( \boldsymbol{\alpha\otimes\beta}) ( \textbf{r}_{1}^{\prime}\otimes\textbf{r}_{2}^{\prime}) $ . This can be done using the computer code presented in Appendix B.

Example 4.1. Let $\phi_{1}$ and $\phi_{2}$ be the structure functions of 2-out-of-3:F and 3-out-of-4:F systems, respectively. Then, from Example 2.2 and Theorem 3.1, the minimal signatures are obtained as $\boldsymbol{\alpha}=(0,3,-2)$ and $\boldsymbol{\beta}=(0,0,4,-3)$ . Then

\begin{equation*}\boldsymbol{\alpha\otimes\beta=}( 0,0,0,0,0,0,12,-9,0,0,-8,6)\end{equation*}

and

\begin{equation*}\textbf{r}_{1}\otimes\textbf{r}_{2}=( r^{2},r^{3},r^{4},r^{5},r^{3},r^{4},r^{5},r^{6},r^{4},r^{5},r^{6},r^{7}) .\end{equation*}

Using the algorithm presented in Appendix 8, the minimal signature of the system which is a series connection of 2-out-of-3:F and 3-out-of-4:F systems is obtained as

\begin{equation*}\boldsymbol{\omega}=(0,0,0,0,12,-17,6).\end{equation*}

5. Systems with two types of components

Consider a coherent system that consists of two types of components. Let $n_{i}$ denote the number of components of type i, $i=1,2$ . Then the reliability of the system can be represented by

\begin{equation*}\mathbb{P}\{ \xi>n\} =\sum_{m_{1}=0}^{n_{1}}\sum_{m_{2}=0}^{n_{2}}\omega_{m_{1},m_{2}}r_{1}^{m_{1}}r_{2}^{m_{2}}, \end{equation*}

where $r_{i}$ denotes the reliability of components of type $i,i=1,2$ and $\xi\sim MG(\textbf{a,Q}_{r_{1},r_{2}}\textbf{,u}_{r_{1},r_{2}})$ . The matrix $\textbf{Q}_{r_{1},r_{2}}$ and the vector $\textbf{u}_{r_{1},r_{2}}$ are constructed with the help of the coefficients in the probability generating function of $\xi$ . The vector

\begin{equation*}\boldsymbol{\omega=}(\omega_{0,0},\omega_{0,1},\ldots,\omega_{0,n_{2}},\ldots,\omega_{n_{1},0},\omega_{n_{1},1},\ldots,\omega_{n_{1},n_{2}})\end{equation*}

of $(n_{1}+1)\times(n_{2}+1)$ elements represents the minimal signature of the system (see e.g. [Reference Eryilmaz, Coolen and Coolen-Maturi9]). Using the method presented in Theorem 3.1, the minimal signature of the system can be computed from

\begin{equation*}\boldsymbol{\omega}=\textbf{A}^{-1}\textbf{b}\end{equation*}

for $i=1,2,\ldots,(n_{1}+1)\times(n_{2}+1)$ , and the (i,j)th element of the matrix $\textbf{A}$ is

\begin{equation*}a_{ij}=\begin{cases}(1-p_{2})^{(j-1)(i-1)} & \text{if }j=1,2,\ldots,n_{2}+1,\\ \\[-8pt](1-p_{1})^{i-1}(1-p_{2})^{(j-(n_{2}+2))(i-1)} & \text{if }j=n_{2}+2,\ldots, 2n_{2}+2,\\ \\[-8pt](1-p_{1})^{2(i-1)}(1-p_{2})^{(j-(2n_{2}+3))(i-1)} & \text{if }j=2n_{2}+3,\ldots,3n_{2}+3,\\ \\[-8pt]\vdots & \\ \\[-8pt](1-p_{1})^{n_{1}(i-1)}(1-p_{2})^{(j-(n_{1}n_{2}+n_{1}+1))(i-1)} & \text{if}j=n_{1}n_{2}+n_{1}+1,\ldots,n_{1}n_{2}+n_{1}+n_{2}+1.\end{cases}\end{equation*}

The vector $\textbf{b}$ is given by

\begin{equation*}\textbf{b}=\begin{bmatrix}1\\ \\[-8pt]p(1)\\ \\[-8pt]\vdots\\ \\[-8pt]p(n_{1}n_{2}+n_{1}+n_{2})\end{bmatrix}\! ,\end{equation*}

where

\begin{equation*}p(t)={a\boldsymbol\Delta}_{t}^{n}(\textbf{I}-\boldsymbol{\Delta}_{t})^{-1}\textbf{d}_{t}^{\prime},\end{equation*}

and $\boldsymbol{\Delta}_{t}$ and $\textbf{d}_{t}$ are obtained by replacing $r_{1}$ and $r_{2}$ in $\textbf{Q}_{r_{1},r_{2}}$ and $\textbf{u}_{r_{1},r_{2}}$ with $(1-p_{1})^{t}$ and $(1-p_{2})^{t}$ , $t=1,2,\ldots,n_{1}n_{2}+n_{1}+n_{2}$ .

Example 5.1. For a 2-out-of-n:F system consisting of $n_{1}$ components of type 1 and $n-n_{1}$ components of type 2, the p.g.f. of the waiting time for the system failure is (see Appendix C for the derivation)

(5.1) \begin{align}\psi(z) & =\dfrac{(1-r_{1})^{2}[ r_{1}z^{2}-n_{1}r_{1}^{n_{1}}z^{n_{1}+1}+(n_{1}-1)r_{1}^{n_{1}+1}z^{n_{1}+2}] }{r_{1}(1-zr_{1})^{2}}\notag\\& \quad\, +\dfrac{n_{1}r_{1}^{n_{1}-1}(1-r_{1})(1-r_{2})z^{n_{1}+1}}{1-zr_{2}}+\dfrac{r_{1}^{n_{1}}(1-r_{2})^{2}z^{n_{1}+2}}{(1-zr_{2})^{2}}.\end{align}

Note that if $n_{1}=0$ and $r_{1}=r_{2}=r$ in (5.1), then

\begin{equation*}\psi(z)=\biggl[ \dfrac{(1-r)z}{1-zr}\biggr] ^{2} ,\end{equation*}

which is the p.g.f. corresponding to the 2-out-of-n:F system with a single type of components.

Example 5.2. For a 2-out-of-3:F system, let $n_{1}=1$ . Then, from (5.1), we have

\begin{equation*}\psi(z)=\dfrac{(1-r_{1})(1-r_{2})z^{2}+(1-r_{2})(r_{1}-r_{2})z^{3}}{1-2r_{2}z+r_{2}^{2}z^{2}}.\end{equation*}

Thus $\xi\sim MG(\textbf{a,Q}_{r_{1},r_{2}}\textbf{,u}_{r_{1},r_{2}})$ with $\textbf{a}=(1,0,0\textbf{)}$ and

\begin{equation*}\textbf{Q}_{r_{1},r_{2}}=\begin{bmatrix}2r_{2} &\quad 0 &\quad 1\\ \\[-7pt]0 &\quad 0 &\quad 0\\ \\[-7pt]-r_{2}^{2} &\quad 1 &\quad 0\end{bmatrix}, \quad \textbf{u}_{r_{1},r_{2}}^{\prime}=\begin{bmatrix}0\\ \\[-7pt](1-r_{2})(r_{1}-r_{2})\\ \\[-7pt](1-r_{1})(1-r_{2})\end{bmatrix}\! .\end{equation*}

Using the method, the minimal signature of the 2-out-of-3:F system consisting of $n_{1}=1$ component of type 1 and $n-n_{1}=2$ components of type 2 is obtained as

\begin{equation*}\boldsymbol{\omega=(}\omega_{0,0},\omega_{0,1},\omega_{0,2},\omega_{1,0},\omega_{1,1},\omega_{1,2}\textbf{)}=(0,0,1,0,2,-2).\end{equation*}

Example 5.3. A consecutive-2-out-of-n:F system is a system that consists of n linearly ordered components, and fails if and only if there exist at least two consecutive failed components. Assume that the first $n_{1}$ components have common reliability $r_{1}$ and the remaining $n-n_{1}$ components have common reliability $r_{2}$ . Then the p.g.f. of the random variable $\xi$ which denotes the waiting time until two consecutive failures is given by the following equations when $n_{1}=1$ , $n_{1}=2$ , and $n_{1}=3$ (see Appendix D for the proofs):

(5.2) \begin{align}\psi(z)&=\dfrac{(1-r_{1})(1-r_{2})z^{2}+(1-r_{2})(r_{1}-r_{2})z^{3}}{1-r_{2}z+r_{2}(r_{2}-1)z^{2}}\quad \text{for }n_{1}=1,\end{align}

(5.3) \begin{align}\psi(z)&=\dfrac{(1-r_{1})^{2}z^{2}+(1-r_{1})(r_{1}-r_{2})z^{3}+(1-r_{2})(r_{1}-r_{2})z^{4}}{1-r_{2}z+r_{2}(r_{2}-1)z^{2}}\quad \text{for }n_{1}=2,\end{align}

and for $n_{1}=3$

(5.4) \begin{align}\psi(z) & =\dfrac{1}{1-r_{2}z+r_{2}(r_{2}-1)z^{2}} [ (1-r_{1})^{2}z^{2}+(1-r_{1})^{2}(r_{1}-r_{2})z^{3} \notag\\& \quad\, +(1-r_{1})(r_{1}-r_{2})(1-r_{2}(1-r_{1}))z^{4} \notag\\& \quad\, + ( r_{1}(r_{1}-r_{2})(r_{1}-2)(r_{2}-1)) z^{5}]\end{align}

The following result is useful for obtaining the p.g.f. corresponding to a series system of two or more modules.

Proposition 5.1. The p.g.f. of the waiting time for the failure of the two-component series system, when one component is of type 1 with reliability $r_{1}$ and the other component is of type 2 with reliability $r_{2}$ , is

(5.5) \begin{equation}\psi(z)=\dfrac{(1-r_{1})z+(r_{1}-r_{2})z^{2}}{1-zr_{2}}.\end{equation}

Proof. The result is immediate from

\begin{equation*}\psi(z)=z(1-r_{1})+zr_{1}\varphi(z),\end{equation*}

where $\varphi(z)$ denotes the p.g.f. of the geometric distribution with probability of success $1-r_{2}$ , that is,

\begin{equation*}\varphi(z)=\dfrac{z(1-r_{2})}{1-zr_{2}}.\end{equation*}

Example 5.4. Consider the system with structure function

(5.6) \begin{equation}\phi(x_{1},x_{2},x_{3},x_{4})=\min(x_{1},\max(x_{2},x_{3},x_{4})).\end{equation}

Assume that component 1 has reliability $p_{1}$ and components 2, 3, and 4 have common reliability $p_{2}$ . Taking $r_{1}=p_{1}$ and $r_{2}=1-(1-p_{2})^{3}$ in (5.5), the p.g.f. of the waiting time for the failure of the system with structure function (5.6) is obtained as

\begin{equation*}\psi(z)=\dfrac{(1-p_{1})z+(p_{1}-1+(1-p_{2})^{3})z^{2}}{1-z (1-(1-p_{2})^{3}) }.\end{equation*}

As illustrated by Example 5.4, the result presented in Proposition 5.2 is useful for obtaining the p.g.f. of a series system consisting of disjoint modules. There are coherent systems which can be written as a series system of two or more systems which may contain common components. In the following, we obtain the p.g.f. of the waiting time for the failure of the series system consisting of two modules that have one common component. The representation of a coherent system as a series system of two modules considerably reduces the complexity in computing the signature.

Proposition 5.2. Let $\phi_{1}$ and $\phi_{2}$ be the structure functions of two coherent systems of sizes n and m, respectively. Suppose that component i is a common component of both systems. If $n+m$ components have common reliability value p, then the p.g.f. corresponding to the failure of the series system of the two systems $\phi_{1}$ and $\phi_{2}$ is given by

\begin{equation*}\psi(z)=\dfrac{c_{1}z+c_{2}z^{2}+c_{3}z^{3}}{1+d_{1}z+d_{2}z^{2}+d_{3}z^{3}},\end{equation*}

where

\begin{align*}c_{1} & =1-pr_{1}-(1-p)r_{1}^{\ast},\\[3pt] c_{2} & =pr_{2}^{\ast}(r_{1}-1)+p(r_{1}-r_{2})+(1-p) [ r_{1}^{\ast}-r_{2}^{\ast}+r_{2}(r_{1}^{\ast}-1)] ,\\[3pt] c_{3} & =pr_{2}^{\ast}(r_{2}-r_{1})+(1-p)r_{2}(r_{2}^{\ast}-r_{1}^{\ast}),\\[3pt] d_{1} & =-(r_{2}+r_{2}^{\ast}),\\[3pt] d_{2} & =r_{2}r_{2}^{\ast},\\[3pt] d_{3} & =0\end{align*}

for $r_{1}^{\ast},r_{2}^{\ast}>0$ , where $r_{j}$ ( $r_{j}^{\ast}$ ) denotes the reliability of the system $\phi_{j}$ when the component i works (fails), $j=1,2$ .

Proof. Manifestly

\begin{equation*}\psi(z) =\mathbb{E}(z^{\xi}\mid X_{i}=1) \,\mathbb{P}\{ X_{i}=1\} +\mathbb{E}(z^{\xi}\mid X_{i}=0) \,\mathbb{P}\{ X_{i}=0\} =p\psi_{1}(z)+(1-p)\psi_{2}(z),\end{equation*}

where $\psi_{1}(z)=\mathbb{E}(z^{\xi}\mid X_{i}=1)$ and $\psi_{2}(z)=\mathbb{E}(z^{\xi}\mid X_{i}=0)$ . When the state of the component i is fixed, the entire system becomes a series system with two disjoint modules such that the first module has reliability $r_{1}$ ( $r_{1}^{\ast}$ ) and the second module has reliability $r_{2}$ ( $r_{2}^{\ast}$ ) when the component is working (has failed). Thus, from Proposition 5.1,

\begin{equation*}\psi_{1}(z)=\begin{cases}1 & \text{if $r_{1}=1$ and $r_{2}=1$,}\\\dfrac{(1-r_{1})z+(r_{1}-r_{2})z^{2}}{1-zr_{2}} & \text{otherwise,}\end{cases}\end{equation*}

and

\begin{equation*}\psi_{2}(z)=\dfrac{(1-r_{1}^{\ast})z+(r_{1}^{\ast}-r_{2}^{\ast})z^{2}}{1-zr_{2}^{\ast}}.\end{equation*}

Thus the proof is complete.

Corollary 5.1. Let $\xi$ denote the waiting time for the failure of the series system of two coherent systems $\phi_{1}$ and $\phi_{2}$ . If component i is a common component of systems $\phi_{1}$ and $\phi_{2}$ , then $\xi\sim MG(\textbf{a,Q,u})$ with $\textbf{a=(}1,0,0\textbf{)}$ and

\begin{equation*}\textbf{Q}=\begin{bmatrix}(r_{2}+r_{2}^{\ast}) &\quad 0 &\quad 1\\ \\[-7pt]0 &\quad 0 &\quad 0\\ \\[-7pt]-r_{2}r_{2}^{\ast} &\quad 1 &\quad 0\end{bmatrix}, \quad \textbf{u}^{\prime}=\begin{bmatrix}c_{1}\\ \\[-7pt]c_{3}\\ \\[-7pt]c_{2}\end{bmatrix}\! ,\end{equation*}

where $c_{1}$ , $c_{2}$ , and $c_{3}$ are given by Proposition 5.2.

Example 5.5. Consider the system with structure function

\begin{equation*}\phi(x_{1},x_{2},x_{3})=\min(\max(x_{1},x_{2}),\max(x_{2},x_{3})).\end{equation*}

In this case $\phi_{1}(x_{1},x_{2})=\max(x_{1},x_{2})$ and $\phi_{2}(x_{2},x_{3})=\max(x_{2},x_{3})$ , and component 2 is common to both systems. Assume that the components have common reliability p. Clearly $r_{1}=r_{2}=1$ and $r_{1}^{\ast}=r_{2}^{\ast}=p$ . Thus, from Corollary 5.1, the p.g.f. of the system is obtained as

\begin{equation*}\psi(z) =p+\dfrac{z(1-p)}{1-zp}(1-p) =\dfrac{p+z(1-2p)}{1-zp}.\end{equation*}

Example 5.6. Consider the system with structure function

\begin{equation*}\phi(x_{1},x_{2},x_{3},x_{4})=\min(\max(x_{1},x_{2}),\max(\min(x_{2},x_{3}),x_{4})).\end{equation*}

In this case $\phi_{1}(x_{1},x_{2})=\max(x_{1},x_{2})$ and $\phi_{2}(x_{2},x_{3},x_{4})=\max(\min(x_{2},x_{3}),x_{4})$ , and component 2 is shared by two systems. Assume that the components have common reliability p. Clearly $r_{1}=1$ , $r_{2}=2p-p^{2}$ , and $r_{1}^{\ast}=r_{2}^{\ast}=p$ . Thus, from Corollary 5.1, the p.g.f. of the system is obtained as

\begin{equation*}\psi(z)=\dfrac{c_{1}z+c_{2}z^{2}+c_{3}z^{3}}{1+d_{1}z+d_{2}z^{2}+d_{3}z^{3}},\end{equation*}

where

\begin{alignat*}{3} c_{1}&=(1-p)^{2},\quad & c_{2}&=-p(1-p)^{3},\quad & c_{3}& =-p^{2}(1-p)^{2},\\[3pt] d_{1}&=p^{2}-3p,\quad & d_{2}&=p(2p-p^{2}),\quad & d_{3}&=0. \end{alignat*}

Example 5.7. Consider the system with structure function

\begin{equation*}\phi(x_{1},x_{2},x_{3},x_{4},x_{5})=\min(\min(x_{1},\max(x_{2},x_{3})),\max(\min(x_{2},x_{4}),x_{5})).\end{equation*}

In this case $\phi_{1}(x_{1},x_{2},x_{3})=\min(x_{1},\max(x_{2},x_{3}))$ and $\phi_{2}(x_{2},x_{4},x_{5})=\max(\min(x_{2},x_{4}),x_{5})$ , and component 2 is common to both systems. Assume that the components have common reliability p. Manifestly $r_{1}=p$ , $r_{2}=2p-p^{2}$ , and $r_{1}^{\ast}=p^{2},r_{2}^{\ast}=p$ . Thus, from Corollary 5.1, the p.g.f. of the system is found to be

\begin{equation*}\psi(z)=\dfrac{c_{1}z+c_{2}z^{2}+c_{3}z^{3}}{1+d_{1}z+d_{2}z^{2}+d_{3}z^{3}},\end{equation*}

where

\begin{alignat*}{3}c_{1} & =1-2p^{2}+p^{3},&\quad c_{2} & =p(p^{4}-3p^{3}+2p^{2}+3p-3),&\quad c_{3} & =p^{2}(2-4p+3p^{2}-p^{3}),\\[3pt] d_{1} & =p^{2}-3p,&\quad d_{2} & =p(2p-p^{2}),&\quad d_{3} & =0.\end{alignat*}

6. Summary and conclusions

In this paper we have developed a method for computing the minimal signature of a coherent system by using matrix-geometric distributions. The method is mainly based on using the p.g.f. of a properly defined waiting time random variable in a sequence of binary trials. If the p.g.f. is rational, then the minimal signature is computed using distributional properties of the matrix-geometric distributions. In some setups (especially if the system is not defined in terms of its structure function), derivation of the probability generating function corresponding to the system failure is easier than obtaining the reliability function directly (see e.g. Examples 3.1 and 3.2). Indeed, the construction of the minimal path sets representation for the systems given in Examples 3.1 and 3.2 is quite a difficult task because of the complexity of the structures.

The proposed method has been modified to calculate the minimal signature of a system with two types of components. Moreover, in Proposition 5.1 and 5.2, respectively, we obtained the p.g.f. of a series system consisting of disjoint models and the p.g.f. of a series system of two modules containing a common component. As has been illustrated by several examples, the method is useful not only for a system with a single type of components but also for a system consisting of two (and possibly more) types of components.

The modification of the method for multi-state systems will be among our future research problems.