1 Introduction

The k-out-of-n (briefly denoted as k / n) systems are important redundancy structures in reliability engineering. A n-component system is said to be a k / n system if it operates as long as at least k components out of the n components operate. In these kind of systems all components have an equal portion to the performance of the entire system. Hence the number of working components specifies the operation of the system. The stochastic and aging properties of k / n systems have been extensively investigated in the reliability literature (see, for example, Kuo and Zuo 2003; Li and Zhao 2008; Eryilmaz 2011). Recently k / n systems are extended to weighted k / n systems where the weight associated with each component can be considered as load/capacity of that component. If the performance of the k / n system is characterized by the total weights of operating components, then the system is said to have a weighted k / n structure. In other words, a weighted k / n system has an operational level of at least k if the total weight of working components is k or more. For a specific example of a weighted k / n system in real life applications, we refer to Samaniego and Shaked (2008). The reliability assessment and stochastic properties of weighted k / n systems under the assumption of independence between the components lifetimes, are studied by several authors (see, among others, Ball et al. 1995; Chen and Yang 2005; Higashiyama 2001; Rushdi 1990; Wu and Chen 1994a, b). Eryilmaz (2015) studied the capacity loss and residual capacity in binary weighted-k-out-of-n:G systems. The mean instantaneous performance of the systems with weighted components was presented by Eryilmaz (2013). For an extensive survey on applications of the systems with weighted components see Samaniego and Shaked (2008). Recently, attempts have been made to study the properties of the weighted k / n systems consisting of dependent components. Copula functions are one of the popular applied methods for modeling dependence among lifetimes of the components. For a comprehensive study on the theory of copulas, we refer the reader to Nelsen (2006). Analysis of the systems reliability using copula functions can be found in Jia and Cui (2012), Navarro et al. (2010, 2017), Tang et al. (2013a, b) and Wang and Pham (2012).

According to the definition of a weighted k / n system, the system lies among the structures that are made of leastwise two different kinds of components (for more description and reliability properties of such systems, we refer the reader to Kochar and Xu 2010; Cui and Xie 2005; Navarro et al. 2015). Recently Eryilmaz and Sarikaya (2013) and Eryilmaz (2014), have considered a weighted k / n system including two kinds of components. Such a system is assumed to be composed of n components in which a fixed number m out of n is from a class \({\mathfrak {C}}_X\) of identically distributed components and the rest of \(n-m\) components are from a class \({\mathfrak {C}}_Y\) of identically distributed components. The cited authors have obtained some results on the stochastic and aging properties of the lifetimes of weighted k / n systems based on various copulas. Li et al. (2016) have studied on weighted k / n system with statistically dependent component lifetimes. Further resources on these subjects can also be found in Coolen and Coolen-Maturi (2012), Eryilmaz et al. (2018) and Samaniego and Navarro (2016).

Crescenzo (2007) presented the random strategy in the field of reliability theory. He showed that, in some situations, a random strategy might be a better option when our choice must be between two kinds of units such that one is more reliable than the other. Some useful applications related to the random strategy are in Navarro and Spizzichino (2010), Crescenzo and Pellerey (2011), Hazra and Nanda (2014), Navarro et al. (2015) and Hazra et al. (2017). Some applications of the above strategy may arise in reality. First, assume that a manufacturer uses a mix of two types of units for the production of its products. The reliability of manufactured units depends directly on the materials of the unit selected from each type and human factors, conditions of production, etc. The second application of the randomness is in physical systems e.g. stochastic resonance, Berdichevsky and Gitterman (1998) and Gammaitoni et al. (1998), Brownian ratchets, Bier (1997). The systems under the influence of the shock processes can be considered as the third example. In justifying the application of random theory Navarro et al. (2015) mentioned “Of course, the best systems are those which include only the best components. However, we shall assume that this option is not possible, maybe simply because we do not know which are the best components, and that we want to use both kinds of components. In these situations, random strategies may lead to the best systems.”

The goal of the present paper is to investigate the reliability and stochastic properties of weighted k / n systems which consist of a random number of components when the components are from two different types. In other words, we consider a weighted k / n system and assume that among the n number of the components of the system, a random number M, \(M=0,1,\dots ,{m},\) of components are chosen from class \({\mathfrak {C}}_X\) and the rest of \(n-M\) components are from class \({\mathfrak {C}}_Y\) where the components in \({\mathfrak {C}}_X\) (and \({\mathfrak {C}}_Y\)) are independent and identically distributed.

The paper is organized as follows: in Sect. 2, we recall some definitions and preliminary results that are useful in other sections. These include the definitions of some partial stochastic orders, the notion of “stochastic precedence” and the notion of copula. In Sect. 3, we first give a precise description of our proposed weighted k / n system. Then, we obtain the reliability of the system lifetime as a mixture representation. Several examples are also examined computationally and graphically in this section. Section 4 is devoted to comparisons between the lifetimes of two weighted k / n systems under different conditions. We first prove that if \(M_i\), \(i=1,2,\) components are chosen from \({\mathfrak {C}}_X\) such that \(M_1\) and \(M_2\) are stochastically ordered, then under some conditions the lifetimes of the systems consisting of \(M_1\) and \(M_2\), respectively, are also stochastically ordered. The rest of Sect. 4 is centered on comparison between two systems in the sense of stochastic precedence. In other words, we study conditions under which for two weighted k / n systems with lifetimes \(T_1\) and \(T_2\), \(P(T_1 \le T_2) \ge {1}/{2}\). Finally, some concluding remarks are presented in Sect. 5.

2 Preliminaries

In this section, we present some notions which are useful in deriving the main results. Throughout the paper, for a continuous random variable (r.v) X, let \({\bar{F}}\), f and \(\lambda _{F}={f}/{{\bar{F}}}\) be survival function, density function and failure rate function, respectively. For a continuous random variable Y, functions \({\bar{G}}\), g and \(\lambda _{G}={g}/{{\bar{G}}}\) are defined similarly.

Definition 1

The random variable X is said to be smaller than the random variable Y in the

  1. (i)

    usual stochastic order (denoted by \(X\le _{st}Y\)) if for all t, \({\bar{F}}(t)\le {\bar{G}}(t)\) ;

  2. (ii)

    hazard rate order (denoted by \(X\le _{hr}Y\)) if for all t, \(\lambda _{F}(t)\ge \lambda _{G}(t)\);

  3. (iii)

    likelihood ratio order (denoted by \(X\le _{lr}Y\)) if g(t) / f(t) is an increasing function of t.

It is well known that the following implications hold between these orderings:

$$\begin{aligned} X\le _{lr}Y\Rightarrow X\le _{hr}Y\Rightarrow X\le _{st}Y. \end{aligned}$$

The concepts of partial orderings are extended to the multivariate case. For n-dimensional random vector X, let \({\bar{F}}\) and f be survival function (i.e., \({\bar{F}}(\mathbf t )= P(\mathbf{X > \mathbf t })\)) and density function, respectively. For n-dimensional random vector Y, functions \({\bar{G}}\) and g are defined similarly. We denote \(\min \{\mathbf{t}_1,\mathbf{t}_2\}=(\min \{t_{1,1},t_{1,2}\}, \ldots , \min \{t_{n,1},t_{n,2}\})\) and \(\max \{\mathbf{t}_1,\mathbf{t}_2\}=(\max \{t_{1,1},t_{1,2}\}, \ldots , \max \{t_{n,1},t_{n,2}\})\).

  1. (i)

    The random vector \(\mathbf{X}\) is smaller than the random vector \(\mathbf{Y}\) in the multivariate stochastic order if

    $$\begin{aligned} P(\mathbf{X}\in \mathfrak {U})\le P(\mathbf{Y}\in \mathfrak {U}), \end{aligned}$$

    for all sets \(\mathfrak {U}\subseteq R^n\) (where \(\mathfrak {U}\) is an upper set).

  2. (ii)

    The random vector \(\mathbf{X}\) is said to be smaller than the random vector \(\mathbf{Y}\) in the multivariate hazard rate order if

    $$\begin{aligned} {\bar{F}}(\mathbf{t}_1){\bar{G}}(\mathbf{t}_2)\le {\bar{F}}(\min \{\mathbf{t}_1,\mathbf{t}_2\}){\bar{G}}(\max \{\mathbf{t}_1, \mathbf{t}_2\}) \quad \ {\ \forall \ \mathbf{t}_1, \mathbf{t}_2 \in {\mathbb {R}}^n,} \end{aligned}$$
  3. (iii)

    The random vector \(\mathbf{X}\) is said to be smaller than the random vector \(\mathbf{Y}\) in the multivariate likelihood ratio order if

    $$\begin{aligned} f(\mathbf{t}_1)g(\mathbf{t}_2)\le f(\min \{\mathbf{t}_1,\mathbf{t}_2\})g(\max \{\mathbf{t}_1,\mathbf{t}_2\}) \quad {\forall \ \mathbf{t}_1, \mathbf{t}_2 \in {\mathbb {R}}^n,} \end{aligned}$$

Standard references for more details on basic features and applications of these orders include Barlow and Proschan (1981), Li and Li (2013) and Shaked and Shanthikumar (2007).

Dealing with vectors of dependent random variables, a common approach to describe the dependence between the random variables is using the copula. Any joint distribution function H of a random vector \((T_{1},\ldots ,T_{n})\) with the marginal distribution functions \(F_{1},\ldots ,F_{n},\) can be written (by the Sklar theorem) as

$$\begin{aligned} H(t_1,\ldots ,t_n)=C_{\alpha }(F_1 (t_1) ,\ldots , F_n (t_n)), \end{aligned}$$
(1)

for all \(t_i \in {\mathbb {R}}, i = 1, \ldots , n\), where the n-dimensional function \(C_{\alpha }:[0,1]^n\rightarrow [0,1]\) is called a copula function. The quantity \(\alpha \) is the parameter of the copula which describes the dependency between \(T_{1},\ldots ,T_{n}\). Also, the copula \(C_{\alpha }\) is unique if marginals \(F_{1},\ldots ,F_{n}\) are continuous and is given by

$$\begin{aligned} C_{\alpha }(u_1, \ldots , u_n) = H(F_1^{-1}(u_1), \ldots , F_n^{-1}(u_n)), \end{aligned}$$

where \(F_i^{-1}(u)=inf\{t: F_i(t)\ge u\},\)\(i=1, \ldots , n.\) Conversely, the joint distribution of random vector \((T_{1},\ldots ,T_{n})\) can be determined by (1) if the marginal distributions of \(T_1,\ldots ,T_n\) and the copula function are known. For more details on copulas and their properties see, Nelsen (2006).

There are many parametric copula families in which these parameters control the strength of dependence between the marginals. Indeed, the copula parameter \(\alpha \) determines the properties of \(C_{\alpha }(u_1,\ldots ,u_n)\). In some cases, there is a one to one relationship between copula parameter and Kendall’s tau that allow to reparameterize parameter space into an easier interpretable one. For an n-dimensional copula \(C_{\alpha }(\mathbf u ), \mathbf u =(u_1,\ldots ,u_n)\), the Kendall’s \(\tau \) correlation is defined as follows

$$\begin{aligned} \tau _{n}(\alpha )=\frac{1}{2^{n-1}-1}\left[ 2^{n}\int _{0}^{1}\ldots \int _{0}^{1} C_{\alpha }(\mathbf u )dC_{\alpha }(\mathbf u )-1\right] , \end{aligned}$$

Genest et al. (2011). In the independence case, \(C_{\alpha }(\mathbf u )=u_1\ldots u_n\) and

$$\begin{aligned} \int _{0}^{1}\ldots \int _{0}^{1} C_{\alpha }(\mathbf u )dC_{\alpha }(\mathbf u )=1/2^n, \end{aligned}$$

which implies that \(\tau _{n}(\alpha )=0\). When the Kendall coefficient is given, the copula parameter can be achieved.

Among the class of copulas, an interesting and important one is the class of Archimedean copulas, which can be written as

$$\begin{aligned} C(u_1,\ldots ,u_n)=\varphi (\varphi ^{-1}(u_1)+\dots +\varphi ^{-1}(u_n)), \end{aligned}$$

where \(\varphi :[0,\infty ]\rightarrow [0,1]\) is a continuous and strictly decreasing function with \(\varphi (0)=1,\varphi (\infty )=0\) and the inverse function is denoted as \(\varphi ^{-1}\). The function \(\varphi \) is named Archimedean generator of C. Many classical copulas such as Ali-Mikhail-Haq, Clayton, Frank and Gumbel families belong to this class. An important member of Archimedean copulas family is the product copula i.e. a copula for the case of independence.

Now, we present the definition of symmetric copula.

Definition 2

For a given copula \(C(u_1,\ldots ,u_n)\), if

$$\begin{aligned}&C(u_1,\ldots ,u_{i-1}, u_i, u_{i+1},\ldots , u_{k-1}, u_k, u_{k+1},\ldots , u_n)\\&\quad =C(u_1,\ldots ,u_{i-1}, u_k, u_{i+1},\ldots , u_{k-1}, u_i, u_{k+1},\ldots , u_n), \end{aligned}$$

then \(u_i\) and \(u_k\) are said to be exchangeable. For any pair \(u_i, u_k \in [0,1]\), if \(u_i\) and \(u_k\) are exchangeable, then the copula \(C(u_1,\ldots ,u_n)\) is a symmetric copula.

In reliability, the dependence structure among the component lifetimes are commonly positive, so this positive dependence should be considered whenever we choose an appropriate copula. In this context we consider two families called Clayton and FGM, the first is an Archimedean copula but the latter is not, while both of them are symmetric copulas. For ease of reference, we recall the exact formula of the Clayton and FGM copulas.

  1. (i)

    The n-variate Clayton copula is defined by

    $$\begin{aligned} C_{\alpha }(u_{1},\ldots ,u_{n})=(u_{1}^{-\alpha }+\cdots +u_{n}^{-\alpha }-n+1)^{{-1}/{\alpha }} \quad \alpha \in [-\,1,\infty ) \backslash \lbrace 0\rbrace . \end{aligned}$$

    For this family of copulas the value of Kendall’s tau is given by

    $$\begin{aligned} \tau _{n}(\alpha )=\frac{1}{2^{n-1}-1}\left[ 2^n\prod _{i=0}^{n-1} \left( \frac{1+i\alpha }{2+i\alpha }\right) -1\right] . \end{aligned}$$
  2. (ii)

    The n-variate form of FGM family of copulas is defined by

    $$\begin{aligned} C_{\theta }(u_1,\ldots ,u_n)=\left( \prod _{i=1}^{n}u_i\right) \left\{ 1+\theta \prod _{i=1}^{n} (1-u_i)\right\} \quad \theta \in (-\,1,1). \end{aligned}$$

    In this case, the value of Kendall’s tau is given by

    $$\begin{aligned} \tau _{n}(\theta )=\frac{\theta }{3^n(2^{n-1}-1)}\{1+(-\,1)^n\}. \end{aligned}$$

    It is necessary to mention that the FGM copula describes positive (negative) dependence for \(\theta >0\)\((\theta <0)\).

3 System description

In the following, we consider a weighted k / n system consists of n dependent components. The components belong to two distinct classes \({\mathfrak {C}}_X=\{X_1,\ldots ,X_m\}\) and \({\mathfrak {C}}_Y=\{Y_{m+1},\ldots ,Y_n\}\) with sizes m and \(n-m\), respectively. Let \(F_X\) and \(F_Y\) denote the distribution functions of the components lifetime in \({\mathfrak {C}}_X\) and \({\mathfrak {C}}_Y\), respectively. We assume that the components have two states at any time, either working or failed. When a selected component from \({\mathfrak {C}}_{X}\)(\({\mathfrak {C}}_{Y}\)) is in a working state, it has the weight w\((w^*)\). So, the system is in performance level k if and only if the total weights of operating components is k or above. In other words, components of the system are chosen from two distinct classes of components such that m of them have the weight w with lifetime distribution \(F_X\), and the remaining \(n-m\) components have the weight \(w^*\) with lifetime distribution \(F_Y\). If \(X_1,\ldots ,X_m,Y_{m+1},\ldots ,Y_n\) denote the lifetimes of the system components in the two classes, then the total weight of the system is expressed by the stochastic process \(W_n(t)\) at time \(t\ge 0\),

$$\begin{aligned} W_n(t)=\sum _{i=1,\ldots ,m}{w I(X_i>t)}+\sum _{i=m+1,\ldots ,n}{w^* I(Y_i>t)}, \end{aligned}$$

where the symbol I(.) denotes the indicator random variable. The lifetime of the weighted \(k\)/\(n\) system, T, is

$$\begin{aligned} T=\inf \{t:W_n(t)<k\}. \end{aligned}$$

Hence the system reliability is

$$\begin{aligned} R(t)=P(T>t)=P(W_n(t)\ge k), \quad \forall t\ge 0. \end{aligned}$$

Suppose that the dependence between \(X_1,\ldots ,X_m,Y_{m+1},\ldots ,Y_n\) is modeled by the n-dimensional copula function C, i.e. the joint distribution function of \(X_1,\ldots ,X_m,Y_{m+1},\ldots ,Y_n\) is displayed as

$$\begin{aligned} H(t_1,\ldots ,t_n)=C(F_X (t_1) ,\ldots ,F_X (t_m) , F_Y (t_{m+1} ),\ldots , F_Y (t_n)). \end{aligned}$$

The reliability properties of a system with two different kind of dependent components, when dependency is modeled by copulas, was studied by Eryilmaz (2014).

In what follows, we consider the case when n-copula is symmetric and the size of two classes \({\mathfrak {C}}_X\) and \({\mathfrak {C}}_Y\) are random. Let M be a random variable with support contained in {0,1,...,n}. If m (the number of components from \({\mathfrak {C}}_X\)) is selected randomly according to the random variable M, then the system reliability function can be represented as

$$\begin{aligned} R_M (t)=\,&P(T_M>t)\\ =&\sum _{m=0}^n{P\left( w\sum _{i=1,\ldots ,m}I({X_i}>t)+{w^*}\sum _{i=m+1, \ldots ,n}I({Y_i}>t)\ge k\right) P(M=m)}\\ =&\sum _{m=0}^n{R_m(t)P(M=m)}\\ =&\sum _{m=0}^n{\mathop {\sum \sum }_{\begin{array}{c} {wy+{w^*}z \ge k}\\ 0\le y \le m\\ 0\le z\le n-m \end{array}}}\left( {\begin{array}{c}m\\ y\end{array}}\right) \left( {\begin{array}{c}n-m\\ z\end{array}}\right) \sum _{l=0}^y \sum _{s=0}^z(-1)^{(l+s)} \left( {\begin{array}{c}y\\ l\end{array}}\right) \left( {\begin{array}{c}z\\ s\end{array}}\right) \\&{\times }C(\underbrace{F_X(t),\ldots ,F_X(t)}_{(m-y+l)\text {times}} , \underbrace{F_Y(t),\ldots ,F_Y(t)}_{(n-m-z+s)\text {times}})\\&{\times } P(M=m). \end{aligned}$$

The last equality follows from relation (4) in Eryilmaz (2014).

Let us look at the following examples which rely on the notations introduced just now.

Fig. 1
figure 1

The systems reliability a related to Example 1, b related to Example 2

Full size image

Example 1

Consider two weighted 3 / 4 systems with \(w=2\), \(w^*=1\), \(F_X (t)=1-\exp \{-\,0.1t\}\), \(F_Y (t)=1-\exp \{-\,0.4t\}\). Let the dependence structure among components be generated by FGM copula. Assume that in the first (second) system the number of components from \(C_X\) is selected randomly according to random variables \(M_1\) (\(M_2\)). If the random variables \(M_1\) and \(M_2\) follow Binomial distribution B(4, 0.1) and B(4, 0.9), respectively, then for \(j=1,2\),

$$\begin{aligned} R_{M_i}(t)=&\sum _{m=0}^4{P\left( {2}\sum _{i=1,\ldots ,m}{I(X_i>t)} +\sum _{i=m+1,\ldots ,n}{I(Y_i>t)}\ge 3\right) P(M_{j}=m)}\\ =&\sum _{m=0}^4{R_m(t) P(M_{j}=m)}\\ =&\sum _{m=0}^4{\mathop {\sum \sum }_{\begin{array}{c} {2y+z \ge 3}\\ 0\le y\le m\\ 0\le z\le 4-m \end{array}}}\left( {\begin{array}{c}m\\ y\end{array}}\right) \left( {\begin{array}{c}4-m\\ z\end{array}}\right) \sum _{l=0}^y\sum _{s=0}^z(-1)^{(l+s)} \left( {\begin{array}{c}y\\ l\end{array}}\right) \left( {\begin{array}{c}z\\ s\end{array}}\right) \\ {}&{\times }(1-\exp \{-0.1t\})^{m-y+l}(1-\exp \{-0.4t\})^{4-m-z+s}\\ {}&{\times }(1+\theta (\exp \{-0.1t\})^{m-y+l}(\exp \{-0.4t\})^{4-m-z+s}) P(M_{j}=m) \end{aligned}$$

Figure 1a, shows reliability results of these systems when \(\tau =0.001\).

The following example considers the case where the components are independent (i.e. the product copula is used).

Example 2

Consider two weighted 3 / 3 systems and assume that \(w=3\), \(w^*=2\), \(F_X (t)=1-\exp \{-\,0.2\sqrt{t}\}\), \(F_Y (t)=1-\exp \{-\,0.3\sqrt{t}\}\) where the components are assumed to be independent. Let the number of components of the first (second) system be selected randomly from \(C_X\) according to the random variable \(M_1\) (\(M_2\)). If \(M_1\) and \(M_2\) are chosen randomly from {0,1,2,3} with probabilities \(\{{5}/{26},{6}/{26},{7}/{26},{8}/{26}\}\) and \(\{{1}/{10},{2}/{10},{3}/{10},{4}/{10}\}\), respectively, then

$$\begin{aligned} {R_{M_1}(t)}=&\sum _{m=0}^3{\mathop {\sum \sum }_{\begin{array}{c} {3a+2b \ge 3}\\ 0\le a\le m\\ 0\le b\le 3-m \end{array}}}\left( {\begin{array}{c}m\\ a\end{array}}\right) \left( {\begin{array}{c}3-m\\ b\end{array}}\right) (\exp \{-0.2\sqrt{t}\})^a(1-\exp \{-0.2\sqrt{t}\})^{m-a}\\ {}&{\times }(\exp \{-0.3\sqrt{t}\})^b(1-\exp \{-0.3\sqrt{t}\})^{3-m-b} {(m+5)}/{26} \end{aligned}$$

and

$$\begin{aligned} {R_{M_2}(t)}=&\sum _{m=0}^3{\mathop {\sum \sum }_{\begin{array}{c} {3a+2b \ge 3}\\ 0\le a\le m\\ 0\le b\le 3-m \end{array}}}\left( {\begin{array}{c}m\\ a\end{array}}\right) \left( {\begin{array}{c}3-m\\ b\end{array}}\right) (\exp \{-0.2\sqrt{t}\})^a(1-\exp \{-0.2\sqrt{t}\})^{m-a}\\ {}&{\times }(\exp \{-0.3\sqrt{t}\})^b(1-\exp \{-0.3\sqrt{t}\})^{3-m-b} {(m+1)}/{10} \end{aligned}$$

In Fig. 1b, the reliability plots of these systems are depicted.

4 Comparison between two systems based on usual stochastic ordering

In this section we consider two independent weighted k / n systems with lifetimes \(T_1\) and \(T_2\). We aim to investigate conditions under which the reliability of the systems are ordered in two cases. In the first case, let \(T_i\) be the lifetime of the system built up in the case when \(M_i\) components are chosen randomly from the class \({\mathfrak {C}}_{X_i},\)\(i=1,2.\) The reliability of the two systems are compared in Theorem 1. In the second case, let \(T_i\) be the lifetime of the system built up in the case when the distribution functions of the components in \({\mathfrak {C}}_{X_i}\) and \({\mathfrak {C}}_{Y_i}\) are stochastically ordered, \(i=1,2.\) A comparison of systems reliability is shown in Theorem 2. To organize the proof of the theorems, we present the following lemmas.

Lemma 1

Let \(X_1,\ldots , X_n\) be continuous random variables with symmetric copula \(C_{X_1,\ldots ,X_n}\), marginal distribution functions \(F_{X_1},\ldots ,F_{X_n}\), and joint distribution function \(H_{X_1,\ldots ,X_n}\). Let \(Z_i=I(X_i>t)\), \(i=1,\ldots ,n\). Then there exists a symmetric selection for the copula \(C_{Z_1,\ldots ,Z_n }\).

Proof

We prove this for the case of 2-copula. The proof for the case of n-copula is similar. First notice that

$$\begin{aligned} H_{Z_1,Z_2} (s_1,s_2)=\,&P(I(X_1>t)\le {s_1},I(X_2>t)\le {s_2})\\ =&{\left\{ \begin{array}{ll} 0 &{} \text {for }{s_1}<0 \, \text {or}\, {s_2}<0,\\ C_{X_1,X_2} (F_{X_1} (t),F_{X_2} (t)) &{} \text {for }0\le {s_1}<1,0\le s_2<1,\\ C_{X_1,X_2} (F_{X_1} (t),1) &{} \text {for }0\le {s_1}<1,{s_2}\ge 1,\\ C_{X_1,X_2} (1,F_{X_2} (t)) &{} \text {for }{s_1}\ge 1,0\le {s_2}<1,\\ C_{X_1,X_2} (1,1) &{} \text {for }{s_1}\ge 1,{s_2}\ge 1. \end{array}\right. } \end{aligned}$$

Define subcopula \(C^\prime _{Z_1,Z_2}\) as

$$\begin{aligned} C^\prime _{Z_1,Z_2}(u_1,u_2)={\left\{ \begin{array}{ll} 0 &{} \text {for }u_1=0\ \text {or}\ u_2=0, \\ C_{X_1,X_2} (F_{X_1} (t),F_{X_2} (t)) &{} \text {for }u_1=F_{X_1} (t), u_2=F_{X_2} (t), \\ C_{X_1,X_2} (F_{X_1} (t),1) &{} \text {for }u_1=F_{X_1} (t),u_2=1,\\ C_{X_1,X_2} (1,F_{X_2} (t)) &{} \text {for }u_1=1,u_2=F_{X_2} (t),\\ C_{X_1,X_2} (1,1) &{} \text {for }u_1=1,u_2=1. \end{array}\right. } \end{aligned}$$

Let D be the set of all permutations of all points in \(RanF_{Z_1} \times RanF_{Z_2}.\) Define subcopula \(C^{\prime \prime }_{Z_1,Z_2}\) on D in the manner that the value of \(C^{\prime \prime }_{Z_1,Z_2}\) is the same for all permutations of a point. It is possible since the copula \(C_{X,Y}\) is symmetric. That is,

$$\begin{aligned} C^{\prime \prime }_{Z_1,Z_2}(u_1,u_2)={\left\{ \begin{array}{ll} 0 &{} \text {for }u_1=0\ \text {or}\ u_2=0, \\ C_{X_1,X_2} (F_{X_1} (t),F_{X_2} (t)) &{} \text {for }u_1=F_{X_1} (t), u_2=F_{X_2} (t), \\ C_{X_1,X_2} (F_{X_1} (t),F_{X_2} (t)) &{} \text {for }u_1=F_{X_2} (t), u_2=F_{X_1} (t), \\ C_{X_1,X_2} (F_{X_1} (t),1) &{} \text {for }u_1=F_{X_1} (t),u_2=1,\\ C_{X_1,X_2} (F_{X_1} (t),1) &{} \text {for }u_1=1,u_2=F_{X_1} (t),\\ C_{X_1,X_2} (1,F_{X_2} (t)) &{} \text {for }u_1=1,u_2=F_{X_2} (t),\\ C_{X_1,X_2} (1,F_{X_2} (t)) &{} \text {for }u_1=F_{X_2} (t),u_2=1,\\ C_{X_1,X_2} (1,1) &{} \text {for }u_1=1,u_2=1. \end{array}\right. } \end{aligned}$$

Then the extension of \(C^{\prime \prime }_{Z_1,Z_2}\) via Sklar theorem is a symmetric copula named it \(C_{Z_1,Z_2}\). The copula \(C_{Z_1,Z_2}\) is a function that links \(F_{Z_1}\) and \(F_{Z_2}\) to \(H_{Z_1,Z_2}\). \(\square \)

Lemma 2

Assume that \(R_m\) represents the reliability of a weighted k / n system with two distinct classes \({\mathfrak {C}}_X\) and \({\mathfrak {C}}_Y\) where the size of the class \({\mathfrak {C}}_X\) is m. Suppose that the components of the system are dependent with a symmetric copula. Let also w and \(w^* \) be the weights of the components and \({\bar{F}}_X\) and \({\bar{F}}_Y\) denote the reliability functions of the components in \({\mathfrak {C}}_X\) and \({\mathfrak {C}}_Y\), respectively. If \(w\le w^*\) and \({\bar{F}}_X(t)\le {\bar{F}}_Y(t)\), for all t, then \(R_{m} \le R_{m-1},\) for all \(m=0,\ldots ,n,\) with convention \(R_{-1}=1,\) where

$$\begin{aligned} R_m(t)=P\left( w\sum _{i=1}^m I(X_i>t) + {w^*}\sum _{i=1}^{n-m} I(Y_i>t) \ge k\right) . \end{aligned}$$

Proof

First note that under the assumptions \(w\le w^*\) and \({\bar{F}}_X(t)\le {\bar{F}}_Y(t)\), for all t, we have for all values \(s\in R^+\)

$$\begin{aligned} P(wI(X>t)\ge s)\le P(w^* I(Y>t)\ge s). \end{aligned}$$
(2)

From (2), it follows that \( V_m (t)\le _{st} W_{n-m+1}(t)\) where \(V_i(t)=wI(X_i>t),\)\(i=1,\ldots ,m\) and \(W_i(t)=w^{*} I(Y_i>t),\)\(i=1,\ldots ,n-m+1.\) It is well-known that for a symmetric n-copula C, \(n\ge 3\), all its \((n -1)\)-margins are symmetric and they are equal. From this fact and Lemma 1

$$\begin{aligned} C_{V_1,\ldots ,V_{m-1},W_1,\ldots ,W_{n-m},W_{n-m+1} }=C_{V_1,\ldots ,V_{m-1},V_m,W_1,\ldots ,W_{n-m}}.\nonumber \end{aligned}$$

Put

$$\begin{aligned} W_{n,m}(t) =\sum _{i=1}^m{V_i(t)} + \sum _{i=1}^{n-m}{W_i(t)}, \end{aligned}$$

and

$$\begin{aligned} W_{n,m-1}(t) =\sum _{i=1}^{m-1}{V_i(t)}+ \sum _{i=1}^{n-m+1}{W_i(t)}. \end{aligned}$$

Then from Theorems 6.B.14 and 6.B.16.(a) of Shaked and Shanthikumar (2007), we have

$$\begin{aligned} W_{n,m}(t)\le _{st} W_{n,m-1}(t). \end{aligned}$$

That is, \(R_m\le R_{m-1},\) for all \(m=1,\ldots ,n\). This completes the proof. \(\square \)

Now, we are ready to prove the main theorems.

Theorem 1

Suppose that \(T_i\), \(i=1,2,\) represent the lifetime of a weighted k / n system where the system components are dependent with a symmetric copula and the components are chosen randomly from two distinct classes \({\mathfrak {C}}_X\) and \({\mathfrak {C}}_Y\). Let also w and \(w^* \) be the weights of the components and \({\bar{F}}_X\) and \({\bar{F}}_Y\) denote the reliability functions of the components in \({\mathfrak {C}}_X\) and \({\mathfrak {C}}_Y\). Assume that the random variables \(M_1\) and \(M_2\) are the number of selected components from the set \(C_X\) in the first and second system, respectively.

  1. (i)

    If \(M_1 \le _{st} M_2, w\le w^*\) and for all t, \({\bar{F}}_X(t)\le {\bar{F}}_Y(t)\), then \(T_1 \ge _{st} T_2\).

  2. (ii)

    If \(M_1 \le _{st} M_2, w\ge w^*\) and for all t, \({\bar{F}}_X(t)\ge {\bar{F}}_Y(t)\), then \(T_1 \le _{st} T_2\).

Proof

We only prove part (i), the proof of part (ii) is similar. We have,

$$\begin{aligned} R_M (t)=&\sum _{m=0}^n{R_m (t)P(M=m)}\\ =&\sum _{m=0}^{n-1}{R_m (t)[P(M\ge m)-P(M\ge m+1)]+R_n(t) P(M=n)}\\ =&\sum _{m=0}^{n}(R_m(t)-R_{m-1}(t)) P(M\ge m), \end{aligned}$$

with convention \(R_{-1}=1\). From, \(M_1 \le _{st} M_2,\) we have for all m, \(P(M_1\ge m)\le P(M_2\ge m).\) Also from Lemma 2, \(R_m\le R_{m-1},\)\(m=1,\ldots ,n\). Hence,

$$\begin{aligned} (R_m (t)-R_{m-1} (t))P(M_1\ge m)\ge (R_m (t)-R_{m-1} (t))P(M_2\ge m), m=1,\ldots ,n, \end{aligned}$$

which, in turn, implies that, for all t, \(R_{M_1} (t)>R_{M_2} (t)\). \(\square \)

Fig. 2
figure 2

The system reliability for system in Example 3

Full size image

The following example gives an illustration of Theorem 1.

Example 3

Consider a weighted 2 / 2 system with \(w=1\), \(w^*=2\), \(F_X (t)=1-\exp \{-\,0.2t\}\), \(F_Y (t)=1-\exp \{-\,0.1t\}\) and assume that the dependence structure among components is generated by Clayton copula. We consider two different cases for random variable M. In the first one we assume that \(M=M_1\) takes its values randomly from {0,1,2} with uniform probabilities and in the second case \(M=M_2\) takes its values on the same set with probabilities \(\{{1}/{6},{2}/{6},{3}/{6}\}\).

We leave it to the reader to verify that \(M_1\le _{st} M_2\) and \({\bar{F}}_{X}(t)\le {\bar{F}}_{Y}(t)\). So the conditions of Theorem 1 are satisfied. Hence \(R_{M_1}(t)\ge R_{M_2}(t)\). The plots of the reliability functions of the systems are depicted in Fig. 2 when \(\tau =0.5\). In the case that the Clayton copula evinces negative dependency, the obtained result is similar to the positive dependency.

To complete the discussion, hazard rate (hr) and likelihood ratio (lr) orders are investigated as well. In this case, it is seen that the ordering for M is not transferred to the system lifetime \(T_M\). This is shown in the next example.

Example 4

Consider a weighted 1 / 1 system with \(w=2\), \(w^*=1\), \(F_X (t)=1-\exp \{-\,0.1t\}\), \(F_Y (t)=1-\exp \{-\,0.2t\}\). If \(M=M_1\) takes its values randomly from {0,1} with probabilities {0.1,0.9} and \(M=M_2\) takes its values on the same set with probabilities {0.00901,0.990990}, it is plain to see \(M_1\le _{hr} M_2\) and \(X \ge _{hr} Y\). However, a situation can be found in which \(T_{M_1}\le _{hr} T_{M_2}\) doesn’t hold, i.e., \(P(T_{M_1}\ge t_2)P(T_{M_2}\ge t_1)\ge P(T_{M_1}\ge t_1)P(T_{M_2}\ge t_2), \forall t_1\le t_2\). It will be established if take \(t_1=3.465\) and \(t_2=4.581\). Since the hr order is the necessary condition for lr order and in this example, \(M_1\le _{lr} M_2\) and \(X \ge _{lr} Y\), hence \(T_{M_1}\le _{lr} T_{M_2}\) doesn’t hold.

In the following, we explore the conditions under which the stochastic ordering between the components of two weighted k / n systems imply the stochastic ordering between the lifetimes of the systems.

Theorem 2

Suppose that \(T_j\) represents the lifetime of a weighted k / n system with two classes \({\mathfrak {C}}_{X_j}\) and \({\mathfrak {C}}_{Y_j}\) where the size of the class \({\mathfrak {C}}_{X_j}\) is random, \(j=1,2\). Let M be a random variable with support contained in \(\{0,1,\ldots ,n\}\). Suppose the size of the class \({\mathfrak {C}}_{X_j}\) is selected randomly according to the random variable M. Moreover, Let \(F_{X_j}\) and \(F_{Y_j}\) indicate the distribution functions of the components in \({\mathfrak {C}}_{X_j}\) and \({\mathfrak {C}}_{Y_j}\). Assume that when a selected component from \({\mathfrak {C}}_{X_j}({\mathfrak {C}}_{Y_j})\) lies in a working situation, it has the weight w\((w^*)\), \(j=1,2\). Let the dependence structure among components be built with a common symmetric copula. If \(X_1\ge _{st} X_2\) and \(Y_1\ge _{st} Y_2,\) then \(T_1 \ge _{st} T_2\).

Proof

Let \(X_{ij}\) and \(Y_{lj}\) be the lifetimes of the components in the classes \({\mathfrak {C}}_{X_j}\) and \({\mathfrak {C}}_{Y_j}\), \(j=1,2, i=1,\ldots ,m\) and \(l=1,\ldots ,n-m\). It suffices to show that

$$\begin{aligned}&\left( w\sum _{\begin{array}{c} i=1\\ i\in {\mathfrak {C}}_{X_1} \end{array}}^mI({X_{i1}}>t)+{w^*}\sum _{\begin{array}{c} i=1\\ i\in {\mathfrak {C}}_{Y_1} \end{array}}^{n-m}I({Y_{i1}}>t)\right) \\&\quad \ge _{st}\left( w\sum _{\begin{array}{c} i=1\\ i\in {\mathfrak {C}}_{X_2} \end{array}}^mI({X_{i2}}>t)+{w^*}\sum _{\begin{array}{c} i=1\\ i\in {\mathfrak {C}}_{Y_2} \end{array}}^{n-m}I({Y_{i2}}>t)\right) ,\nonumber \end{aligned}$$

for each fixed m. According to the assumption

$$\begin{aligned} {X_{i1}\ge _{st} X_{i2},\quad Y_{l1}\ge _{st} Y_{l2},\quad i=1,\ldots ,m,l=1,\ldots ,n-m.} \end{aligned}$$
(3)

Then from what we have argued in Theorem 1, \(wI({X_{i1}}>t)\ge _{st} wI({X_{i2}}>t)\) and \(w^*I({Y_{l1}}>t)\ge _{st} w^*I({Y_{l2}}>t).\) From Lemma 1 the components in random vectors

$$\begin{aligned} (I({X_{11}}>t),\ldots ,I({X_{m1}}>t),I({Y_{11}}>t), \ldots ,I({Y_{(n-m)1}}>t)) \end{aligned}$$

and

$$\begin{aligned} (I({X_{12}}>t),\ldots ,I({X_{m2}}>t),I({Y_{12}}>t), \ldots ,I({Y_{(n-m)2}}>t)) \end{aligned}$$

are dependent with a symmetric common copula. Then, from the inequality (3), the result follows using Theorems 6.B.14 and 6.B.16.(a) of Shaked and Shanthikumar (2007). \(\square \)

Remark 1

Note that the random variables \(T_1\) and \(T_2\) in part (i) of Theorem 1 and Theorem 2 are independent and since

$$\begin{aligned} P(T_1 \ge T_2)=\int _{0}^{\infty }P(T_1 \ge t) d{F_{T_2} (t)}, \end{aligned}$$

then it is easy to see that the usual stochastic order implies the sp order (see Arcones et al. 2002; Santis et al. 2015). Recall that for two random variables \(X_1\) and \(X_2\), the random variable \(X_1\) is said to be stochastically precede \(X_2\) (written \(X_1 \le _{sp} X_2\)) if and only if \(P(X_1 \le X_2)\ge {{1}/{2}}\). The random variables \(X_1\) and \(X_2\) are sp-equivalent if and only if they satisfy \(P(X_1 \le X_2)={{1}/{2}}\).

In the following, we give two examples to measure the difference between \(P(T_1\ge T_2)\) and 1 / 2 in two cases. The first case is when the dependence structure is Clayton and the second one is when the component lifetimes are independent.

Example 5

Consider a weighted k / 10 system for \(k=1,\ldots ,11\) with \(w=1\), \(w^*=2\), \(F_{X_1} (t)=1-\exp \{-\,0.1t\}\), \(F_{Y_1} (t)=1-\exp \{-\,0.2t\}\), \(F_{X_2} (t)=1-\exp \{-\,0.5t\}\), \(F_{Y_2} (t)=1-\exp \{-\,0.6t\}\). So the conditions of Theorem 2 are satisfied. Thus by Remark 1, we have \(P(T_1 \ge T_2)\ge {1}/{2}\). Now, assume that the dependence structure among components is generated by Clayton copula when \(\tau =0.2\) and \(\tau =0.5\). In the case that the random variable M follows Binomial distribution B(10, 0.7), the values of \(P(T_1\ge T_2)\) for \(k=1,\ldots ,11\) are calculated. The corresponding results are presented in Table 1.

Example 6

Consider a weighted k / 10 system for \(k=1,\ldots ,11\) with \(w=1\), \(w^*=2\), \(F_{X_1} (t)=1-\exp \{-\,0.1t\}\), \(F_{Y_1} (t)=1-\exp \{-\,0.2t\}\), \(F_{X_2} (t)=1-\exp \{-\,0.5t\}\), \(F_{Y_2} (t)=1-\exp \{-\,0.6t\}\). Again the conditions of Theorem 2 are satisfied. Then Remark 1 gives \(P(T_1 \ge T_2)\ge {1}/{2}\). Here, assume that the components are independent and the random variable M follows Binomial distribution B(10, 0.7). The third column in Table 1 shows \(P(T_1\ge T_2)\) for \(k=1,\ldots ,11\).

Table 1 The values of \(P(T_1\ge T_2){:}\) the first column (\(\tau =0.2\)), the second column (\(\tau =0.5\)), for system in Example 5, the third column (independence) for system in Example 6
Full size table

5 Conclusion

In this paper, we studied some reliability and stochastic properties of weighted k-out-of-n systems. We assumed that the system is built up from two different classes of components that are classified with respect to the weights and reliability functions of the components. It was assumed that a random number of components, say M, are chosen from the first class and the rest of components are selected from the second class. Different copula functions were considered for modeling the structure of dependency of the system component lifetimes. The reliability of the system, was expressed as a mixture of the reliability of weighted k-out-of-n systems, with fixed number of components of the two types, in terms of the probability mass function of the random variable M. We showed that when the random mechanism of the chosen components for different two weighted k-out-of-n systems are ordered in usual stochastic (st) order then, under some conditions on the weights and the distributions of components lifetime, the lifetimes of the two systems are also ordered in st order. In addition, the lifetimes of two different systems were compared in the sense of stochastic precedence concept. We presented some illustrative examples for the results under different conditions.