2. The FR and RFR for coherent systems

In the present section, some results on the behavior of the FR and the RFR functions of coherent systems are provided. In a k-out-of-n system, it is known that if the component lifetimes are i.i.d. according to an IFR distribution, then the system’s lifetime is also IFR (see [Reference Barlow and Proschan1]). Samaniego [Reference Samaniego18] proved the result for a mixed system under a condition on the structure of the system. Navarro et al. [Reference Navarro, Sordo and Suárez-Llorens15] studied conditions for the preservation of the IFR (and other reliability classes) under the formation of coherent systems based on the domination function. We now begin by establishing a theorem in this regard. The result gives a simpler sufficient condition for a coherent (or mixed) system with IFR (DFR) component lifetimes to be IFR (DFR). An analogous result determining the behavior of the RFR function of a coherent structure can also be obtained; see Theorem 2.2 below.

Proof. The FR of a mixed system can be rephrased as [Reference Samaniego18]

\begin{equation*}h_{T}(t)=\dfrac{\sum_{i=0}^{n-1}(n-i)s_{i+1}\binom{n}{i}u_{t}^{i}}{\sum_{i=0}^{n-1} (\!\sum_{j=i+1}^{n}s_{j})\binom{n}{i}u_{t}^{i}}h(t),\end{equation*}

where $u_{t}=F(t)/\bar{F}(t)$ represents the odds of failure versus survival. To prove the result, we need to show that the function $\xi$ , given by

\begin{equation*}\xi(x,k)=\sum_{i=0}^{n-1}c_{i,k}\binom{n}{i}x^{i} {,}\end{equation*}

is $\mathrm{TP}_{2}$ $(\mathrm{RR}_{2})$ in $(x,k)\in[0,\infty)\times\{0,1\}$ , where

\begin{equation*}c_{i,k}=\begin{cases} \sum_{j=i+1}^{n}s_{j}& k=0, \\ (n-i)s_{i+1}& k=1.\end{cases}\end{equation*}

It can be easily observed that $x^{i}$ is $\mathrm{TP}_{2}$ in $(x,i)\in[0,\infty)\times\{0,1,\ldots,n-1\}$ . Under the assumption of the theorem, $c_{i,k}$ is $\mathrm{TP}_{2}$ $(\mathrm{RR}_{2})$ in $(i,k)\in\{0,1,\ldots,n-1\}\times\{0,1\}$ . The required result then follows from Lemma 1.1.

An analog of Theorem 2.1 about the preservation of DRFR class under the formation of coherent systems is as follows.

Proof. One can show that the RFR of the system can be expressed as

\begin{equation*}r_{T}(t)=\dfrac{\sum_{i=1}^{n}i s_{i}\binom{n}{i}u_{t}^{i}}{\sum_{i=1}^{n} \big(\!\sum_{j=1}^{i}s_{j}\big)\binom{n}{i}u_{t}^{i}} r(t),\end{equation*}

where $u_t=F(t)/\bar{F}(t)$ and r(t) denotes the common RFR of component lifetimes. Define the function $\tilde{\xi}(x,k)=\sum_{i=1}^{n}\tilde{c}_{i,k}\binom{n}{i}x^{i}$ , where

\begin{equation*}\tilde{c}_{i,k}=\begin{cases} \sum_{j=1}^{i}s_{j} & k=0, \\ is_{i}& k=1.\end{cases}\end{equation*}

The result then follows from Lemma 1.1 on noting that under the assumption of the theorem, $x^{i}$ is $\mathrm{TP}_{2}$ in $(x,i)\in[0,\infty)\times\{1,2,\ldots,n\}$ and $\tilde{c}_{i,k}$ is $\mathrm{RR}_{2}$ in $(i,k)\in\{1,2,\ldots,n\}\times\{0,1\}$ .

Some important facts regarding Theorems 2.1 and 2.2 are demonstrated in the following examples. In the first example, it is seen that the conditions of the cited theorems on the system signature are not necessary for the results to hold.

Example 2.1. Consider the bridge system pictured in Figure 1. Suppose the component lifetimes follow a Weibull distribution $W(\alpha,\beta)$ with reliability function $\bar{F}(t)=\exp\{-(t/\beta)^\alpha\}$ , $t\geq 0$ , where $\alpha,\beta>0$ . Figure 2 depicts the system FR for different values of $\alpha$ and $\beta$ . It can be shown that the system signature $\boldsymbol{s}=(0,0.2,0.6,0.2,0)$ satisfies neither the condition of Theorem 2.1 nor that of Theorem 2.2. However, as seen in Figure 2, the FR of the system with IFR (DFR) Weibull components may be increasing (decreasing). Moreover, one can observe in Figure 3 that the RFR of the system with such components is decreasing. Note that for the case where the component lifetimes are distributed as $W(0.8,0.9)$ , the system has an upside-down bathtub-shaped FR.

Figure 1:

The bridge system.

Figure 2:

The FR of the bridge system with different Weibull component lifetimes.

Figure 3:

The RFR of the bridge system with different Weibull component lifetimes.

For all coherent structures with 1–4 components, the signature vectors of order 4 are provided by Navarro and Rubio [Reference Navarro and Rubio13]. They are given in Table 1, in which we have determined whether each system satisfies the assumptions of Theorems 2.1 and 2.2. It can be seen that of 28 coherent systems listed in Table 1, only six systems do not fulfill the stated assumption in Theorem 2.1. The same is also true for the assumption of Theorem 2.2.

Table 1.

Signature vectors of order 4 for all coherent systems with 1–4 components

It is well known that there does not exist any lifetime distribution whose RFR function is strictly increasing on its interval of support (see [Reference Block, Savits and Singh2]). In the following theorem, we have proved a stronger result implying that the RFR of any continuous lifetime distribution must be first decreasing at the beginning of the interval of support. In other words, such a distribution is initially DRFR.

Proof. For $\delta=b-a$ , the theorem is essentially a result of Block, Savits, and Singh [Reference Block, Savits and Singh2]. Thus, assume that $\delta<b-a$ , which implies that $\delta<\infty$ .

Let r(t) denote the RFR of X and assume that it is increasing on $(a,a+\delta)$ . It follows that

\begin{equation*}\int_{t}^{a+\delta}r(x)\,{\mathrm{d}} x \leq \int_{t}^{a+\delta}r(a+\delta)\,{\mathrm{d}} x=(a+\delta-t)r(a+\delta)\end{equation*}

for all $t\in(a,a+\delta)$ . This, in turn, implies that

\begin{equation*}F(a+\delta)\leq F(t)\,{\mathrm{e}}^{(a+\delta-t)r(a+\delta)}.\end{equation*}

Taking the limit as $t\rightarrow a^{+}$ and using the continuity of F, we have $F(a+\delta)\leq 0$ . Therefore $F(a+\delta)=0$ , which is a contradiction to the assumption $a=\inf\{x\colon F(x)>0\}$ .

It follows from Theorem 2.3 that there is no distribution (with a non-negative left extremity of the support) with an upside-down bathtub-shaped RFR.

If $\bar{F}(t)$ is strictly concave at any point $t_0$ , then the FR h(t) is strictly increasing at $t_0$ . Similarly, the RFR r(t) is strictly decreasing at $t_0$ when F(t) is strictly concave at $t_0$ . The following lemma reveals that, under some conditions, the mixture of distributions is IFR and DRFR. We shall use this lemma to determine the local behavior of the FR and RFR of a coherent system.

Lemma 2.1. Consider the mixture distribution

\begin{equation*}F(t)=\int_AF_\theta(t) \,{\mathrm{d}} G(\theta),\end{equation*}

where $\{F_\theta,\ \theta\in A\}$ is a family of sufficiently smooth (continuous, second time-derivatives) distributions. Let $\bar{F}$ and $\bar{F}_\theta$ denote the reliability functions corresponding to F and ${F}_\theta$ , respectively.

1. (a) If each $\bar{F}_\theta$ is concave in a neighborhood of any point $t_0$ , and if at least one of the $\bar{F}_\theta$ is strictly concave in this neighborhood, then $\bar{F}$ is strictly concave in this neighborhood and the corresponding failure rate is strictly increasing at $t_0$ .

2. (b) If each $F_\theta$ is concave in a neighborhood of $t_0$ , and if at least one of the $F_\theta$ is strictly concave in this neighborhood, then F is strictly concave in this neighborhood and the corresponding reversed failure rate is strictly decreasing at $t_0$ .

Proof. Part (a) is due to Klutke, Kiessler, and Wortman [Reference Klutke, Kiessler and Wortman7], but for the sake of completeness, we sketch the proof here. Let $h_\theta$ and $r_\theta$ denote the failure rate and the reversed failure rate functions corresponding to $F_\theta$ , respectively. First, for each $\theta\in A$ , observe that

\begin{equation*}h^{\prime}_\theta(t_0)=h^2_\theta(t_0)-\dfrac{\bar{F}^{\prime\prime}_\theta(t_0)}{\bar{F}_\theta(t_0)}.\end{equation*}

If $\bar{F}_\theta$ is strictly concave at $t_0$ , then $\bar{F}^{\prime\prime}_\theta(t_0)<0$ and hence $h^{\prime}_\theta(t_0)>0$ . Therefore, under the assumptions of the lemma, $\bar{F}$ is strictly concave at $t_0$ . Similarly,

\begin{equation*}r^{\prime}_\theta(t_0)=\dfrac{{F}^{\prime\prime}_\theta(t_0)}{{F}_\theta(t_0)}-r^2_\theta(t_0),\end{equation*}

and if ${F}_\theta$ is strictly concave at $t_0$ , then ${F}^{\prime\prime}_\theta(t_0)<0$ and $r^{\prime}_\theta(t_0)<0$ . Therefore, under the stated assumptions in part (b), F is strictly concave at $t_0$ .

The bathtub-shaped FR appears in many reliability practices such as environmental-stress-screening to manufactured products or burn-in. In this case the curve has the property that in the so-called ‘early failure’ period, the FR decreases over time. To be more precise, a bathtub-shaped FR h(t) is strictly decreasing on $(0,t_1)$ , constant on $(t_1,t_2)$ , and strictly increasing on $(t_2,\infty)$ , where it is assumed that $0\leq t_1\leq t_2\leq\infty$ . This means that the FR is strictly decreasing as the unit becomes stronger in early life, but finally increasing as the unit begins to deteriorate. In such a situation, h(t) is called a bathtub-shaped FR with two change points $t_1$ and $t_2$ . Klutke et al. [Reference Klutke, Kiessler and Wortman7] noticed some limitations of the bathtub-shaped FR in mixture models, and showed that a sufficient condition for the mixture of distributions with concave reliability functions in a neighborhood of 0 to have an IFR at 0 is that at least one of the distributions has a strictly concave reliability function in a neighborhood of 0; see Lemma 2.1. This means that the mixture of such distributions is initially IFR and thus cannot follow the classical bathtub shape. The following theorem reveals an application of this result to coherent systems.

Proof. Under the assumption of the theorem, the system reliability function has the form

\begin{equation*}\bar{F}_T(t)=\sum_{i=k}^{n}s_i\bar{F}_{i:n}(t).\end{equation*}

Observe that if $\bar{F}_{k:n}(t)$ is strictly concave in a neighborhood of $t_0$ , then $f^{\prime}_{k:n}(t)$ (the derivative of the corresponding density with respect to t) is strictly increasing in that interval. We conclude that

\begin{equation*}f^{\prime}_{k+1:n}(t)=\dfrac{n-k}{k}\{[1+\phi(t)]h(t)f_{k:n}(t)+\phi(t)f^{\prime}_{k:n}(t)\},\end{equation*}

where $\phi(t)=F(t)/\bar{F}(t)$ , is positive and hence $\bar{F}_{i:n}(t)$ , $i=k,k+1,\ldots,n$ , is strictly concave in the neighborhood of $t_0$ . The proof is completed by using Lemma 2.1(a).

It can be concluded from the proof of Theorem 2.4 that a sufficient condition for $\bar{F}_{k:n}(t)$ to be strictly concave on an interval is that $\bar{F}^n(t)$ be strictly concave on that interval.

Proof. By a similar argument to that used in the proof of Theorem 2.4, we conclude that if $F_{\ell:n}(t)$ is strictly concave at $t_0$ , then so are $F_{i:n}$ , $i=1,2,\ldots,\ell$ . The result then follows from the mixture representation

\begin{equation*}F_T(t)=\sum_{i=1}^{\ell}s_i\bar{F}_{i:n}(t),\end{equation*}

and part (b) of Lemma 2.1.

As a consequence of Theorem 2.4, the next corollary shows that the initial behavior of the density function of components determines whether the system is initially improving or getting worse.

Proof. By Theorem 2.8, it is sufficient to show, under the assumption of the corollary, that $\bar{F}_{1:n}(t)=\bar{F}^n(t)$ is strictly concave in a neighborhood of 0. Let u(t) denote the second derivative of $\bar{F}_{1:n}(t)$ . It is readily observed that the function

\begin{equation*}u(t)=n\bar{F}^{n-2}(t)[(n-1)f(t)-f'(t)\bar{F}(t)]\end{equation*}

is continuous and $\lim_{t\rightarrow 0}u(t)<0$ . This implies that $u(t)<0$ for all t in a sufficiently small neighborhood of 0 and hence $\bar{F}_{1:n}(t)$ is strictly concave in that neighborhood.

Example 2.3. Consider a beta distribution with strictly concave reliability function

\begin{equation*}\bar{F}(t)=1-t^2,\quad 0\leq t\leq 1.\end{equation*}

It is easy to verify that $\bar{F}^n(t)$ is strictly concave in the interval $[0,{{1}/{\sqrt{2n-1}}}]$ and therefore, using Theorem 2.4, any n-component system with $\bar{F}(t)$ as the component reliability function is initially IFR.

3. Doubly truncated lifetimes

Let $D=\{(x,y)\colon F(x)<F(y)\}$ . Kotlarski [Reference Kotlarski8] and Shanbhag and Rao [Reference Shanbhag and Rao21] studied the doubly censored (truncated) mean function $m(x,y)=\mathbb{E}\{X\mid x<X\leq y\}$ defined for all $(x,y)\in D$ , which represents the expected lifetime for an item that was operating at time x and failed at or before time y. The mean residual life of that unit is defined as $\mathbb{E}\{X-x\mid x<X\leq y\}$ , which may be referred to as doubly censored mean residual life. In this section we study the stochastic properties of doubly truncated lifetimes for coherent structures. The following theorem gives a comparisons of two systems with different structures in the sense of likelihood ratio order.

Theorem 3.1. Let $T_{1}$ and $T_{2}$ be the lifetimes of two coherent (or mixed) systems with component lifetimes $X_{1},X_{2},\ldots,X_{k}$ , $k\leq n$ , and $X_{1},X_{2},\ldots,X_{m}$ , $m\leq n$ . If ${\boldsymbol{s}}^{(1)}$ and ${\boldsymbol{s}}^{(2)}$ are the respective signatures of order n such that $\boldsymbol{s}^{(1)}\leq_{\rm lr}\boldsymbol{s}^{(2)}$ , then for any $(t,y)\in D$ ,

\begin{equation*}(T_{1}-t\mid t<T_{1}\leq y)\leq_{\rm lr}(T_{2}-t\mid t<T_{2}\leq y).\end{equation*}

Proof. For $m=1,2$ , the reliability function of $(T_{m}-t\mid t<T_{m}\leq y)$ can be rephrased as

\begin{equation*}\mathbb{P}\{T_{m}-t>x\mid t<T_{m}\leq y\}=\sum_{i=1}^{n}s_{i}^{(m)}(t,y)\mathbb{P}\{X_{i:n}-t>x\mid t<X_{i:n}\leq y\},\end{equation*}

where

(1) \begin{equation}s_{i}^{(m)}(t,y)= \mathbb{P}\{T_{m}=X_{i:n}\mid t<T_{m}\leq y\}= \dfrac{s_{i}^{(m)}\mathbb{P}\{t<X_{i:n}\leq y\}}{\sum_{j=1}^{n}s_{j}^{(m)}\mathbb{P}\{t<X_{j:n}\leq y\}}.\end{equation}

It is obvious that ${\textbf{\textit{s}}}^{(1)}(t,y)\leq_{\rm lr}{\textbf{\textit{s}}}^{(2)}(t,y)$ , where ${\textbf{\textit{s}}}^{(m)}(t,y)=(s_1^{(m)}(t,y),\ldots,s_n^{(m)}(t,y))$ , $m=1,2$ . On the other hand, note that

\begin{equation*}\dfrac{\tfrac{{\mathrm{d}}}{{\mathrm{d}} x}\mathbb{P}\{X_{i+1:n}-t>x\mid t<X_{i+1:n}\leq y\}}{\tfrac{{\mathrm{d}}}{{\mathrm{d}} x}\mathbb{P}\{X_{i:n}-t>x\mid t<X_{i:n}\leq y\}}=\dfrac{n-i}{i}\dfrac{\mathbb{P}\{t<X_{i:n}\leq y\}}{\mathbb{P}\{t<X_{i+1:n}\leq y\}}\dfrac{F(t+x)}{\bar{F}(t+x)},\end{equation*}

which is an increasing function of x in $[0,\infty)$ for all $t\geq 0$ , and hence

\begin{equation*}(X_{i:n}-t\mid t<X_{i:n}\leq y)\leq_{\rm lr}(X_{i+1:n}-t\mid t<X_{i+1:n}\leq y).\end{equation*}

The rest of the proof can be established from Theorem 1.C.7 in [Reference Shaked and Shanthikumar20, page 49].

Let T be the lifetime of a coherent system consisting of n i.i.d. components with ordered lifetimes $X_{i:n}$ . Assume that ${\textbf{\textit{s}}}(t,y)$ is the vector whose ith element is

(2) \begin{equation}s_i(t,y)=\mathbb{P}\{T=X_{i:n}\mid t<T\leq y\},\quad i=1,2,\ldots,n\end{equation}

(see equation (1)). This probability vector is, in fact, the signature vector of a coherent structure when we have taken into account some partial information about the system’s lifetime. In the literature, such kinds of signature vectors are often called the dynamic signature of the system. Navarro, Balakrishnan, and Samaniego [Reference Navarro, Balakrishnan and Samaniego14], Samaniego, Balakrishnan, and Navarro [Reference Samaniego, Balakrishnan and Navarro19], Zhang [Reference Zhang23], Mahmoudi and Asadi [Reference Mahmoudi and Asadi10], and Tavangar [Reference Tavangar22] are among the references in which different types of dynamic signature are defined. In the following theorem, sufficient conditions are presented for the likelihood ratio ordering of the dynamic signature (2) of two structures with different component lifetimes.

Proof. Let $F_{1}$ and $F_{2}$ denote the distributions of $X_{1}$ and $Y_{1}$ , respectively. Note that

\begin{equation*}\mathbb{P}\{t<X_{i:n}\leq y\}=i\binom{n}{i}\int_{0}^{1}I_{[F_{1}(t),F_{1}(y)]}(u)u^{i-1}(1-u)^{n-i}\,{\mathrm{d}} u \end{equation*}

and

\begin{equation*}\mathbb{P}\{t<Y_{i:n}\leq y\}=i\binom{n}{i}\int_{0}^{1}I_{[F_{2}(t),F_{2}(y)]}(u)u^{i-1}(1-u)^{n-i}\,{\mathrm{d}} u,\end{equation*}

where I denotes the indicator function. We prove that

\begin{equation*}P_m(t,y)=i\binom{n}{i}\int_{0}^{1}I_{[F_{m}(t),F_{m}(y)]}(u)u^{i-1}(1-u)^{n-i}\,{\mathrm{d}} u\end{equation*}

is $\mathrm{TP}_{2}$ in (i, m) in $\{1,2,\ldots,n\}\times \{1,2\}$ .

One can easily prove that $u^{i-1}(1-u)^{n-i}$ is $\mathrm{TP}_{2}$ in $(i,u)\in\{1,2,\ldots,n\}\times[0,1]$ . Now we show that $I_{[F_{m}(t),F_{m}(y)]}(u)$ is $\mathrm{TP}_{2}$ in (u, m) in $[0,1]\times\{1,2\}$ . It follows from $Y_{1}\leq_{\rm st}X_{1}$ that $F_1(x)\leq F_2(x)$ , for all x, and hence only two cases may arise.

1. (i) $F_{1}(t)\leq F_{2}(t)\leq F_{1}(y)\leq F_{2}(y)$ . In this case it can be shown that

   \begin{equation*} I_{[F_{2}(t),F_{2}(y)]}(u)/I_{[F_{1}(t),F_{1}(y)]}(u) \end{equation*}
   is increasing in $u\in[0,1]$ for all $(t,y)\in D$ .

2. (ii) $F_{1}(t)\leq F_{1}(y)\leq F_{2}(t)\leq F_{2}(y)$ . In this case it can be easily checked that

   \begin{equation*}I_{[F_{1}(t),F_{1}(y)]}(u_2)I_{[F_{2}(t),F_{2}(y)]}(u_1)\leq I_{[F_{1}(t),F_{1}(y)]}(u_1)I_{[F_{2}(t),F_{2}(y)]}(u_2).\end{equation*}

Therefore $I_{[F_{m}(t),F_{m}(y)]}(u)$ is $\mathrm{TP}_{2}$ in (u, m) in $[0,1]\times \{1,2\}$ . Then it follows from Lemma 1.1 that $P_m(t,y)$ is $\mathrm{TP}_2$ in $(i,m)\in\{1,2,\ldots,n\}\times\{1,2\}$ .

Now, since the product of two $\mathrm{TP}_2$ functions is a $\mathrm{TP}_2$ function, we conclude that $s_i^{(m)}(t,y)$ is $\mathrm{TP}_2$ in $(i,m)\in\{1,2,\ldots,n\}\times\{1,2\}$ .