2. Family of r-modified reliability systems

The discussion in the last section for more sustainable operation of a classical lighting system reveals that if one replaces the assumption of independent components in the classical reliability framework by components that exhibit dependency according to the modified sequence, the family of r-modified reliability systems will be generated.

Hence, any system will belong to the family of r-modified reliability systems if and only if it obeys the following basic assumptions:

1. (a) The system can be seen as a linear system with n ( $n \geq 1$) ordered components;

2. (b) Each component has three possible states, operational (working), failed (non-working), or locked;

3. (c) After the occurrence of an operational component, the system locks the next r − 1 ( $r\geq 1$) components, i.e., a working component guarantees that the next r − 1 components can also be considered as non-failed ones. This assumption would permit the operator to automatically force these r − 1 components to a non-working state without affecting the system’s reliability;

4. (d) At a specific instance, the reliability (working probability) of the t-th component that is not locked is equal to p_t ( $q_t=1-p_t$ is the component’s failure probability);

5. (e) The system fails according to a pre-fixed criterion, similar to the ones in the case of classical reliability systems. Typical criteria considered are as follows: the system fails if

   1. (i) a single component fails, or

   2. (ii) all n components fail, or

   3. (iii) at least k of the n components fail, or

   4. (iv) at least consecutive k of the n components fail.

Depending on the exact form of system’s failure criterion (i.e., the different cases of assumption (e)), we will have a different member of the family of r-modified reliability systems. Specifically,

• • When in assumption (e), (i) is valid, then we will have an r-modified series system. For $r=1,$ an 1-modified series system, reduces to a typical series system;

• • When in assumption (e), (ii) is valid, then we will have an r-modified parallel system. For $r=1,$ an 1-modified parallel system, corresponds to a typical parallel system;

• • When in assumption (e), (iii) is valid, then we will have an r-modified-k-out-of-n:F reliability system. For $r=1,$ an 1-modified-k-out-of-n:F system, reduces to a typical k-out-of-n:F system;

• • When in assumption (e), (iv) is valid, then we will have an r-modified-consecutive-k-out-of-n:F reliability system. For $r=1,$ an 1-modified-consecutive-k-out-of-n:F system, reduces to a typical consecutive-k-out-of-n:F system.

An r-modified-(consecutive-)k-out-of-n:F system has three design parameters, n, k, and r. The extra parameter r (in comparison to the typical two-parameter (consecutive-)k-out-of-n:F system) can be seen as a “sustainability” parameter. However, both r-modified series and r-modified parallel systems are special cases of r-modified-k-out-of-n:F system or r-modified-consecutive-k-out-of-n:F system, since an r-modified-(consecutive-)1-out-of-n:F system corresponds to an r-modified series system and an r-modified-(consecutive-)n-out-of-n:F system corresponds to an r-modified parallel system. Therefore, in this work, we will focus on the family of r-modified-(consecutive-)k-out-of-n:F systems and the results developed here will naturally cover their special cases as well. We shall denote the reliability of an r-modified-(consecutive-)k-out-of-n:F system by $R_{n,k,r}(\mathbf{p})$.

For a clear understanding, let us consider the following example. Let us consider the case r = 2 (a system locks $r-1=1$ component after the occurrence of a working one) and assume that at a specific instance, the state of an r-modified reliability system with n = 10 ordered components can be represented as follows:

\begin{equation*}S \ast F S \ast S \ast F F S,\end{equation*}

where S stands for a working component, F stands for a failed component, and $\ast$ stands for a component (following a working one) when the system is locked. We can then readily observe the following:

• • If the above representation corresponds to a 2-modified(-consecutive)-1-out-of-10:F system (2-modified series system), then the system fails;

• • If the above representation corresponds to a 2-modified(-consecutive)-10-out-of-10:F system (2-modified parallel system), then the system does not fail;

• • If the above representation corresponds to a 2-modified-3-out-of-10:F system, then the system fails;

• • If the above representation corresponds to a 2-modified-4-out-of-10:F system, then the system does not fail;

• • If the above representation corresponds to a 2-modified-consecutive-2-out-of-10:F system, then the system fails;

• • If the above representation corresponds to a 2-modified-consecutive-3-out-of-10:F system, then the system does not fail.

The above representation could, for example, model the system of street lights, described in the Introduction section, at a specific instance during daytime when there is limited visibility. This would mean that the first working street light can guarantee that there is sufficient light in the area next to it and, therefore, the operator can turn off the second street light (from a reliability viewpoint, the second street light is considered as an operating one). This is also the case for the pair of the 4-th and 5-th street-lights, and the 6-th and 7-th ones. It is worth noting that at different time instances, different system components are locked. If, for example, at a different time instance the first street light failed, the second one would not be locked being either a working or failed one.

Having formally introduced the family of r-modified reliability systems, it should be noted that each component of the system has three possible states, denoted by S, F, and $\ast$. As the systems’ failure criteria are dependent only on the occurrence of failed components (i.e., we study F: systems), the states S and $\ast$ can be considered, from a reliability viewpoint, as non-failed components. Having this in mind, we can see that the following conditions hold:

1. (a) Any r-modified reliability system works when there are no failures (all its components are S’s or $\ast$’s);

2. (b) Any r-modified reliability system fails when all its components are “failures” (F’s);

3. (c) The structure function of any r-modified reliability system is nondecreasing.

The validity of the above (a), (b), and (c) conditions classifies r-modified reliability systems in the class of monotone systems (see [Reference Kuo and Zhu22, p. 299]). It is worth noting that examples of monotone systems are found in different fields such as communication, multiprocessor and transportation systems (see [Reference Pham28]).

In the following two sections, we study the reliability of two members of this family of r-modified:F systems. For this purpose, we will use two different approaches. First, we will take advantage of the fact that the sequence that models any r-modified system exhibits a particular kind of dependence (as described in the current section) which can be modeled by the use of higher-order Markov chains. Therefore, we shall construct such chains and classify the systems under study as Markov chain imbeddable structures (MIS’s; [Reference Koutras20]). For more information on the finite Markov chain imbedding technique, we refer to [Reference Fu and Koutras17]; see also [Reference Balakrishnan and Koutras2, Reference Fu and Lou16] and the references therein. Our second approach is combinatorial, taking also advantage of the structure of the sequence under study and the specific failure-patterns of interest.

3. Reliability of r-modified-k-out-of-n:F system

In this section, we study the reliability of r-modified-k-out-of-n:F system, i.e., any system that obeys assumptions (a)–(d) and part (iii) of (e) listed in Section 2.

Proposition 3.1. The reliability $R_{n,k,r}(\mathbf{p})$ of r-modified-k-out-of-n:F system ( $k\geq 1$ and $r\geq 1$) is given by

(1)\begin{equation} R_{n,k,r}(\mathbf{p})= \boldsymbol{\pi}_0\left(\prod_{t=1}^{n}\Lambda_t\right)\mathbf{1}^{\prime},~~n\geq k, \end{equation}

where $\boldsymbol{\pi}_0$ denotes an $1\times k(r + 1)$ vector with all its entries being 0 except for the r-th entry which is 1, $\mathbf{1}^{\prime}$ denotes a $k(r + 1)\times 1$ vector with all its entries being 1, and $\Lambda_t$ is a $k(r + 1)\times k(r + 1)$ matrix which has all its entries 0 except for the following entries:

• • $(r,1)$, which is p_t;

• • $(1 + ir + j, 2 + ir + j)$, $i=0,\dots, \min \{k-1,1\},$ $j=0,\dots,r-2,$ which are all 1;

• • $(2r + 2 + (r + 1)i + j, 2r + 3 + (r + 1)i + j)$, for $k\geq 3,$ $i=0,\dots, k-3,$ $j=0,\dots,r-2,$ which are all 1;

• • $(2r + i, r + 1)$, for $k\geq 2,$ $i=0,1,$ which are all p_t;

• • $(3r + 1 + (r + 1)j + i, 2r + 2 + (r + 1)j)$, for $k\geq 3,$ $i=0,1,$ $j=0,\dots,k-3,$ which are all p_t;

• • $(r,\min \{k,2\}r+1)$, which is q_t;

• • $(2r + i, 3 r + 2)$, for $k\geq 3,$ $i=0,1,$ which are all q_t;

• • $(3r + 1 + (r + 1)j + i, 4r + 3 + (r + 1)j)$, for $k\geq 4,$ $i=0,1,$ $j=0,\dots,k-4,$ which are all q_t;

• • $(k(r + 1) - i, k(r + 1))$, for $k\geq 2,$ $i=1,2,$ which are all q_t;

• • $(k(r + 1), k(r + 1))$, which is 1.

It is worth noting that when the Markov chain imbedding technique is applied, the typical convention is that $\boldsymbol{\pi}_0=(1,0,\dots,0)$. However, in the present case, such a choice would result in a Markov chain $\{Y_t, t\geq 0\}$ that would be locked for $t=1,\dots, r-1$. Instead, we do assume that at time t = 0, the Markov chain is at state $0^{(r-1)}$ (which ensures that the system is not locked at time t = 1) and use the vector $\boldsymbol{\pi}_0$ of Proposition 3.1. This is also the case later on in Proposition 4.1.

In order to get a clear understanding of Proposition 3.1, we now present two examples of the transition probability matrix $\Lambda_t$ for two different parametrizations. For k = 3 and $r=2,$ $\Lambda_t$ takes on the form

\begin{equation*} {\Lambda_t=\left( \begin{array}{cccccccc|c} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ p_t & 0 & 0 & 0 & q_t & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & p_t & 0 & 0 & 0 & 0 & q_t & 0\\ 0 & 0 & p_t & 0 & 0 & 0 & 0 & q_t & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & p_t & 0 & 0 & q_t\\ 0 & 0 & 0 & 0 & 0 & p_t & 0 & 0 & q_t\\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{array} \right) },\end{equation*}

while for k = 2 and $r=3,$ we have

\begin{equation*} {\Lambda_t=\left( \begin{array}{ccccccc|c} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ p_t & 0 & 0 & 0 & 0 & 0 & q_t & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & p_t & 0 & 0 & 0 & q_t\\ 0 & 0 & 0 & p_t & 0 & 0 & 0 & q_t\\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{array} \right) }.\end{equation*}

We should note that a careful observation of the one-state transition probability matrix $\Lambda_t$ of Proposition 3.1 reveals that each state $i^{(F)}$, $i=1,\dots,k-1,$ of the Markov chain can be accumulated at state $i^{(r-1)}$. Such a procedure would reduce the dimension of the square matrix $\Lambda_t$ by k − 1 (i.e., it would result in a $(kr+1)\times (kr+1)$ matrix $\Lambda_t$). Although we maintained all states of the Markov chain for the construction of the Markov chain to be clearer, such a procedure could be useful for numerical calculations when the values of the design parameters k and r are large.

It is clear that for r = 1, Proposition 3.1 provides the reliability of a classical k-out-of-n:F system. Moreover, for k = 1, it provides the reliability of an r-modified series system, while, for k = n, it provides the reliability of an r-modified parallel system.

Proposition 3.1 provides a closed and numerically efficient formula for the calculation of the reliability of an r-modified reliability system through the use of the transition probability matrix in the general case when its non-locked components do not necessarily have equal reliabilities. In the special case when the reliabilities of the non-locked components are equal, different formulas for the calculation of the reliability of the system, which do not involve matrices, can also be derived. We shall first derive a closed-form expression for the reliability of the system based on combinatorial arguments.

Proposition 3.2. The reliability $R_{n,k,r}(p)$ of r-modified-k-out-of-n:F system ( $k\geq 1$ and $r\geq 1$), when the non-locked components have constant and equal reliabilities $p_t=p$, is given, for $n\geq k$, by

(2)\begin{eqnarray}\nonumber R_{n,k,r}(p)&=&\sum_{j=0}^{r-1}\sum_{y=\lceil\frac{n-k-j}{r} \rceil} ^{\lfloor\frac{n-1-j}{r}\rfloor}{n-1-j-y(r-1)\choose y}p^{y+1}q^{n-1-j-yr}\nonumber\\ &&+\sum_{i=1}^{k-1}\sum_{y=\lceil\frac{n-r-k+1}{r} \rceil} ^{\lfloor\frac{n-r-i}{r}\rfloor}{n-r-i-y(r-1)\choose y}p^{y+1}q^{n-(y+1)r}, \end{eqnarray}

where $\lfloor x\rfloor $ ( $\lceil x\rceil$) is the greatest (least) integer less (greater) than or equal to x.

In the case of equal (non-locked) component reliabilities, we can also derive the following simple recursive relation for $R_{n,k,r}(p)$ by conditioning on the number of failed components before the first occurrence of a working one in the system of n components.

Proposition 3.3. The reliability $R_{n,k,r}(p)$ of the r-modified-k-out-of-n:F system ( $k\geq 1$ and $r\geq 1$), when the non-locked components have constant and equal reliabilities $p_t=p$, satisfies the recursive scheme

(3)\begin{equation} R_{n,k,r}(p)=p\sum_{i=0}^{k-1}q^{i}R_{n-i-r,k-i,r}(p),~~n\geq k, \end{equation}

with initial conditions $R_{n,k,r}(p)=1$, for n < k.

4. Reliability of r-modified-consecutive-k-out-of-n:F system

In this section, we study the reliability of r-modified-consecutive-k-out-of-n:F system, i.e., any system that obeys the assumptions (a)–(d) and part (iv) of (e) listed earlier in Section 2.

Proposition 4.1. The reliability $R_{n,k,r}(\mathbf{p})$ of r-modified-consecutive-k-out-of-n:F system ( $k\geq 1$ and $r\geq 1$) is given by

(4)\begin{equation} R_{n,k,r}(\mathbf{p})= \boldsymbol{\pi}_0\left(\prod_{t=1}^{n}\Lambda_t\right)\mathbf{1}^{\prime}, \end{equation}

where $\boldsymbol{\pi}_0$ denotes an $1\times (k + r)$ vector with all its entries being 0 except for the r-th entry which is 1, $\mathbf{1}^{\prime}$ denotes a $(k + r)\times 1$ vector with all its entries being 1 and $\Lambda_t$ is a $(k + r)\times (k + r)$ matrix which has all its entries as 0 except for the following entries:

• • $(i,i+1)$, $i=1,\dots, r-1,$ which are all 1;

• • $(i,1)$, $i=r,\dots, k+r-1,$ which are all p_t;

• • $(i,i+1)$, $i=r,\dots, k+r-1,$ which are all q_t;

• • $(k+r,k+r)$, which is 1.

Proof. Let us introduce a Markov chain $\{Y_t, t\geq 0\},$ with a finite state space $S=\{0,0^{(1)},\dots, 0^{(r-1)},1,\dots, k\}$ and state k representing an absorbing state as it corresponds to the failure of the system, as follows:

• • $Y_t=0$ if the t-th component is in a working state;

• • $Y_t=0^{(i)}$, $i=1,\dots, r-1,$ if the t-th component is the r-th consecutive component which is locked and the t − i-th component is in a working state;

• • $Y_t=i$, $i=1,\dots, k,$ if the t-th component is the i-th consecutive component which is in a failed state and the t − i-th component (if there is one) is a locked one.

Then, the one-state transition probability matrix can be written as

\begin{equation*} {\Lambda_t=\left( \begin{array}{cccccccc|c} 0 & 1 & 0 & \dots & 0 & 0 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 & 0 & \dots & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ p_t & 0 & 0 & \dots & 0 & q_t & 0 & \dots & 0\\ p_t & 0 & 0 & \cdot & 0 & 0 & q_t & \dots & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ p_t & 0 & 0 & \dots & 0 & 0 & 0 & \dots & q_t\\ \hline 0 & 0 & 0 & \dots & 0 & 0 & 0 & \dots & 1\\ \end{array} \right) }_{(k+r)\times (k+r)}, \end{equation*}

which has the entries as described above. Under this setup, Theorem 3.1 of [Reference Koutras20] for an MIS readily yields (4), where $\boldsymbol{\pi}_0$ is the vector of initial probabilities of the Markov chain $\{Y_t, t\geq 0\}.$

For illustrative purposes, let us consider a specific configuration of the one-state transition probability matrix $\Lambda_t.$ For k = 3 and r = 2, $\Lambda_t$ takes on the form

\begin{equation*} {\Lambda_t=\left( \begin{array}{cccc|c} 0 & 1 & 0 & 0 & 0 \\ p_t & 0 & q_t & 0 & 0\\ p_t & 0 & 0 & q_t & 0\\ p_t & 0 & 0 & 0 & q_t\\ \hline 0 & 0 & 0 & 0 & 1\\ \end{array} \right). } \end{equation*}

Special cases of the r-modified-consecutive-k-out-of-n:F system are also of interest. For r = 1, Proposition 4.1 provides the reliability of a classical consecutive-k-out-of-n:F system. For k = 1, it provides the reliability of an r-modified series system and for k = n, it provides the reliability of an r-modified parallel system.

We now focus on the special case in which the reliabilities of the non-locked components of an r-modified-consecutive-k-out-of-n:F system are constant and equal to p. A combinatorial approach can also be applied in this case in order to derive a closed-form expression for the reliability of the system not involving matrices. For this purpose, we first recall the following well-known result.

Corollary 4.2. [Reference Riordan30]. The number of allocations of α indistinguishable balls into m distinguishable cells, so that each cell is occupied by at most k balls, is given by

\begin{equation*}C(\alpha, m, k)=\sum_{j=0}^{\lfloor\frac{a}{k+1}\rfloor}(-1)^{j}{m\choose j}{\alpha-(k+1)j+m-1\choose \alpha-(k+1)j}.\end{equation*}

With Corollary 4.2, we now proceed to obtain the following proposition.

Proposition 4.3. The reliability $R_{n,k,r}(p)$ of r-modified-consecutive-k-out-of-n:F system ( $k\geq 1$ and $r\geq 1$), when the non-locked components have constant and equal reliabilities $p_t=p$, is given, for $n\geq k$, by

(5)\begin{eqnarray} \nonumber R_{n,k,r}(p)&=&\sum\limits_{j=0}^{r-1}\sum\limits_{y=1}^{1+\lfloor\frac{n-1-j}{r}\rfloor} C(n-(y-1)r-j-1,y,k-1)p^{y}q^{n-1-(y-1)r-j}\\ && +\sum\limits_{j=1}^{k-1}\sum\limits_{y=1}^{\lfloor\frac{n-j}{r}\rfloor} C(n-yr-j,y,k-1)p^{y}q^{n-yr}. \end{eqnarray}

We mention that $\binom{n}{m}$ here denotes the extended binomial coefficient; see [Reference Feller15, p. 50].

Moreover, the following proposition (analogous to Proposition 3.3) can be established by conditioning on the number of failed components before the first occurrence of a working one in the sequence of n components.

Proposition 4.4. The reliability $R_{n,k,r}(p)$ of r-modified-consecutive-k-out-of-n:F system ( $k\geq 1$ and $r\geq 1$), when the non-locked components have constant and equal reliabilities $p_t=p$, satisfies the recursive scheme

(6)\begin{equation} R_{n,k,r}(p)=p\sum_{i=0}^{k-1}q^{i}R_{n-i-r,k,r}(p),~~n\geq k, \end{equation}

with initial conditions $R_{n,k,r}(p)=1$, for n < k.

5. Numerical results

In this section, we present some illustrative numerical examples for the members of the family of r-modified reliability systems.

Let us first consider a system consisting of n = 32 micro fans along its surface that protect it from overheating. Under specific environmental circumstances (e.g., when the temperature is below a pre-fixed threshold), an operating micro fan can substitute the operation of the next r − 1 micro fans and, therefore, their operation is automatically terminated. We assume that the reliability of each micro fan is constant and equal to p = 0.90, unless it is automatically forced to a non-working state. The system fails if and only if at least k of the n micro fans fail. It is then evident that this system corresponds to an r-modified-k-out-of-n:F system.

As a second example, let us recall the lighting system described in the Introduction. Let us assume that it also consists of n = 32 components (street lights). When there is restricted visibility during daytime, an operational street light can substitute the operation of the next r − 1 lights, which are automatically turned off. We assume again that the reliability of each street light is constant and equal to p = 0.90, unless it is automatically turned off. The system fails if quite a large area of the road is not sufficiently lighted, say, there are k consecutive failed lights. This system corresponds to an r-modified-consecutive-k-out-of-n:F system.

We now provide illustrative numerical results for the whole family of r-modified reliability systems by using the results established in Sections 3 and 4. Tables 1 and 2 provide the reliabilities of an r-modified-k-out-of-n:F system and an r-modified-consecutive-k-out-of-n:F system, respectively, for the specific configurations of the aforementioned examples and different choices of design parameters k and r.

Table 1.

Exact reliabilities of an r-modified-k-out-of-n:F system with n = 32 components and equal component reliabilities $p_{t}=p=0.90$ (when the components are not locked).

Table 2.

Exact reliabilities of an r-modified-consecutive-k-out-of-n:F system with n = 32 components and equal component reliabilities $p_{t}=p=0.90$ (when the components are not locked).

The numerical results presented in Tables 1 and 2 verify the following:

• • For n, r, and p fixed, the reliabilities of r-modified-k-out-of-n:F system and r-modified-consecutive-k-out-of-n:F system are both increasing functions of k;

• • For n, k, and p fixed, the reliabilities of r-modified-k-out-of-n:F system and r-modified-consecutive-k-out-of-n:F system are both increasing functions of r;

• • For n, k, r, and p fixed, the reliability of r-modified-consecutive-k-out-of-n:F system is greater than or equal to the reliability of r-modified-k-out-of-n:F system.

A graphical display of the last statement is provided in Figure 1, which plots the reliability of two different members of the family of r-modified reliability systems for a specific configuration of the design parameters n, k, and r with respect to the constant component reliability p (when the component is not locked).

Figure 1.

The reliability of the 3-modified-consecutive-4-out-of-32 reliability system (upper yellow line) and the reliability of the 3-modified-4-out-of-32 reliability system (lower blue line) with respect to constant component reliability p (when the component is not locked).

On the other hand, Figures 2 and 3 depict the effect of the increase in the sustainability parameter r on the increase in the reliability of the system. In each plot, the lower line depicts the reliability of the classical 4-out-of-16 and the consecutive-4-out-of-16 reliability systems, respectively. Therefore, one may easily observe that the new modified systems outperform the classical non-modified ones. Last observations do pose another interesting question: Could a guidance be given to practitioners about the appropriate value of the sustainability parameter in every practical application? Undoubtedly, the answer has to do with the specific system reliability one wishes to achieve. To make that statement clearer, we provide in Table 3 the reliabilities of r-modified-4-out-of-16 and r-modified-consecutive-4-out-of-16 systems for constant component reliability p = 0.75, as r increases. The entries of Table 3 reveal that if, for example, a practitioner is interested in achieving a system reliability of at least $98\%$ when modeling a real-life engineering system by the use of an r-modified-4-out-of-16 system (e.g., the micro fans systems described in the beginning of Section 5), then she/he should use a value of r of at least 7. On the other hand, if the real-life engineering system was modeled by the use of an r-modified-consecutive-4-out-of-16 system (e.g., the lighting system described before in the current section), then using r = 3 would achieve the practitioner’s goal. It should be stressed that, theoretically speaking, larger values could be assigned to sustainability parameter r (than the ones presented in Table 3), but the results do not get significantly affected.

Figure 2.

The reliability of the r-modified-4-out-of-16 reliability system for r = 1, r = 2, r = 3, and r = 4 (from lower to the upper line, respectively) with respect to constant component reliability p (when the component is not locked).

Figure 3.

The reliability of the r-modified-consecutive-4-out-of-16 reliability system for r = 1, r = 2, r = 3, and r = 4 (from lower to the upper line, respectively) with respect to constant component reliability p (when the component is not locked).

Table 3.

Exact reliabilities of an r-modified-4-out-of-16:F system and an r-modified-consecutive-4-out-of-16:F system with equal component reliabilities $p_{t}=p=0.75$ (when the components are not locked) for various values of r.

A careful observation to be noted from both Figures 2 and 3 is that the interval in which each system’s reliability is greater than the common reliability of its components increases as r increases.

6. Concluding remarks

In this paper, we have introduced and studied the family of r-modified reliability systems, which generalizes the (consecutive-)k-out-of-n:F system by adding an extra “sustainability” parameter r to the set of the system’s design parameters n and k. We have derived the reliability of two basic members of this family by constructing suitable Markov chains or by using a direct combinatorial approach. We have presented examples of “sustainable” systems, which can be modeled using this new theoretical framework, along with illustrative numerical results, which reveal that the members of the new family outperform the classical reliability systems studied in the literature.

This is the first and introductory work on the family of r-modified reliability systems. Different traditional reliability systems can also be studied by assuming that their components exhibit dependence generated by the modified sequence. A straightforward example is the introduction and study of the r-modified-m-within-consecutive-k-out-of-n: F systems (for the m-within-consecutive-k-out-of-n: F systems, one may see [Reference Griffith18]). Furthermore, the present work can also be extended to the case of circular r-modified reliability systems. We are currently working on these problems and hope to report the findings in a future paper.