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Consecutive-k_{1} and k_{2}-out-of-n: F Structures with a Single Change Point

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## Abstract

**:**

_{1}and k

_{2}-out-of-n: F system with a single change point is launched. In addition, the number of path sets of the proposed structure is determined with the aid of a combinatorial approach. Moreover, two crucial performance characteristics of the proposed model are studied. The numerical investigation carried out reveals that the behavior of the new structure is outperforming against its competitors.

## 1. Introduction

_{1}and k

_{2}-out-of-n: F structure ([17]).

_{1}and k

_{2}-out-of-n: F structure consisting of two different types of units, namely having a single change point. In Section 2, we describe the general framework of the proposed reliability model. An intriguing result is provided in Section 3 and refers to the determination of the number of path sets of the proposed structure. Moreover, two reliability characteristics of it are also studied. After carrying out a numerical investigation, we provide some evidence for the performance of the consecutive-k

_{1}and k

_{2}-out-of-n: F structure with a single change point (see Section 4). Finally, the Section 5 summarizes the contribution of the present paper, while some practical concluding remarks are also highlighted.

## 2. The Constructing Framework for Consecutive-k_{1} and k_{2}-out-of-n: F Systems with a Single Change Point

_{1}and k

_{2}-out-of-n structure contains n components in a line and fails if, and only if, there exist at least non-overlapping consecutive k

_{1}failed components and consecutive k

_{2}failed components. Kindly note that parameters k

_{1}, k

_{2}are interchangeable, namely, their role is not distinguished. In that sense, we assume hereafter and without loss of generality that ${k}_{1}<{k}_{2}$.

_{1}components of the consecutive-k

_{1}and k

_{2}-out-of-n: F system share a common reliability p

_{1}(units of Group 1, hereafter), while the remaining ones, namely the remaining ${n}_{2}=n-{n}_{1}$ units have reliability p

_{2}(units of Group 2, hereafter). Note that in our general approach $(1\le {k}_{1}<{k}_{2}\le n)$, the aforementioned probabilities ${p}_{2},{p}_{1}$ are not necessarily equal. Therefore, in our framework the location of the ${n}_{1}$-th unit is considered as the change point of the proposed model, which shall be noted hereafter as consecutive-k

_{1}and k

_{2}-out-of-n: F system with a single change point.

_{1}and k

_{2}-out-of-n: F (see [17] or [18]). Note that the common consecutive-k

_{1}and k

_{2}-out-of-n: F, which has been introduced in [17] does not contain change points. In that sense, the proposed model generalizes the latter structure, but it also offers a new system which seems to be more flexible and applicable to real-life problems. For studies concerning alternative models with a single change point, the interested reader is referred to [19] or [20].

_{1}and k

_{2}-out-of-n: F system with a single change point.

_{1}non-overlapping failed units (of either Group) and k

_{2}non-overlapping failed ones (of either Group). Note that the restriction ${k}_{1}+{k}_{2}\le n$ is quite obvious, while either ${k}_{1}$ or ${k}_{2}$ could be equal to or larger than $\mathrm{max}\left({n}_{1},{n}_{2}\right)$. For example, let us consider the special case where the design parameters of the proposed model are determined as ${n}_{1}=5,{n}_{2}=3,{k}_{1}=2,{k}_{2}=3$. The resulting consecutive-2 and 3-out-of-8: F structure can be illustrated as

- Two consecutive failed units of Group 1 and three consecutive failed units of Group 2;
- Three consecutive failed units of Group 1 and two consecutive failed units of Group 2;
- Two consecutive failed units of Group 1 and three consecutive failed units of Group 1;
- Four consecutive failed units of Group 1 located at places 2 to 5 (out of 8) and one failed unit of Group 2 located at place 6 (out of 8).

## 3. Main Results

_{1}and k

_{2}-out-of-n: F system with a single change point. In particular, we aim at calculating the number of path sets of the structure including i units of Group 1 and j units of Group 2 (${r}_{{n}_{1},{n}_{2},{k}_{1},{k}_{2}}\left(i,j\right)$, hereafter). Based on this outcome, the reliability function and the mean time to failure of the proposed reliability scheme are also studied.

_{1}components of the structure. It goes without saying that ${x}_{1}$ expresses the length of the first run of 0s in the same side of the underlying structure. It is easily deduced that quantities ${x}_{r},r=2,3,\dots ,i$ obey the next restrictions

_{2}units of Group 2. The following conditions should be satisfied by the above-mentioned random quantities.

_{1}and k

_{2}-out-of-n: F system with a single change point that we should take into account. Our focus remains to the status of the first unit of Group 2 located in the structure line. Under Scheme 2 we assume that the particular unit has failed. Consequently, a binary sequence of n elements (under Scheme 2), including i working units of Group 1 (w.u.G1) and j working units of Group 2 (w.u.G2) is illustrated as follows.

_{1}and k

_{2}-out-of-n: F system with a single change point.

**Proposition**

**1.**

_{1}and k

_{2}-out-of-n: F system with a single change point, consisting of n

_{1}(n

_{2}) units of Group 1 (2) with common reliability p

_{1}(p

_{2}). The number of path sets of the structure including i units of Group 1 and j units of Group 2 is determined by the aid of the following

**Proof.**

_{1}and k

_{2}-out-of-n: F system with a single change point operates if and only if

- there is no run of 0s of a length equal to or larger than ${k}_{2}$ (Scenario 1)

- there is exactly one run of 0s of a length equal to or larger than ${k}_{2}$ (and in any case less than ${k}_{1}+{k}_{2}$) and simultaneously the length of the remaining runs of 0s is smaller than ${k}_{1}$ (Scenario 2). □

_{1}and k

_{2}-out-of-n: F system with a single change point follows Scheme 1 illustrated in Figure 4. In other words, we assume that the first unit of Group 2 appearing in the structure line is working. Then, the total number of binary sequences of units of Group 1 and Group 2 under Scheme 1 shall be determined for each one of the above-mentioned scenarios separately. More specifically,

- under Scenario 1, the total number of binary sequences of units of Group 1 and Group 2 under Scheme 1 equals to the number of integer solutions of the following linear equations$${x}_{1}+{x}_{2}+\dots +{x}_{i+1}={n}_{1}-i$$$${y}_{1}+{y}_{2}+\dots +{y}_{j}={n}_{2}-j$$

- under Scenario 2, there exist two possible choices as mentioned earlier (Choices 1 and 2). Under Choice 1, the unique run of 0s having a length equal to or larger than ${k}_{2}$ contains exclusively failed units of Group 1. Therefore, the total number of binary sequences of units of Group 1 and Group 2 under Scheme 1, Scenario 2 and Choice 1 equals to the number of integer solutions of the following linear equations$${x}_{{d}_{1}}+{x}_{{d}_{2}}+\dots +{x}_{{d}_{i}}={n}_{1}-i-{x}_{{d}_{i+1}}$$$${y}_{1}+{y}_{2}+\dots +{y}_{j}={n}_{2}-j$$$${r}_{1,2,1}=\left(i+1\right)\xb7{C}_{{k}_{1}}\left(j,{n}_{2}-j\right)\xb7{{\displaystyle \sum}}_{\begin{array}{c}x={k}_{2}\\ \end{array}}^{{n}_{1}-i}{C}_{{k}_{1}}\left(i,{n}_{1}-i-x\right).$$

_{2}contains exclusively failed units of Group 2. Therefore, the total number of binary sequences of units of Group 1 and Group 2 under Scheme 1, Scenario 2 and Choice 2 equals to the number of integer solutions of the following linear equations

- under Scenario 1, the total number of binary sequences of units of both groups under Scheme 2 equals to the number of integer solutions of the following linear equations$${x}_{1}+{x}_{2}+\dots +{x}_{i}={n}_{1}-i-{x}_{i+1}$$$${y}_{2}+{y}_{3}+\dots +{y}_{j+1}={n}_{2}-j-{y}_{1}$$$${r}_{2,1}={{\displaystyle \sum}}_{\begin{array}{c}x=0\\ \end{array}}^{{k}_{2}-2}{{\displaystyle \sum}}_{\begin{array}{c}y=1\\ \end{array}}^{{k}_{2}-x-1}{C}_{{k}_{2}}\left(i,{n}_{1}-i-x\right)\xb7{C}_{{k}_{2}}\left(j,{n}_{2}-j-y\right).$$
- under Scenario 2, there exist three possible choices as mentioned earlier (Choices 1, 2 and 3). Under Choice 1, the unique run of 0s having a length equal to or larger than ${k}_{2}$ contains exclusively failed units of Group 1, namely it coincides to one of the ${x}_{1},{x}_{2},\dots ,{x}_{i}$. Therefore, the total number of binary sequences of units of both groups under Scheme 2, Scenario 2 and Choice 1 equals to the number of integer solutions of the following linear equations$${x}_{{d}_{1}}+{x}_{{d}_{2}}+\dots +{x}_{{d}_{i-1}}={n}_{1}-i-{x}_{{d}_{i}}-{x}_{i+1}$$$${y}_{2}+{y}_{3}+\dots +{y}_{j+1}={n}_{2}-j-{y}_{1}$$$${r}_{2,2,1}=i\xb7{{\displaystyle \sum}}_{\begin{array}{c}z={k}_{2}\\ \end{array}}^{{n}_{1}-i}{{\displaystyle \sum}}_{x=0}^{{k}_{1}-2}{{\displaystyle \sum}}_{y=1}^{{k}_{1}-x-1}{C}_{{k}_{1}}\left(j,{n}_{2}-j-y\right)\xb7{C}_{{k}_{1}}\left(i-1,{n}_{1}-i-x-z\right).$$

_{1}and k

_{2}-out-of-n: F system with a single change point can be readily determined. We next make the common assumption that the number of units of each type in the underlying structure is fixed. That practically means that the design parameters ${n}_{1},{n}_{2}$ are pre-determined. Under this assumption, the reliability of the consecutive-k

_{1}and k

_{2}-out-of-n: F system with a single change point can be computed via the following formula

_{1}and k

_{2}-out-of-n: F system with a single change point, namely the expected time till the system no longer operates, can be determined as

## 4. Numerical Results

_{1}and k

_{2}-out-of-n: F system with a single change point. The numerical results and figural representations displayed throughout the next lines, are based on the mathematical outcomes which have been presented and proved in the previous section.

_{1}and k

_{2}-out-of-n: F system with a single change point. In particular, we study the performance of the proposed structure under different values of the design parameters ${n}_{1},{n}_{2},{k}_{1},{k}_{2}$. We mainly aim at delivering remarks about how these parameters reflect the performance of the resulting structure.

_{1}under specific choices of the remaining parameters.

_{1}, the corresponding reliability becomes smaller. The following figure sheds light on the performance of the proposed structure with respect to the parameter k

_{1}. Figure 7 illustrates the reliability of the consecutive-k

_{1}and k

_{2}-out-of-n: F system with a single change point for different values of k

_{1}.

_{1}is, the better the performance becomes of the corresponding system.

_{1}and k

_{2}-out-of-n: F system with a single change point in comparison with the consecutive-k-out-of-n: F structure with a single change point proposed in [19] and the m-consecutive-k-out-of-n: F structure with a single change point proposed in [20]. In order to provide fair comparisons, we consider for both systems the same design parameters p

_{1}, p

_{2}, n

_{1}, n

_{2}and we evaluate the corresponding reliability values.

_{1}and k

_{2}-out-of-n: F system with a single change point achieves larger reliability values and consequently outperforms its competitors, namely the corresponding consecutive-k-out-of-n: F system with a single change point and the m-consecutive-k-out-of-n: F system with a single change point. For instance, for the special case p

_{1}= 0.7, p

_{2}= 0.6, n

_{1}= 10, n

_{2}= 5 the resulting consecutive-k

_{1}and k

_{2}-out-of-15: F system for k

_{1}= 3, k

_{2}= 4 achieves reliability value equal to 96.8151% and 98.9616%, respectively. At the same time, the consecutive-k-out-of-15: F system with a single change point and k = 3 or k = 4 seems weaker since its corresponding reliability is equal to 69.0048% or 89.5755%, respectively. The same conclusion is drawn if we look at the performance of the corresponding m-consecutive-k-out-of-15: F system with a single change point, whose reliability for (m, k) = (3, 2) or (4, 2) equals to 90.5603% and 88.2080%, respectively.

## 5. Discussion

_{1}and k

_{2}-out-of-n: F system with a single change point was established and studied in some detail. Two reliability characteristics of the proposed system were investigated and the corresponding explicit expressions for determining them were also deduced. The main contribution of the manuscript refers to the determination of the number of path sets of a given size for the proposed reliability structure. Based on the abovementioned result, one may readily reach closed formulae for the computation of the corresponding reliability function and mean time to failure of the system. An intriguing extension of the proposed model emerges under the assumption that the number of components of each type is random. Such a case occurs if the single change point of the underlying structure has not been emplaced at a certain location, but contrariwise the change point is supposed to be random. The latter case simply expresses the dynamic version of the proposed consecutive-k

_{1}and k

_{2}-out-of-n: F system with a single change point. Finally, an analogous reliability study of structures with two common failure criteria and a single change point and maintenance policy could be an interesting topic for future research.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Derman, C.; Lieberman, G.J.; Ross, S.M. On the consecutive-k-out-of-n: F system. IEEE Trans. Reliab.
**1982**, 31, 57–63. [Google Scholar] [CrossRef] - Triantafyllou, I.S.; Koutras, M.V. On the signature of coherent systems and applications for consecutive–k–out–of–n: F systems. In Advances in Mathematical Modeling for Reliability; Bedford, T., Quigley, J., Walls, L., Alkali, B., Daneshkhah, A., Hardman, G., Eds.; IOS Press: Amsterdam, The Netherlands, 2008; pp. 119–128. [Google Scholar]
- Triantafyllou, I.S.; Koutras, M.V. On the signature of coherent systems and applications. Probab. Eng. Inf. Sci.
**2008**, 22, 19–35. [Google Scholar] [CrossRef] - Griffith, W.S. On consecutive–k–out–of–n: Failure systems and their generalizations. In Reliability and Quality Control; Basu, A.P., Ed.; Elsevier: Amsterdam, The Netherlands, 1986; pp. 157–165. [Google Scholar]
- Tong, Y.L. A rearrangement inequality for the longest run, with an application to network reliability. J. Appl. Probab.
**1985**, 22, 386–393. [Google Scholar] [CrossRef] - Triantafyllou, I.S.; Koutras, M.V. Signature and IFR preservation of 2-within-consecutive-k-out-of-n:F systems. IEEE Trans. Reliab.
**2011**, 60, 315–322. [Google Scholar] [CrossRef] - Triantafyllou, I.S. Signature-based Analysis of the weighted-r-within-consecutive-k-out-of-n: F systems. Mathematics
**2022**, 11, 2554. [Google Scholar] [CrossRef] - Eryilmaz, S.; Koutras, M.V.; Triantafyllou, I.S. Signature based analysis of m-consecutive k-out-of-n: F systems with exchangeable components. Nav. Res. Logist.
**2011**, 58, 344–354. [Google Scholar] [CrossRef] - Dafnis, S.D.; Makri, F.S.; Philippou, A.N. The reliability of a generalized consecutive system. Appl. Math. Comput.
**2019**, 359, 186–193. [Google Scholar] [CrossRef] - Chang, J.G.; Cui, L.; Hwang, F.K. Reliabilities for (n,f,k) systems. Stat. Probab. Lett.
**1999**, 43, 237–242. [Google Scholar] [CrossRef] - Zuo, M.J.; Lin, D.; Wu, Y. Reliability evaluation of combined k–out–of–n: F, consecutive–k–out–of–n: F and linear connected–(r,s)–out–of–(m,n):F system structures. IEEE Trans. Reliab.
**2000**, 49, 99–104. [Google Scholar] [CrossRef] - Triantafyllou, I.S.; Koutras, M.V. Reliability properties of (n,f,k) systems. IEEE Trans. Reliab.
**2014**, 63, 357–366. [Google Scholar] [CrossRef] - Cui, L.; Kuo, W.; Li, J.; Xie, M. On the dual reliability systems of (n,f,k) and <n,f,k>. Stat. Probab. Lett.
**2006**, 76, 1081–1088. [Google Scholar] - Triantafyllou, I.S. Reliability study of <n,f,2> systems: A generating function approach. Int. J. Math. Eng. Manag. Sci.
**2021**, 6, 44–65. [Google Scholar] - Eryilmaz, S.; Zuo, M.J. Constrained (k,d)-out-of-n systems. Int. J. Syst. Sci.
**2010**, 41, 679–685. [Google Scholar] [CrossRef] - Triantafyllou, I.S. On the lifetime and signature of the constrained (k,d) out-of-n: F reliability systems. Int. J. Math. Eng. Manag. Sci.
**2021**, 6, 66–78. [Google Scholar] [CrossRef] - Zhao, X.; Cui, L.; Yang, X. A consecutive k1 and k2-out-of-n system and its reliability. In Advanced Reliability Modeling II; Yun, W.Y., Dohi, T., Eds.; World Scientific: Singapore, 2006; pp. 97–103. [Google Scholar]
- Triantafyllou, I.S. On the consecutive-k
_{1}and k_{2}-out-of-n Reliability Systems. Mathematics**2020**, 8, 630. [Google Scholar] [CrossRef][Green Version] - Eryilmaz, S. Consecutive k-out-of-n: Lines with a change point. Proc. IMechE Part O J. Risk Reliab.
**2016**, 230, 545–550. [Google Scholar] [CrossRef] - Triantafyllou, I.S. m-consecutive-k-out-of-n: F structures with a single change point. Mathematics
**2020**, 8, 2203. [Google Scholar] [CrossRef]

**Figure 3.**Failure scenarios with exactly 5 failed units for the consecutive-2 and 3-out-of-8: F system with a single change point.

**Figure 4.**The consecutive-k

_{1}and k

_{2}-out-of-n: F system with a single change point under Scheme 1.

**Figure 5.**The consecutive-k

_{1}and k

_{2}-out-of-n: F system with a single change point under Scheme 2.

**Figure 6.**The reliability of the consecutive-k

_{1}and k

_{2}-out-of-n: F system with a single change point (${n}_{2}=4,{k}_{1}=3,{k}_{2}=3,{p}_{1}=0.8$).

**Figure 7.**The reliability of the consecutive-k

_{1}and k

_{2}-out-of-n: F system with a single change point (${n}_{1}=6,{n}_{2}=6,{k}_{2}=5,{p}_{1}=0.5$).

**Table 1.**Numerical comparisons between m-consecutive-k-out-of-n: F with a change point systems and consecutive-k-out-of-n: F with a change point systems.

Consecutive-k-out-of-n: F with a Change Point | m-Consecutive-k-out-of-n: F with a Change Point | Consecutive-k_{1} and k_{2}-out-of-n: F System with a Single Change Point | |||||
---|---|---|---|---|---|---|---|

(n1, n2) | (p1, p2) | k | Reliability | (m, k) | Reliability | (k_{1}, k_{2}) | Reliability |

(10, 5) | (0.8, 0.7) | 3 | 0.868297 | (3, 2) | 0.958571 | (3, 4) | 0.994680 |

4 | 0.969613 | (4, 2) | 0.953963 | (4, 3) | 0.991787 | ||

(0.7, 0.6) | 3 | 0.690048 | (3, 2) | 0.905603 | (3, 4) | 0.968151 | |

4 | 0.895755 | (4, 2) | 0.882080 | (4, 3) | 0.989616 | ||

(12, 8) | (0.8, 0.7) | 4 | 0.950400 | (3, 2) | 0.916497 | (3, 4) | 0.991902 |

5 | 0.987742 | (3, 3) | 0.994343 | (3, 5) | 0.998053 | ||

(0.7, 0.6) | 4 | 0.842588 | (3, 2) | 0.837928 | (3, 4) | 0.954128 | |

5 | 0.945907 | (3, 3) | 0.969490 | (3, 5) | 0.984154 | ||

(15, 10) | (0.8, 0.7) | 5 | 0.983599 | (3, 2) | 0.888646 | (4, 5) | 0.997824 |

6 | 0.995840 | (3, 3) | 0.990176 | (4, 6) | 0.999482 | ||

(0.7, 0.6) | 5 | 0.929143 | (3, 2) | 0.826399 | (4, 5) | 0.985979 | |

6 | 0.975524 | (3, 3) | 0.950575 | (4, 6) | 0.995288 |

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**MDPI and ACS Style**

Triantafyllou, I.S.; Chalikias, M.
Consecutive-*k*_{1} and *k*_{2}-out-of-*n*: *F* Structures with a Single Change Point. *Stats* **2023**, *6*, 438-449.
https://doi.org/10.3390/stats6010027

**AMA Style**

Triantafyllou IS, Chalikias M.
Consecutive-*k*_{1} and *k*_{2}-out-of-*n*: *F* Structures with a Single Change Point. *Stats*. 2023; 6(1):438-449.
https://doi.org/10.3390/stats6010027

**Chicago/Turabian Style**

Triantafyllou, Ioannis S., and Miltiadis Chalikias.
2023. "Consecutive-*k*_{1} and *k*_{2}-out-of-*n*: *F* Structures with a Single Change Point" *Stats* 6, no. 1: 438-449.
https://doi.org/10.3390/stats6010027