On the Lifetime and Signature of Constrained (k, d)-out-of-n: F Reliability Systems

In the present paper we carry out a reliability study of the constrained (k, d)-out-of-n: F systems with exchangeable components. The signature vector is computed by the aid of the proposed algorithm. In addition, explicit signaturebased expressions for the corresponding mean residual lifetime and the conditional mean residual lifetime of the aforementioned reliability system are also provided. For illustration purposes, a well-known multivariate distribution for modelling the lifetimes of the components of the constrained (k, d)-out-of-n: F structure is considered. KeywordsConstrained (k,d)-out-of-n: F structures, Samaniego’s signature, Monte-Carlo simulation, Mean residual lifetimes, Conditional mean residual lifetimes, Multivariate Pareto model.


Introduction
In the field of Reliability Engineering, an extensive activity is recorded referring to the consecutive-type reliability systems, mainly due to their applications in optimization of telecommunication networks, complex infrared detecting systems.
The constrained (k, d)−out−of−n: F system, which was established by Eryilmaz and Zuo (2010), fails if and only if there exist at least k failed components or less than d consecutive working components between any of two successive failures. It is straightforward that for d=0, the aforementioned reliability model behaves as the traditional k-out-of-n: F structure. Additional intriguing generalizations of the classical k−out−of−n: F or consecutive−k−out−of−n: F systems include the m−consecutive−k− out−of−n: F structure ( mk n  ) (see, e.g. Makri and Philippou, 1996;Eryilmaz et al., 2011), the sparsely connected consecutive-k structures (see, e.g. Zhao et al., 2007;Shen and Cui, 2015), the m-consecutive-k-out-of-n:F-r-S-interrupted systems (see Dafnis et al., 2019), the r−within−consecutive k−out−of−n: F model (see, e.g. Tong, 1985;Griffith, 1986;Triantafyllou and Koutras, 2011), the (n, f¸ k) and the < n, f¸ k > systems (see, e.g. Chang et al., 1999;Cui et al., 2006;Triantafyllou and Koutras, 2014) or the m-consecutive-k, l-out-of-n systems (see Cui et al., 2015).
For a detailed review of the consecutive−k−out−of−n: F structures and their extensions the works of Chao et al. (1995), Eryilmaz (2010) and Triantafyllou (2015) seem to be useful, while the surveys provided by Kuo and Zuo (2003), Ram (2013) and Chang et al. (2000) are suitable too.
Throughout the lines of the present manuscript, the reliability properties of the constrained (k, d)−out−of−n: F systems are studied. In Section 2, the proposed algorithm for determining the signature vector of the aforementioned reliability structures is presented, while several numerical results are displayed and appropriately commented. In Section 3, we deliver some signature-based expressions for the mean residual lifetime (MRL) and the conditional mean residual lifetime (CMRL) of the constrained (k, d)−out−of−n: F structure, while an illustrative example is discussed in some detail.

The Proposed Algorithmic Procedure for Computing the Signature of the Constrained (k, d)−out−of−n: F Systems
In the present section, we present the step-by-step procedure which can be followed in order to determine the coordinates of the signature of the constrained (k, d)−out−of−n: F models with exchangeable components. We first express by T the lifetime of a reliability structure consisting of n components and by n T Since the signature vector of a coherent system is promptly connected to its reliability performance, several well-known characteristics, such as the reliability polynomial of a structure or stochastic orderings among different reliability systems, can be approached by the aid of their signature (see, Samaniego, 1985;Kochar et al., 1999;Koutras, 2008a, 2008b). For some recent advances on the evaluation of the signature vector of a reliability structure, one may refer to Ram (2018, 2019), Yi and Cui (2018) and Kumar et al. (2019).
We next illustrate the detailed procedure for determining the signature vector Step 2. Generate a random sample of size n from an arbitrary continuous distribution F, which correspond to the lifetimes of the n components of the underlying constrained (k, d)−out−of−n: F structure.
Step 3. Determine the input parameters of the algorithm, namely the design parameters , dk, where 0, d k n  .
Step 4. Define the random variable M as the chronologically ordered lifetime that leads to the breakdown of the C(k,d,n) model. The random variable M takes on values from 1 to n depending on which component proved to be the fatal one for the operation of the reliability structure.
Step 5. Determine the value of variable M for the specific random sample of n observations which has been generated in Step 2. Each time the random variable M takes on the value i, where 1, 2,..., in = , the corresponding coordinate of vector a increases accordingly, namely the value In order to ascertain the validity of the proposed simulation procedure described above, we shall first apply the algorithm for small values of the design parameters n, d, k and compare the simulation-based outcomes to the corresponding results produced by the aid of the theoretical approach based on the definition of the signature vector mentioned previously. The simulation study has been accomplished based on the Java software environment and involves 10.000 replications for each result. The purpose for constructing the following table is to confirm numerically the correctness of the algorithmic method proposed in the present manuscript. In other words, before applying the new procedure for determining several reliability characteristics of the underlying structure, such as the signature vector or the MRL, we first proceed to implement the proposed algorithm in cases where the numerical results we are chasing for were already known by using alternative methods. In such way, we have the opportunity to assure that our proposal seems to produce correct outcomes.  As it is easily observed, the simulation-based results seem to be close enough to the exact values in all cases considered. For example, let us assume that a constrained (4,1)−out−of−5: F system is implemented. Under the aforementioned design, the exact non-zero coordinates of the corresponding vector, namely the 2 nd , 3 rd and 4 th coordinate equal to 40%, 50% and 10% respectively, while the simulation-based results by the aid of the proposed procedure are 39.97%, 50.28% and 9.75%.
We next apply the abovementioned algorithm for computing the signatures of several members of the class of constrained (k, d)−out−of−n: F systems. Table 2 displays the numerical results for the signature vector of the constrained (k, d)−out−of−n: F structure for different designs. For example, let us consider the case ( , , ) (1,5,7) d k n = . The constrained (5,1)−out−of−7: F structure fails at the 2 nd ordered failure component with probability 27.97%, at the 3 rd ordered component lifetime with probability 43.90%, while its 4 th and 5 th failed component leads to the breakdown of the whole system with probability 25.12% and 3.01% respectively.
In practical terms, the MRL function coincides to the expected (additional) survival time of the underlying structure of age t. Since the representation for the reliability function of the C(k,d,n) model holds true Note that the quantity : () in mt is defined as Given that the lifetimes of the components are exchangeable, the probabilities appearing in expressions (3)-(5), can be readily determined by the aid of the following general result (see, e.g. David and Nagaraja, 2003)

r E T t T t P T t x T t dx
Since Please note that a parallel argumentation for the determination of MRL and CMRL functions of some other reliability structures has been followed by Eryilmaz et al. (2011); Triantafyllou and Koutras (2014).
For illustration purposes, we next consider an example for implementing the aforementioned formulae. Let us assume that the random vector     It is noteworthy that, as it can be readily confirmed, the MRL function of the C(k,d,n) model under the multivariate Pareto model satisfies the following monotonicity attributes: • is dropping off in regard to n (for a specified group of , , , t d k a ) (see Figure 3). • is dropping off in regard to d (for a specified group of , , , t n k a ) (see Figure 4). • is dropping off in regard to t (for a specified group of , , , n d k a ). • is dropping off in regard to k (for a specified group of , , , t d n a ) (see Figure 2). • is dropping off in regard to a (for a specified group of , , , t d k n ) (see Figure 1).

Conclusion
In the present manuscript, the constrained (k, d)−out−of−n: F structure consisting of exchangeable components has been investigated. The main contribution of the paper refers to the development of a new algorithmic procedure for determining several reliability characteristics of the underlying system, such as its signature, mean residual lifetime or conditional mean residual lifetime. The figures and numerical results which are illustrated reveal the impact of the design parameters n, k and d on the performance of the constrained (k, d)−out−of−n: F system. The reliability study of alternative consecutive-type structures based on the proposed algorithm could be an interesting topic for future research.