10. Signature Reliability of k-out-of-n Sliding Window System

The sliding window system (SWS) is a generalized form of k-out-of-n:F system which has n linearly ordered multistate elements. Each window can have two states: completely working and totally failed. Application of SWS is found in quality control, service system, manufacturing, radar, and military system. Chiang and Nui [1] discussed the consecutive k-out-of-n:F system in cases when consecutive elements are failed and computed the reliability in lower and upper form. Levitin [2] discussed a linear multistate SWS, which is the generalized form of the consecutive k-out-of-r-from-n:F system, in case of multiple failure and evaluated the reliability of the considered system with the help of universal generating function (UGF). Koucky [3] evaluated the reliability of the k-out-of-n system with failure elements, and concluded that elements not need to be independent and identically distributed (i.i.d.). Levitin [4] considered a linear multistate multiple SWS, which is the generalized form of the linear consecutive k-out-of-r-from-n:F system, in case of multiple failures and computed the system reliability with the help of UGF. Habib et al. [5] discussed the reliability of a linear consecutive k-out-of-r-from-n:G system in case of multistate failure using the total probability theorem. Ram and Singh [6] considered a complex system with common cause failure and where each element could have constant failure rate. They determined the system reliability and cost analysis using the supplementary variable technique. Levitin and Ben-Haim [7] determined the reliability of a consecutive SWS using an algorithm based on the UGF technique; the considered system fails if the sum of the performance rate is lower than the total allocation weight. Levitin and Dai [8] discussed the reliability of linear m-consecutive k-out-of-r-from-n:F systems in case of multiple failure elements. Levitin and Dai [9] considered the k-out-of-n SWS in case of multiple failures and computed the reliability of the proposed system using UGF. Xiang and Levitin [10] generalized the linear multistate SWS which consisted of n linearly multistate windows. They evaluated the reliability of a combined m-consecutive and k-out-of-n SWS using the UGF technique. Ram and Singh [11] discussed the reliability, availability, and cost analysis of two independent repairable subsystems using the supplementary variable technique, Laplace transformation, and Gumbel-Hougaard family copula technique. Ram [12] discussed and reviewed the engineering system and physical science and provided different methods for computing system reliability. Pham [13] discussed the modeling of complex systems both hardware and software and calculated the reliability of the considered system. Negi and Singh [14] studied the non-repairable complex system which had two binary subsystems, namely, weighted A-out-of-G:g and weighted l-out-of-b:g and computed the reliability and sensitivity using UGF. Ram and Davim [15] measured the reliability of multistate systems, optimization of multistate systems, and continuous multistate systems using new computational techniques applied to probabilistic and non-probabilistic safety assessment.

In the context of signatures, Shapley [16] and Owen [17,18] discussed the game theory on the basis of random variable and evaluated the probability by extending the game theory. Samaniego [19] introduced the concept of signatures on the basis of a coherent system. A coherent system can have monotone and its elements relevant to each other. Signatures are widely used in communication networks, reliability economics, etc. Kocher et al. [20] compared the coherent systems when elements having i.i.d. and computed signatures and expected lifetime. Boland and Samaniego [21] described the properties of signature reliability of complex systems and compared the signatures of the different systems. They presented applications of signatures in reliability economics and communication networks. Samaniego [22] examined the properties of a coherent system in terms of signatures. He developed a new approach of coherent system and defined its application in networks and communications. Navarro and Rychlik [23] obtained the reliability function in upper and lower form and computed expected lifetime of a coherent system by using the Samaniego concept of i.i.d elements. Navarro et al. [24] discussed the reliability function based on signatures having i.i.d. elements. They computed the system signature with the help of stochastic ordering properties for the coherent system. Samaniego et al. [25] discussed the system properties on the basis of dynamic signatures with the help of ordered statistical methods. Da et al. [26] evaluated the signature of the considered system and the redundancy system also. They determined the signature of a coherent system which has a large number of elements. Marichal and Mathonet [27] determined the system signature on the basis of reliability function. They used a different formula for evaluating signatures with the help of structure function using i.i.d. components. Coolen [28] discussed the nature of signatures with the help of system structure function and described the definition, properties, and applications of signatures. Kumar and Singh [29, 30 and 31] determined the signature of sliding window coherent system, k-out-of-n system, and linear multistate SWS having an i.i.d. component and calculated different measures such as signature, expected lifetime, cost, and Barlow-Proschan index.

From the above discussion, it becomes clear that many researchers computed the system reliability of binary and MSS SWS with different methods. We also studied a k-out-of-n SWS with i.i.d. component to evaluate the reliability characteristic such as signature, expected lifetime, and Barlow-Proschan index with the help of Owen’s method and UGF technique.

10.2 Algorithm for Evaluating the UGF of All the Groups of r Consecutive Elements (See Levitin and Dai [9])

Step 1. Compute UGF $U_{1 - r} (z)$ as follows:

$\begin{matrix} U_{1 - r} (z) = z^{x_{0}} (x_{0} consists of r zeros) . \end{matrix} (10.1)$
Step 2. Obtain UGF of individual MSE u(z) using $\underset{\leftarrow}{\otimes}$ as follows:

$\begin{matrix} \begin{array}{l} u_{a} (z) \underset{\leftarrow}{\otimes} u_{j} (z) = \sum_{l = 1}^{A_{l}} q_{a, l} z^{x_{a, l}} \underset{\leftarrow}{\otimes} \sum_{b = 1}^{B_{l}} P_{i, b} z^{g_{i, l}} \\ = \sum_{l = 1}^{A_{l}} \sum_{b = 1}^{B_{l}} q_{a, l} P_{i, b} z^{ϕ (^{x_{a, l}, g_{i, b}})} \end{array} \end{matrix} (10.2)$

where ϕ is an arbitrary vector of x and g shift all vector elements one position left.
Step 3. Calculate $U_{i + 1 - r} (z)$ using operator $\underset{\leftarrow}{\otimes}$ in a sequence as follows:

$U_{i + 1 - r} (z) = U_{i - r} (z) \underset{\leftarrow}{\otimes} u_{i} (z) for i = 1, 2, . . ., n .$
Step 4. Evaluate all possible groups r consecutive of MSE applying operator $\otimes$ as follows:

$U_{i} (z) = U_{1} (z), \dots, U_{n + 1 - r} (z) .$

10.3 Assessment of m Consecutive Failed Groups to a k-out-of-n SWS

The UGF of r consecutive groups is given by

$U_{a} (z) = \sum_{a = 1}^{A_{a}} q_{a, l} z^{y_{a, l}} .$

Modifying U_a(z) within an integer counter C_a,l, we have

$U_{a} (z) = \sum_{l = 1}^{A_{a}} q_{a, l} z^{C_{a, l}, y_{a, y}}$

where m_a = total number of combination of C_a,l and x_a,l.

Now, assign an initial value 0 and modify the equations (10.1) and (10.2):

$U_{1 - r} (z) = z^{0, y_{0}}$

$U_{a} (z) \otimes u_{i} (z) = \sum_{l = 1}^{A_{a}} q_{a, l} z^{x_{a, l}, x_{a, l}} \underset{\leftarrow}{\otimes} \sum_{b = 1}^{B_{i}} P_{i, b} z^{g, b}$

$U_{a} (z) = \sum_{l = 1}^{A_{a}} \sum_{b = 1}^{B_{e}} q_{a, l} P_{i, b} z^{ρ (C_{a, l}, σ (φ (y_{i, l}, g_{i, b})), ϕ (y_{a, l}, g_{i, b}))}$

where $ρ (C_{g}, x) = {\begin{cases} C_{g + 1} i f x < w \\ 0 i f x \geq w \end{cases} .$

UGF of failure probability is expressed as follows:

$\partial (U_{a} (z)) = \sum_{l = 1}^{m_{a}} q_{a, l} 1 (C_{a, l} = m) .$

The failure probability (E_i) of the system can be computed as the sum of the probabilities of mutually exclusive events as follows:

$E = E_{1} + E_{2} (1 - E_{1}) + \dots + E_{n - r - M + 2} Π_{i = 1}^{n - r - M + 1} (1 - E_{i}) .$

To obtain the probability $E_{e} \prod_{i = 1}^{e = 1} (1 - E_{i})$ one can remove all term with $C_{m + e - 1, l} = m$ from $U_{m + e - 1} (z)$ to get

$U_{m + e} (z) = U_{m + e - 1} (z) \otimes u_{m + r + e - 1} (z) .$

Further, any term l of U_a(z) with the counter value can be obtained as (see Levitin and Dai [9])

$\begin{matrix} C_{a, l} < m - n + a + r - 1 . \end{matrix} (10.3)$

The system reliability of the sets of B consecutive groups of D consecutive MSE $E_{1}, E_{2}, . . ., E_{g - D + B + 2}$ can be expressed as follows:

$\begin{matrix} R = P {\prod_{l = 1}^{g - D - B + 2} [I (\sum_{i = l}^{l + B - 1} [I (\sum_{j = i}^{i + D - 1} b_{j} < w)] < A)] = 1} . \end{matrix} (10.4)$

With the help of equation (10.4), we can evaluate the signature of the system having i.i.d. components as $s_{A} = p_{D} (T_{s} = T_{A : g})$ , where T is the system lifetime and s_A is the probability of the system failure.

Boland [32] obtained the structure function R of the system having i.i.d. components as follows:

$s_{A} = \frac{1}{(\begin{array}{l} g \\ g - A + 1 \end{array})} \sum_{l \subseteq [g]} φ (R) - \frac{1}{(\begin{array}{l} g \\ g - A \end{array})} \sum φ (R) . (10.5)$

10.4 Proposed Algorithms

10.4.1 Algorithm for Evaluating the Reliability of a k-out-of-n SWS (See Levitin and Dai [9])

Step 1. Initialization:

$F = 0; U_{1 - r} (z) = z^{0, x_{0}} .$
Step 2. Compute $U_{j + 1 - r} (z) = U_{j - r} (z) \otimes u_{j} (z)$ , and collect the like terms in the obtained u-function.
Step 3. If $j \geq k + r - 1$ , then add $δ (U_{j + 1 - r} (z))$ to F and eject all the terms t with $c_{j - r + 1, t} = k$ from $U_{j + 1 - r} (z) .$
Step 4. Remove from $U_{j + 1 - r} (z)$ all the terms with $c_{j - r + 1, t} < k - m + j$ .
Step 5. Evaluate the reliability of a k-out-of-n SWS as R = 1 –F.

10.4.2 Algorithm for Calculating the Signature of a k-out-of-n SWS with its Reliability Function

Step 1. Calculate the system signature of the structure function (Boland [32]).

$\begin{matrix} B_{l} = \frac{1}{(\begin{array}{l} m \\ m - n + 1 \end{array})} \sum_{\begin{array}{l} H \subseteq [m] \\ | H | = m - n + 1 \end{array}} ϕ (H) - \frac{1}{(\begin{array}{l} m \\ m - n \end{array})} \sum_{\begin{array}{l} H \subseteq [m] \\ | H | = m - n \end{array}} ϕ (H) . \end{matrix} (10.6)$

Evaluate reliability polynomial of a k-out-of-n SWS by

$H (P) = \sum_{ε = 1}^{m} C_{e} (\begin{matrix} m \\ e \end{matrix}) P^{ε} q^{n - ε}$

where $C_{j} = \sum_{j = m - e + 1}^{m} B_{j}, e = 1, 2, . . ., m .$
Step 2. Compute the tail signature of a k-out-of-n SWS, i.e., (m + 1)-tuple $B = (B_{0}, . . ., B_{m})$ using

$\begin{matrix} B_{l} = \sum_{j = a + 1}^{m} b_{j} = \frac{1}{(\begin{array}{l} m \\ m - a \end{array})} \sum_{| H | = m - a} ϕ (H) . \end{matrix} (10.7)$
Step 3. Evaluate the reliability function in the form of a polynomial by using Taylor expansion about x = 1 by

$\begin{matrix} P (x) = y^{m} h (\frac{1}{y}) . \end{matrix} (10.8)$
Step 4. Assess the tail signature of the k-out-of-n SWS reliability function with the help of equation (10.6) (see Marichal and Mathonet [27]):

$\begin{matrix} B_{a} = \frac{(m - a)!}{m!} D^{a} P (1), a = 0, \dots, m . \end{matrix} (10.9)$
Step 5. Obtain the signature of the k-out-of-n SWS using equation (10.8) as follows:

$\begin{matrix} b = B_{a - 1} - B_{a}, a = 1, \dots, m . \end{matrix} (10.10)$

10.4.3 Algorithm to Assess the Expected Lifetime of a k-out-of-n SWS with Minimum Signature

Step 1. Determine the expected lifetime of an i.i.d. component k-out-of-n SWS which are exponentially distributed with mean μ = 1.
Step 2. Calculate the minimum signature of a k-out-of-n SWS with the expected lifetime of the reliability function by using

$\begin{matrix} {\bar{h}}_{T} (t) = \sum_{j = 1}^{n} C_{j} h_{1 : j} (t) \end{matrix} (10.11)$

where $h_{1 : j} (t) = P_{r} (Z_{1 : j} > t)$ and $h_{j : j} (t) = P_{r} (Z_{j : j} > t)$ for $j = 1, 2, \dots, n$ .
Step 3. Obtain E(T) of a k-out-of-n SWS of i.i.d. components by (see Navarro and Rubio [33]).

$\begin{matrix} E (T) = μ \sum_{j = 1}^{n} \frac{C_{j}}{j} \end{matrix} (10.12)$

where $C_{j} (j = 1, . . ., n)$ is a vector coefficient of minimal signature.

10.4.4 Algorithm for Evaluating the Expected Value of the Component X and Expected Cost Rate of a k-out-of-n SWS When Working Elements Are Failed

Step 1. Calculate the number of failed elements at the time of system failure with signature (Eryilmaz [34]):

$\begin{matrix} E (X) = \sum_{j = 1}^{n} j \cdot b_{j}, j = 1, 2, . . ., n . \end{matrix} (10.13)$
Step 2. Compute the E(X) and E(X)/E(T) of a k-out-of-n SWS with minimum signature.

10.5 Illustration

Consider a 2-out-of-3 SWS for m = 4, r = 2, w = 3 and each window has two states: completely working and total failure along with some performance rate 1, 2, 2, and 2, respectively. A diagram of the proposed system is shown in Figure 10.1.

The probability of the inner parallel component can be expressed as follows:

$P_{e} = 1 - \prod_{m = 1}^{n} (1 - R_{e m})$

where e = 1, 2, 3, 4, m = 1, 2.

The probability P_e, e = 1, 2, 3, 4, of the parallel system is obtained as follows:

$\begin{matrix} P_{e} = R_{e 1} + R_{e 2} - R_{e 1} R_{e 2} . \end{matrix} (10.14)$

Now, the u-function of the k-out-of-n SWS is given by

$U_{e} (z) = P_{e} z^{e} + (1 - P_{e}) z^{0}$

where e = 1, 2, 3, 4, P_e is the probability function and z^e is the performance rate and z⁰ non performance rate.

Thus, the u-function of the k-out-of-n SWS components u_i(z) is given by

$u_{e} (z) = P_{e} z^{a} + (1 - P_{e}) z^{0}$

where a = 1, 2, 2, 2.

Images

FIGURE 10.1
Diagram of k-out-of-n SWS with k = 2, n = 3, m = 4, and r = 2.

In the initial step of the algorithm, the value of 0 is assigned to F. The initial u-function takes the form

$U_{- 1} (z) = z^{0, (0, 0)} .$

Using the algorithm 10.4.1, we get the u-function of the k-out-of-n SWS as follows:

For j = 1

$\begin{matrix} U_{0} (z) = (U_{- 1} (z) \otimes u_{1} (z)) \\ = z^{0, 0} \otimes P_{1} z^{1} + (1 - P_{1}) z^{0} \\ = P_{1} z^{0, (0, 1)} + (1 - P_{1}) z^{0, (0, 0)} \end{matrix}$

For j = 2

$\begin{matrix} U_{1} (z) = U_{0} (z) \otimes u_{2} (z) \\ = P_{1} z^{0, (0, 1)} + (1 - P_{1}) z^{0, (0, 0)} \otimes P_{2} z^{2} + (1 - P_{2}) z^{0} \end{matrix}$

$U_{1} (z) = P_{1} P_{2} z^{0, (1, 2)} + P_{1} (1 - P_{2}) z^{1, (1, 0)} + P_{2} (1 - P_{1}) z^{1, (0, 2)} + (1 - P_{1}) (1 - P_{2}) z^{1, (0, 0)}$

For j = 3

$\begin{matrix} U_{2} (z) = U_{1} (z) \otimes u_{3} (z) \\ = P_{1} P_{2} z^{0, (1, 2)} + P_{1} (1 - P_{2}) z^{1, (1, 0)} + P_{2} (1 - P_{1}) z^{1, (0, 2)} \\ + (1 - P_{1}) (1 - P_{2}) z^{1, (0, 0)} \otimes P_{3} z^{2} + (1 - P_{3}) z^{0} \end{matrix}$

$\begin{array}{l} U_{3} (z) = P_{1} P_{2} P_{3} z^{0, (2, 2)} + P_{1} (1 - P_{2}) P_{3} z^{2, (0, 2)} + (1 - P_{1}) P_{2} P_{3} z^{1, (2, 2)} \\ + (1 - P_{1}) (1 - P_{2}) P_{3} z^{2, (0, 2)} + P_{1} P_{2} (1 - P_{3}) z^{1, (2, 0)} \\ + P_{1} (1 - P_{2}) (1 - P_{3}) z^{2, (0, 0)} + (1 - P_{1}) P_{2} (1 - P_{3}) z^{2, (2, 0)} \\ + (1 - P_{1}) (1 - P_{2}) (1 - P_{3}) z^{2, (0, 0)} . \end{array}$

The like terms in U₂(z) are collected. The value of unreliability F can be expressed as follows:

$\begin{matrix} F = (1 - P_{2}) P_{3} + (1 - P_{1}) P_{2} (1 - P_{3}) + (1 - P_{2}) (1 - P_{3}) . \end{matrix} (10.15)$

After removing the terms in which the counter equals k = 2, U₂(z) can be written as follows:

$U_{2} (z) = P_{1} P_{2} P_{3} z^{0, (2, 2)} + (1 - P_{1}) P_{2} P_{3} z^{1, (2, 2)} + P_{1} P_{2} (1 - P_{3}) z^{1, (2, 0)} .$

After removing the terms that satisfy the condition (10.3) for k = 2, m = 4, and j = 3,

U₂(z) can be obtained as follows:

$U_{2} (z) = (1 - P_{1}) P_{2} P_{3} z^{1, (2, 2)} + P_{1} P_{2} (1 - P_{3}) z^{1, (2, 0)} .$

For j = 4

$\begin{matrix} U_{3} (z) = U_{2} (z) \otimes u_{4} (z) \\ = (1 - P_{1}) P_{2} P_{3} z^{1, (2, 2)} + P_{1} P_{2} (1 - P_{3}) z^{1, (2, 0)} \otimes P_{4} z^{2} + (1 - P_{4}) z^{0} \end{matrix}$

$\begin{array}{l} U_{3} (z) = (1 - P_{1}) P_{2} P_{3} P_{4} z^{1, (2, 2)} + P_{1} P_{2} (1 - P_{3}) P_{4} z^{2, (0, 2)} \\ + (1 - P_{1}) P_{2} P_{3} (1 - P_{4}) z^{2, (2, 0)} + P_{1} P_{2} (1 - P_{3}) (1 - P_{4}) z^{2, (0, 0)} . \end{array}$

Again like terms in U₃(z) are collected. The value of unreliability F can be expressed as follows:

$\begin{matrix} F = P_{1} P_{2} (1 - P_{3}) P_{4} + (1 - P_{1}) P_{2} P_{3} (1 - P_{4}) + P_{1} P_{2} (1 - P_{3}) (1 - P_{4}) . \end{matrix} (10.16)$

Now adding the equations (10.15) and (10.16), we finally get the unreliability:

$F = 1 - P_{1} P_{2} P_{3} - P_{2} P_{3} P_{4} + P_{1} P_{2} P_{3} P_{4}$

Reliability of the k-out-of-n SWS:

$\begin{matrix} R = 1 - F = P_{1} P_{2} P_{3} + P_{2} P_{3} P_{4} - P_{1} P_{2} P_{3} P_{4} . \end{matrix} (10.17)$

Hence, substituting the values of P_e (e = 1, 2, 3, 4, 5) in equation (10.17) from equation (10.14), we obtain the reliability function R of the k-out-of-n SWS as follows:

$\begin{matrix} \begin{array}{l} R = (R_{11} + R_{12} - R_{11} R_{12}) (R_{21} + R_{22} - R_{21} R_{22}) (R_{31} + R_{32} - R_{31} R_{32}) \\ + (R_{21} + R_{22} - R_{21} R_{22}) (R_{31} + R_{32} - R_{31} R_{32}) (R_{41} + R_{42} - R_{41} R_{42}) \\ - (R_{11} + R_{12} - R_{11} R_{12}) (R_{21} + R_{22} - R_{21} R_{22}) (R_{31} + R_{32} - R_{31} R_{32}) \\ \times (R_{41} + R_{42} - R_{41} R_{42}) . \end{array} \end{matrix} (10.18)$

When elements are identically distributed $(R_{e m} \equiv R)$ , reliability function $R (R_{1}, . . ., R_{8})$ of the components of the k-out-of-n SWS and structure function h of the proposed system are given by

$R (R_{1}, . . ., R_{8}) = 16 R^{3} - 40 R^{4} + 44 R^{5} - 26 R^{6} + 8 R^{7} - R^{8}$

and

$H (P_{1}, . . ., P_{8}) = 16 P^{3} - 40 P^{4} + 44 P^{5} - 26 P^{6} + 8 P^{7} - P^{8} .$

10.5.1 Signature of the k-out-of-n SWS

By using Owen’s method on the components of the k-out-of-n SWS, we get reliability function in the form of H(y) as follows:

$\begin{matrix} H (y) = 16 y^{3} - 40 y^{4} + 44 y^{5} - 26 y^{6} + 8 y^{7} - y^{8} . \end{matrix} (10.19)$

Now using equations (10.8) and (10.19), structure function can be expressed as follows:

$P (y) = y^{8} H (\frac{1}{y}) = - 1 + 8 y - 26 y^{2} + 44 y^{3} - 40 y^{4} + 16 y^{5} .$

With the help of step 4 of algorithm 10.4.2, we get the tail signature B for individual element of the k-out-of-n SWS as follows:

$B_{0} = 1, B_{1} = 1, B_{2} = \frac{13}{14}, B_{3} = \frac{11}{14}, B_{4} = \frac{4}{7},$

$B_{5} = \frac{2}{7}, B_{6} = 0, B_{7} = 0, B_{8} = 0 .$

Hence, the tail signature of the k-out-of-n SWS is given by

$B = (1, 1, \frac{13}{14}, \frac{11}{14}, \frac{4}{7}, \frac{2}{7}, 0, 0, 0) .$

Again using step 5 of algorithm 10.4.2, we obtain the signature of the k-out-of-n SWS as follows:

$b = (0, \frac{1}{14}, \frac{1}{7}, \frac{3}{14}, \frac{2}{7}, \frac{2}{7}, 0, 0) .$

10.5.2 MTTF of the k-out-of-n SWS

To compute MTTF using equation (10.19), we get a minimal signature M as follows: $M = (0, 0, 16, - 40, 44, - 26, 8, - 1)$ of the k-out-of-n SWS elements.

Using the steps 2 and 3 of algorithm 10.4.3, we get MTTF

$E (t) = 0.818 .$

10.5.3 Expected Cost

Using step 1 of algorithm 10.4.4, the expected value of X of the k-out-of-n SWS can be computed as follows:

$E (X) = \sum_{j = 1}^{8} j \cdot B_{j}, j = 1, 2, . . ., 8 .$

Hence, the expected value of X is given by

$E (X) = 4.57 .$

Now, using step 2 of algorithm 10.4.4, we can compute the expected cost of the k-out-of-n SWS as follows:

$Expected cost = \frac{E (X)}{E (T)} = 5.5867 .$

10.6 Conclusion

In this study, we considered a k-out-n SWS consisting of n linearly ordered multistate components with m parallel components. We evaluated signature, tail signature, expected lifetime, and expected cost. Signatures increased with increasing parallel component, expected lifetime was 0.818, and expected cost was 5.5867.

Nomenclature

n = number of multistate element (MSE) in the system
m = consecutive i.i.d. components
r = number of consecutive window
w = total allocation weight
k = maximal allocation consecutive groups.
$U_{a} (z)$ = u-function of the r consecutive MSE a
$u_{i} (z)$ = u-function of the system
$\otimes$ = composition operator
g_i,b = the performance state of MSE in state b
y_a,l = random vector in lth state of the r consecutive groups
C_a,l = integer counter of consecutive failed groups
P_i,b = the probability of MSE i is in state b
q_i,b = probability in bth state of the r consecutive groups
E_a = probability of m consecutive groups
σ(y) = sum of element vector y
ϕ(y, g) = shifting operator
R/F/S/s/H = reliability/unreliability/tail signature/signature/reliability function of the system
E(T)/E(X) = expected lifetime/expected value of X of the system components
C_e =minimum signature for components e

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Table of Contents for
10. Signature Reliability of k-out-of-n Sliding Window System

10

Signature Reliability of k-out-of-n Sliding Window System

CONTENTS

10.1 Introduction