Signature of A-Within-B-From-D/G Sliding Window System

In this study, we have proposed a model of the sliding window coherent system in case of multiple failures. The considered model consists of G linearly required multi-state elements and G number of parallel elements in A-within-Bfrom-D/G for each multi-state. The system fails if at least A group elements out of B consecutive of D consecutive multi-state elements have performance lower than the weight w. We have evaluated the signature reliability, expected value and system sensitivity on the basis of the extended universal generating function of the considered system. KeywordsSliding window system, Universal generating function, Signature reliability, Sensitivity.


Introduction
In the real life situation binary state system (BSS) depends on mainly two states namely completely working or total failure. To compute the reliability of any binary system many algorithms have been used including universal generating function (UGF). Levitin (2005) analyzed the computation of reliability of different binary and consecutive A-out-of-G systems by UGF. Levitin and Ben-Haim (2011) computed the reliability of the consecutive sliding window system (SWS) which have many possible states namely total failure and completely working using UGF algorithm. Sun et al. (2012) obtained the optimal solution for a transportation system by the analytical method. Ram (2013) discussed the survey of reliability evaluation of engineering system by various methods. Xiao et al. (2014) considered a B-gap-consecutive A-outof-D-from-G:F system and computed the reliability for various elements causing failure. Negi and Singh (2015) studied the non-repairable complex system which has two binary subsystems namely weighted A-out-of-G:G and weighted l-out-of-b:G and evaluated the reliability, mean time to failure and sensitivity using UGF.
Further, in the context of the signature reliability of the coherent system is widely used to calculate the expected lifetime of any kind of system with independent and identically distributed (i.i.d.) elements. Navarro et al. (2007) introduced the family of univariate distribution. They computed the minimal and the maximal signature of a coherent system along with distribution, bounds and moments of lifetime distribution. Samaniego (2007) discussed the signature of different systems and applied signature in many engineering fields. Bhattacharya and Samaniego (2008) appraised the optimal arrangement of the element in the coherent system and evaluated the optimal solution of parallel and series-parallel systems. Navarro and Hernandez (2008) studied the mean residual lifetime functions of the finite mixture, ordering properties and limiting behaviors. They evaluated the meantime and signature of the coherent system. Navarro and Rubio (2009) computed the signature reliability and expected a lifetime of the coherent system with n elements. Eryimaz (2010) evaluated the reliability of consecutive k-system with some exchangeable element with the help of order stochastic of mixture representation. Navarro and Rychlik (2010) compared the expected lifetime of different systems and estimated the lifetime of i.i.d. elements in the lower and upper form. Mahmoudi and Asadi (2011) considered a coherent system and studied the dynamic signature with different properties of the signature. Marichal and Mathonet (2013) evaluated the weighted mean in case of an independent continuous lifetime and obtained the signature reliability of extension dependent lifetime of the coherent system. Eryilmaz (2012) determined the signature of a coherent system with the repairable element and calculated the expected lifetime for systems like linear consecutive A-within-B-out-of-G:F and Bconsecutive A-out-of-G:F. Da et al. (2012) computed the signature of the coherent system which decomposed into two or more subsystems and also using the redundancy of the backup system. Marichal and Mathonet (2013) evaluated the reliability function, signature, tail signature of the coherent system from the diagonal section by derivatives and with the help of structure function. Da Costa Bueno (2013) obtained the structure function of the multi-state monotone system by using a decomposition of multi-state systems. Franko and Tutuncu (2016) computed the reliability of repairable weighted A-out-of-G:G system in case of signature. Singh (2017a, 2017b) studied the complex A-out-of-G coherent system and sliding window coherent system (SWCS) with i.i.d. elements and calculated various reliability measures such as signature, mean time to failure (MTTF), Barlow-Proschan index using UGF technique. It is clear from the above discussions that many researchers computed the reliability, MTTF, cost of binary and multi-state systems with various techniques, but the signature of SWCS is yet to be studied. Keeping this fact in view, in the present work we propose to study the A-within-B-from-D/G SWCS with G parallel i.i.d. elements consisting of the multi-state element (MSE) of the system. In this study, we have used UGF and Owen's method to estimate the different characteristics such as signature, tail signature, sensitivity, Barlow-Proschan index and expected lifetime having structure or reliability function.

Assessment of Signature Reliability of A-Within-B-From-D/GSWCS (Xiao et al., 2015).
Consider an A-within-B-from-D/G SWCS which contains G ordered MSE in which every element consists of G number of parallel elements. The failure element of the A-within-B-from-D/G system is presented when at least A-out of B-consecutive groups of D consecutive elements are greater than supply w. If the performance of D consecutive elements is less than weight w and a consecutive element consists of MSE, then the system fails if  (1) From equation (1), we can evaluate the signature of the system having i.i.d. elements as , where T is system lifetime and A s is the probability of the system failure. Boland (2001) obtained structure function R of the system having i.i.d. elements as

Evaluating the Failure Reliability of A-Within-B-From-D/G Failure Groups (Xiao et al., 2015)
The failure element of the A-within-B-from-D/G system is discussed as if at least A-groups of Bconsecutive groups of D consecutive elements are not less than supply w.
UGF of the failure element of the system with probability i a R , and state performance i a g , is given Now, with the help of equation (3) Using equation (8) .
Similarly, one can evaluate the value of and collect the like terms.
Step 4. For Step 5. Find and collect the like terms.
Step 6. Calculate Step 7. Find system reliability F R  1 .

Algorithm for Evaluating Signature of A-Within-B-From-D/G SWCS with Its Reliability Function
Step 1: Calculate the signature of the reliability function by (Boland, 2001)  Step 2: Evaluate the tail signature of the system, i.e., (B+1)-tuple Step 3: Calculate the reliability function with the help of Taylor expansion from polynomial function about x=1 by Step 4: Find the tail signature of the system reliability function from equation (12) as (Marichal and Mathonet, 2013 Step 5: Determine the signature of the system using equation (14) . ,..., 1 ,

Algorithm to Determine Expected Lifetime of A-Within-B-From-D/G System with Minimum Signature
Step 1: Evaluate the expected lifetime of i.i.d. element system, which is exponentially distributed with mean  .
Step 2: Calculate the minimum signature of the A-within-B-from-D/G system with the expected lifetime of the reliability function by using ,..., 2 , 1 n i  Step 3: Compute the expected lifetime E(T) of the systems, which have i.i.d. elements by (Navarro and Rubio, 2009).

Algorithm to Calculate Barlow-Proschan index of SWCS
Estimate the Barlow and Proschan (1975) index of the i.i.d. elements are given by its reliability function in equation (2) as (Shapley, 1953;Owen, 1975Owen, , 1988

Algorithm for Determining Expected Value of Element X and Expected Cost Rate of System When Working Elements are Failed (Eryilmaz, 2012)
Step 1: Evaluate the amount of failed elements at the time of system failure from signature Step 2: Calculate the E(X) and E(X)/E(T) of A-within-B-from-D/GSWCS with minimum signature.

Sensitivity of A-Within-B-From-D/GSWCS
The sensitivity of reliability function is defined as the rate of change in output due to an input of the system. If R and  are the reliability and parameter of the system respectively, then sensitivity S with the parameter is expressed as
The probability function   where, i p is the probability function and i z is the working rate and 0 z failure rate.
Now, using step1 of algorithm 1, we have UGF as

Expected Cost Rate of 2-Within-3-From-3/5 SWCS
By using equation (20), we get the expected value of X is E(X)   X E as E(X)=4.5. Barlow-Proschan index of the considered system is obtained using equation (19)  Similarly, obtain all Barlow-Proschan index of the 2-within-3-from-3/5 SWCS as

Result and Discussion
In this study, we deal with A-within-B-from-D/G SWCS and computed different reliability measures , viz. signature, expected lifetime, Barlow-Proschan Index and sensitivity of the proposed system.

Conclusion
In the present paper, we have studied the A-within-B-from-D/G SWCS incorporating multiple failures. An algorithm for evaluating signature estimation on the basis of Owen's method and UGF technique has been used for the considered system. The algorithm is based on structure function, which has been used to evaluate the signature of the proposed system. The results show that the system signature is increasing w.r.t. the price value of the expected cost. Sensitivity with respect to parameters R32 and R42 is found to be highest and lowest respectively.