The Reliability of a System Involving Change Points

The reliability of a system having some change points is presented. The technique of calculation is based on a previously developed TFCF procedure for evaluating the reliability for i.i.d. component. It involves the use of some auxiliary functions to set up a set of recursive relations. The resultant equations are solved numerically. An extension to the more general TSCSTFCF procedure and its application to start-up demonstration tests is given. Also, in case of testing, the possibility of carrying out simultaneous tests on a set of units is considered. KeywordsReliability, Change points, Start-up demonstration tests, Point mass distribution. Notation TSCSTFCF total successes consecutive successes total failures consecutive failures ks total number of successes kf total number of failures kcs length of run of successes kcf length of run of failures Tn,s the total number of successes up to the n'th test Tn,f the total number of failures up to the n'th test Ln,s the length of the largest run of successes up to n Xn the output at the n-th test: =1 success, =0 failure N the random number of tests till stopping u[.] the unit step function [.]  the delta function.


Introduction
The way of evaluating the reliability of systems consisting of n identical components is commonly known. An example of a reference book on the basic theory of consecutive k-out-of-n systems is of Kuo and Zou (2003). A comparison of various procedures for performing tests that are based on this theory has been presented by Gera (2018). Matters become more cumbersome when we must deal with systems that have various parts with different reliabilities. Some work has been carried out for systems with non-identical elements (see for instance Smith and Griffith, 2011). The application of the theory to start-up demonstration tests has been presented by Balakrishnan et al. (2014) and others. We divide the system into several parts (at so-called "change points") so that the components within each part are identical from the point of view of their reliabilities. Eryilmaz (2016) handled the case of a single change point. An extension to the case of two change points has been presented by Peng and Xiao (2018). They developed non-recursive formulas for calculating the survival function and for the reliability of such systems. An example of a pipeline which transports oil from one location to the other has been shown. Due to different environmental conditions along the various parts of the pipeline, there are assumed different reliabilities for each of its parts.
A general method for analyzing the survival function and the point mass distribution function for systems having components (or tests) with identical and independent reliabilities (probabilities of success) has been introduced by Gera (2010Gera ( , 2011Gera ( , 2019)the TSCSTFCF procedure. It has been used for start-up demonstration tests. The procedure is based on some auxiliary functions which lead to a set of recursive relations that are normally solved numerically. This method is extended here to include change points.

A Single Change Point
A change point is given at n=n1 within the distribution of the probabilities of success throughout the different tests: A consecutive kcs-out-of-n G:system is considered.
Let Ln,s be the length of the largest run of successes up to n and let Tn,s be the total number of successes up to the n. Xn is the output at the n'th test: Define the auxiliary functions for r=0,1: Gera (2011Gera ( , 2019 presented interconnecting relations for a more general case (TSCSTFCF). Meanwhile we confine the presentation to the CS (consecutive successes) procedure.  and the reliability of a G: system up to n is given by Example A: Calculation of R (G) (n) for p1=0.75, p2=0.5; change point at n1=3, kcs=3. 's' stands for success and 'f' stands for failure, '1' may be any of them (Table 1).  Obviously, the TSCS procedure exhibits increased values of the reliability compared to those of the CS system.

Two Change Points
A generalization to two change points at n1, n2 is presented. Accordingly, where, The boundary conditions for the system of equations (12)-(15) are: It may be noted that for a CF system (F: system), the reliability is given by: The auxiliary functions are evaluated as before but with the proper interchanging of the roles of the pr and qr variables.
Example C: The reliability of an oil pipeline system. It is considered as an F: system composed of three parts. Considering various combinations of the parameters of the system, we get identical results to those of Peng and Xiao (2018) as presented in their Table 1.

Extension of the Procedure
Like for the case of a single change point, we may generalize the above procedure to include also the total number of failures (the TFCF procedure).
Let kf denote the total number of failures. The tail distribution of N is given in this case by: Example D: The reliability of a TFCF system, n=20, various values of kf, kcf (Table 3).

Application to Start-Up Demonstration Tests
Start-up demonstration sets of tests are often assumed to be i.i.d.(identical independent). In contrast, the probabilities of success of each test may change due to different environmental conditions. Thus, we may apply the above theory also to such tests. Here we may apply a generalized version of the former CSCF, TSCS and TFCF proceduresthe TSCSTFCF procedure, Gera (2011Gera ( , 2018. The set of tests is stopped if either there is a total number of successes or a run of consecutive successes or if we observe a certain total number of failures or a run of failures. Accordingly, a decision regarding the acceptance or rejection of the tested unit is then taken. The aim here is to shorten as much as possible the number of tests subject to some constraints. Referring to (19), the point distribution of N will be given by (for n>1): The generalization of the auxiliary functions and the interconnecting relations between them has already been given before for the i.i.d. case.
Like before, appropriate boundary conditions must be taken into account. Then, use is made of (19) and (20).
Example E: CS procedure. The point mass distribution for some values of n. p1=0.75, p2=0.5, n1=2, kcs=2 (Table 4). Example F: TSCSTFCF procedure. p1=0.75, p2=0.5, n1=3, kcs=kcf=2, without (1) and with the addition of the constraint ks=kf=3 (2) ( Table 5). Considering two change points, we derive the following generalization of the previous interconnecting relations: For n > i ≥ 1: n n n S a n n i f q q a n i f q n n n n a n i f q q a n i f q n n a n i f q n i f n n n k a n n n n k a a n n n a n n k a n n k n a a n n a

Several Parallel Tests
Instead of testing a single unit, it is often preferable to test more units in parallel if we have such spare units at our disposal Gera (2015Gera ( , 2018. The number of units is denoted by M. Instead of the previous runs of successes or failures, here we deal with planar squares and rectangles. We thus look for the occurrence of a rectangle Mxkcs of successes and/or a rectangle Mxkcf of failures. Let RM,n,s be the maximal area of rectangles formed by squares till performing the n'th stage of parallel tests and let RM,n,f be the appropriate area for failures. In analogy to before, Tn,s presents the total number of successes till the n'th stage for all M units and likewise for Tnf. Like before, consider a single change point (1).

Conclusion
The TSCSTFCF procedure for evaluating the reliability of consecutive k-out-of-n systems of components (or tests) has been generalized to include change points within the model. Whereas initially it has been assumed that all components (or tests) have identical reliabilities (or probabilities of success), here we considered systems which are comprised of sets of components (tests) each of which have different reliabilities. The change points indicate the distinction between the parts.
A previously developed method of using some auxiliary functions to evaluate the reliability of a CF system with change points has been presented. A set of recursive equations is created which are solved numerically. A generalization to TFCF systems has been given. A further extension to start-up demonstration tests involving the TSCSTFCF procedures has been added.
A practical application of the technique may be for instance for a long distance oil pipeline system. Such a line may be divided into several sections according to different environmental conditions. Each part exhibits a different value for its reliability. Likewise, sets of tests may be set up according to the difficulty of passing each test.
It should be added that the components (tests) were assumed to be independent. However, normally we meet systems that involve some dependence between the components. Also, we assumed that the components either work or fail. Actually, we often observe degradation in the