Reliability evaluation of systems with interdependent components

ABSTRACT

In this paper an attempt has been made to develop a model to calculate the reliability of a r-out-of-n: G system with interdependent components. The model considered here is particularly suitable for load-sharing devices in which failure of components causes higher load and hence induces a higher failure rate in each of the surviving components. The components are however non-repairable and failed components are not replaced. The model is quite flexible and realistic in the sense that it incorporates several dependency parameters and by giving them different values one may bring within its ambit systems of different kinds, ranging from systems with independent failures (no load-sharing) to systems which fail with the failure of a single component. It is assumed that failure time distribution of the components can be represented by Weibull probability model. The expression for reliability reported here is computationally tractable as it can be expressed in the form of incomplete Beta function.

1. INTRODUCTION

In most investigations related to the evaluation of reliability and other life characteristics of various types of systems, it has been tacitly assumed that components work independently in the sense that as long as the system functions, failure of some of the components does not affect the failure rate (or other life characteristics) of the surviving ones and hence their inherent failure time distributions have no mutual interdependence. More specifically, most researchers have assumed that the random variables representing the lifetime of components in a r-out-of-n:G system are identically and independently distributed and follow exponential probability law (Crowder et al., 1991, Hassett et al., 1995, Kapur and Lamberson, 1977, Lewis, 1987 Shao and Lamberson, 1991, Jones and Hays, 2001). In the real-world, such an assumption may not be tenable. For example, if one component fails, the same work-load is supposed to be shared by the surviving components and consequently there will be an increase in the load shared by each surviving component (Liu, 1998). Increase in load, in turn, affects in a very significant way the failure probabilities of the surviving components (Carter, 1986, Castillo and Siewiorek, 1982, lyer and Rosetti, 1986).

For evaluating the reliability of load-sharing r-out-of-n: G systems, some of the past researchers have considered the joint density function approach (Bhattacharya and Bhattacharji, 1989). This approach has the advantage that it enables one to go beyond the framework of exponentiality and random variables representing the lifetime of surviving components can be supposed to follow Weibull probability law. In order to reduce the number of independent parameters involved in expression for reliability, it is assumed that there is a specific mathematical relationship between the number of failed components and the scale parameter of the Weibull distribution. This relationship takes into consideration the systems in which failure rate of the surviving components decreases with an increase in the degree of dependence (that is, larger the value of dependency parameter). In a more realistic situation, however, the former is expected to increase with an increase in the degree of dependence. The objective of the present investigation is to evaluate reliability of systems which can take into consideration such a situation. For the purpose, 'polynomial type model' has been proposed in this study. Interestingly, the reliability function reported here is mathematically quite elegant in the sense that it can be expressed in the form of incomplete Beta function.

2. CHARACTERISTICS OF SYSTEM UNDER CONSIDERATION

Let N denote the number of parallel subsystems forming a system and [r.sub.j] the number of essential components in subsystem j, j = 1, 2 ..., N. The jth subsystem initially consists of [n.sub.j] components and it fails when ([n.sub.j] - [r.sub.j] + 1) of its components fail. The subsystem j is thus characterized by a [r.sub.j]-out-of-[n.sub.j]: G configuration. The failed components are not replaced. The time of ith failure in the jth subsystem has been denoted by the random variable [X.sub.(i) j]; I = 1, 2 ..... [n.sub.j] and j = 1,2 .... N. Each parallel subsystem is supposed to work independently. With the passage of time, the number of failed components in subsystem j goes on increasing. The subsystem j may be said to be in the state [m.sub.j] if [m.sub.j] out of [n.sub.j] components have failed. The state of the system can then be denoted by an N-tuple ([m.sub.1], [m.sub.2] ..., [m.sub.N]). A system may fail with the failure of a single subsystem or it may require failure of several subsystems. It is further assumed that for a given system configuration, the system reliability at a point of time t can be expressed as a known function of subsystem reliabilities, [R.sub.j](t); j = 1, 2 ..., N. For example, in the case of a series-cum-parallel system, the system reliability [R.sub.s](t) at a point of time t can be expressed as

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In general, [R.sub.j](t) is given by

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [f.sup.j] (t) is the failure density function of subsystem j, that is, [f.sup.j] (t) dt gives the probability of failure of jth subsystem in the time interval (t, t + dt). For deriving an expression for [f.sup.j](t), the joint density function f ([x.sub.(1)j], [x.sub.(2)j], ..., [X.sub.(nj)j]) of the random variables [X.sub.(i)j]; i = 1, 2 ... [n.sub.j] is first obtained for a specific inequality:

(3) 0 [less than or equal to] [x.sub.(1)j] [less than or equal to] [x.sub.(2)j] [less than or equal to] [x.sub.(3)j] [less than or equal to] ... [less than or equal to] [x.sub.(mj)j] [less than or equal to] ... [less than or equal to] [x.sub.(nj)j] < infinity

For the sake of clarity and brevity, the proposed model is first explained for evaluating the reliability [R.sub.j](t) of a subsystem having a 2-out-of-4 :G configuration.

3. EVALUATION OF [R.sub.j](t) BASED ON PROPOSED MODEL FOR 2-OUT-OF-4: G SUBSYSTEM

A 2-out-of-4: G subsystem fails when any three of its four components fail. It is assumed that random variable [X.sub.(i)j]; i = 1,2,3,4 denoting the time of failure of ith component in jth subsystem, follows Weibull probability law with probability density function given by:

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [p.sub.j] are the scale and shape parameters respectively. The joint density function of the random variables [X.sub.(1)j], [X.sub.(z)j], [X.sub.(3)j], [X.sub.(4)j] for the inequality

0 [less than or equal to] [x.sub.(1)j] [less than or equal to] [x.sub.(2)j] [less than or equal to] [x.sub.(3)j] [less than or equal to] [x.sub.(4)j] < [infinity]

can then be expressed as:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since the subsystem fails with the failure of any three of its four components, the random variables [X.sub.(i)j] therefore have to satisfy any one of the [sup.4][p.sub.3] = 24 possible inequalities of the type:

(6) 0 [less than or equal to] [x.sub.(1)j] [less than or equal to] [x.sub.(2)j] [less than or equal to] [x.sub.(3)j] < [infinity]

For example, denoting the four components in the subsystem by [C.sub.1], [[C.sub.2], [C.sub.3] and [C.sub.4], the failure of subsystem occurs when any one of the following 24 failure sequences of components is observed

{[C.sub.1], [C.sub.2], [C.sub.3]}, {[C.sub.1], [C.sub.2], [C.sub.4]}, {[C.sub.1], [C.sub.3], [C.sub.4]}, {[C.sub.1], [C.sub.3], [[C.sub.2]}, {[C.sub.1], [C.sub.4], [C.sub.2]}, {[[C.sub.1], [C.sub.4], [C.sub.3]}

{[C.sub.2]. [C.sub.1], [C.sub.3]}, {[C.sub.2], [C.sub.1], [C.sub.4]}, {[C.sub.2], [C.sub.3], [C.sub.4]}, {[C.sub.2], [C.sub.3], [C.sub.1]}, {[C.sub.2], [C.sub.4], [C.sub.1]}, {[C.sub.2], [C.sub.4], [C.sub.3]}

{[C.sub.3], [C.sub.2], [C.sub.1]}, {[C.sub.3], [C.sub.2], [C.sub.4]}, {[C.sub.3], [C.sub.1], [C.sub.4]}, {[C.sub.3], [C.sub.1], [C.sub.2]}, {[C.sub.3], [C.sub.4], [C.sub.2]}, {[C.sub.3], [C.sub.4], [C.sub.1]}

{[C.sub.4], [C.sub.2], [C.sub.3]}, {[C.sub.4], [C.sub.2], [C.sub.1]}, {[C.sub.4], [C.sub.3], [C.sub.1]}, {[C.sub.4], [C.sub.3], [C.sub.2]}, {[C.sub.4], [C.sub.1], [C.sub.2]}, {[C.sub.4], [C.sub.1], [C.sub.3]}

As these are mutually exclusive cases, the probability of failure of the subsystem is obtained by adding the probabilities of 24 failure sequences stated above. These probabilities, for the Weibull density function (4), can be obtained by integrating the joint density function (5) as follows:

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII.]

that is,

(7) [MATHEMATICAL NOT REPRODUCIBLE IN ASCII.]

In their study relating to the life characteristics of systems with interdependent failures, Bhattacharya and ([m.sub.j]) Bhattacharji (1989), have considered the following relationship between [a.sub.(i)j] and the number of failed

components, [m.sub.j] for a r-out-of-n :G system

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where, [[beta].sub.fj] is the value of scale parameter of the Weibull density function, given by (4) and [K.sub.j] is the dependency parameter; [K.sub.j] = 0 corresponds to the case of independent working of components. The linear dependence is reflected by [K.sub.j] = 1. It may, however, be observed from Equation (7), that larger the value of [K.sub.j], lower is the value of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and hence lower is the failure rate of surviving components. Physically, the failure rate is supposed to increase with an increase in the degree of dependence among different surviving components. Equation (8) may therefore be replaced by a more general 'polynomial type model' of the form:

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where, [k.sub.ij], 1 = 0, 1, 2 ..., [w.sub.j] are dependency parameters and [K.sub.oj] > [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. The advantage of the model (9) lies in its flexibility. The case of independently working components can be considered by taking [K.sub.lj] = 0. By suitably choosing different values of [K.sub.lj] and [w.sub.j]), one can obtain the expressions (5) and (7) for evaluating the reliability of systems with interdependent components. For example, for [w.sub.j] = 1, the equation (9) becomes a linear function of (1 / ([n.sub.j] - [m.sub.j])) and it takes the form:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For the relationship (10) and for [r.sub.j] = 2, [n.sub.j] = 4, the values of a[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] are:

[a.sup.(0).sub.(1)j] = [a.sup.(0).sub.(2)j] = [a.sup.(0).sub.(3)j] = [a.sup.(0).sub.(4)j] = [k.sub.0j] + [k.sub.1j]/4

[a.sup.(1).sub.(1)j] = [a.sup.(1).sub.(2)j] = [a.sup.(1).sub.(3)j] = [a.sup.(1).sub.(4)j] = [k.sub.0j] + [k.sub.1j]/3

[a.sup.(2).sub.(1)j] = [a.sup.(2).sub.(2)j] = [a.sup.(2).sub.(3)j] = [a.sup.(2).sub.(4)j] = [k.sub.0j] + [k.sub.1j]/2

With these values, the function g(.) given by (6) can be expressed as (after suppressing the subscript 'j'):

(11) g([x.sub.(1)], [x.sub.(2)], [x.sub.(3)]) = 1/24 ([k.sub.1] + 4[K.sub.0]) ([k.sub.1] + 3[K.sub.0])([k.sub.1] + 2[K.sub.0]). [p.sub.3] ([x.sub.(1)*[x.sub.(2)]*[x.sub.(3)])[sup.-1] l exp(-[k.sub.0]([x.sup.p.sub.(1)] + [x.sup.p.sub.(2)])) * exp (-([k.sub.1] + 2[K.sub.0]) [x.sup.p.sub.(3))

The failure density function f(t) in Equation (2) for a given subsystem is then expressed as the sum of 24 terms corresponding to 24 possible failure sequences stated above. Each term of the sum is being obtained by integrating the g-function (Equation 11) over the ranges specified by (6). As the model (10) assumes identical components, f(t) works out to be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where, B =([k.sub.l]+4 [k.sub.0])([k.sup.l] + 3[k.sub.0]([k.sup.l]+2[k.sub.0])/24

and, finally from (1),

(12) R(t) = [B.sub.0] exp (-([k.sub.1] + 4[k.sub.0] [t.sup.p] + [B.sub.1] exp (-([k.sub.i] + 3[K.sub.0] [t.sup.p]) + [B.sub.2] exp(-([k.sub.1] + 2[k.sub.0] [t.sub.p)

where,

[B.sub.0] = ([k.sub.1] + 3 [k.sub.0])([k.sub.1] + 2 [k.sub.0]); [B.sub.1] = - ([k.sub.1] + 4 [k.sub.0])([k.sub.1] + 2 [k.sub.0]); [B.sub.2] + ([k.sub.1] + 4 [k.sub.0])([k.sub.1] + 4 [k.sub.0])

The reliability function (12) can also be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which is in the form of an incomplete Beta function with truncation point q = exp (- [k.sub.0] [t.sup.p]) and parameters ([k.sub.1] / [k.sub.0]) + 2 and 3.

4. EVALUATION OF [R.sub.j](t) BASED ON PROPOSED MODEL FOR A GENERAL [r.sub.j]-OUT-OF-[n.sub.j]; G SYSTEM

In a general [r.sub.j]-out-of-[n.sub.j] :G subsystem, the random variables [X.sub.(1)j] ; 1 + 1,2, ..., nj; j = 1, 2,. N, representing the time of failure of [i.sup.th] component in [j.sup.th] subsystem satisfy the inequality (3). It is further assumed that the random variables follow the Weibull probability distribution (4) and load sharing mechanism is described by the proposed model (10). Under these assumption and proceeding in the way explained in Section 3, it may be seen that, after algebraic simplification, the reliability function [R.sub.j] (t) can be expressed as:

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It may be observed that the expression (13) is a straightway extension of (12). The reliability function (13) can also be expressed in the form of an incomplete Beta function:

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where, q = exp (- [k.sub.0] [t.sup.p])

Once the reliability function for [j.sup.th] subsystem is evaluated by using (13) or (14), one can obtain the system reliability [R.sub.s] (t) from (1) for a series-cum-parallel system with interdependent components.

CONCLUDING REMARKS

A more realistic situation has been visualized in this investigation for evaluating the reliability of systems with interdependent components. The system is supposed to consists of independently working subsystems, each with an r-out-of-n: G configuration. For a given subsystem the 'polynomial type' model proposed here is quite flexible in the sense that it may be adopted to study various kinds of subsystems, ranging from complete independent to highly dependent. Mathematical expressions for reliability are however quite cumbersome for higher degree of polynomial. Interestingly, it has been shown here that for a special case the reliability function can be expressed in the form of incomplete Beta function.

REFERENCES

Bhattacharya, A. and Bhattacharji, A. K. "Life Characteristics of Systems with Dependent Failures", Microelectronics and Reliability, Vol. 29, 1989, 517-521.

Carter, A. D. S., Mechanical Reliability, Macmillan, 1986.

Castillo, X and Siewiorek, D. P., "A Workload Dependent Software Reliability Prediction Model", Proc. 12th International Symp. Fault-Tolerant Computers, IEEE Computer Soc. Press, 279-286, 1982.

Crowder, M. J., Kimber, A. C., Smith, R. L. and Sweeting, T. J., Statistical Analysis of Reliability Data, Chapman & Hall, 1991.

Hassett, T. F., Dietrich, D.L. and Szidarovszky, F., "Time-varying Failure Rates in the Availability and Reliability Analysis of Repairable Systems", IEEE Trans. Reliability, Vol. 44, 1995, 155-160.

Iyer, R. K. and Rosetti, D. P., "A Measurement-Based Model for Work Dependency of CPU Errors", IEEE Trans. Computers, Vol. C-35, 1986, 511-519.

Jones, J. and Hayes, J., "Estimation of System Reliability Using a Non-Constant Failure Rate Model", IEEE Trans. Reliability, Vol. 50, 2001, 286-288.

Kapur, K. C. and Lamberson, L. R., Reliability in Engineering Desiqn, John Wiley & Sons, 1977. Lewis, E.E., Introduction to Reliability Enqineering, John Wiley & Sons, 1987.

Liu, H., "Reliability of a Load-Sharing k-out-of-n:G System: Non-iid Components with Arbitrary Distributions", IEEE Trans. Reliability, Vol, 47, 1998, 279-284

Shao, J. and Lamberson, L. R., "Modelling a Shared-load k-out-of-n: G System", IEEE Trans. Reliability, Vol. 40, 1991, 205-209.

Author Profile:

Dr. Abhijit Bhattacharya earned his Ph.D. at Indian Institute of Technology, Kharagpur, India in 1987. He is a Fellow of the Royal Statistical Society, London. Currently, he is a Professor in Indian Institute of Management, Lucknow, India.

Abhijit Bhattacharya, Indian Institute of Management, Lucknow, India

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