use of stochastic networks in reliability analysis

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Use of Stochastic Networks in Reliability AnalysisJ. K. Byers a & R. W. Skeith ba University of Missouri-Rolla ,b University of Arkansas ,Published online: 09 Jul 2007.

To cite this article: J. K. Byers & R. W. Skeith (1972) Use of Stochastic Networks in Reliability Analysis, A I I ETransactions, 4:3, 169-177, DOI: 10.1080/05695557208974846

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Use of Stochastic Networks in Reliability Analysis1

J. K. BYERS Senior Member, AIIE

University of Missouri-Rolla

R. W. SKEITH Senior Member, AIIE University of Arkansas

Abstract: The application of stochastic networks t o reliability analysis is discussed. The article il- lustrates how the conventional "reliability block diagram" can be replaced by a stochastic network. Once the stochastic network is developed the analyst is able t o determine the mean and variance of the reliability of complex systems with relative ease by using the computer program presented. The program is coded in FORTRAN IV and will evaluate the reliability of any mixture of series and parallel subsystems. An example problem is presented in order t o illustrate the technique and the use of the program.

.The use of networks and network analysis is playing an increasingly important role in the analysis of complex systems. Previous articles published by Pritsker (2, 3, 4) have been primarily concerned with a solution to stochastic networks which have additive properties. GERT, Graphical Evaluation and Review Technique, is a method developed for analyzing this type of network. Whitehouse (6) investigated the use of GERT for analyzing reliability problems. He was interested in obtaining information other than the system reliability, e.g., expected life of an expensive part that is repeatedly used. The purpose of this article is to apply the use of stochastic networks to determine the reliability of systems which consist of an arbitrary mixture of series and parallel subsystems.

Traditionally, reliability analyses have been per- formed using the "reliability block diagram." Al- though the block diagram is sufficient for hand cal- culations, it is not convenient to use for computer solutions to complex systems. Additional information is required beyond that given in the block diagram to develop a computer program to analyze a system which consists of a mixture of series and parallel subsystems. This additional information is provided by using a stochastic network. A computer program written in Fortran IV has been developed to analyze the network

to obtain the system reliabilit~.~ Because of various valid reasons, it is realized that some analysts and their companies still desire to use the block diagram. By adding additional information to the block diagram, the computer program can be used to determine the system reliability without the transformation to the stochastic network.

The objective in analyzing the stochastic network is to obtain the mean and variance of the system reliability. To do this the mean and variance of the reliability of the individual subsystems must be known. The equations that are used to obtain the mean and variance of the subsystem reliability for certain failure time density functions are derived. Although the techniques are not new, e.g., Lloyd and Lipow (1) have an excellent discussion on the subject, the results are not normally found in one compilation. A table is included which gives all of the equations for the failure time density functions considered.

'This research was partially supported by NASA contract NAS-12-2084.

=The computer program can be obtained by contacting either one of the authors.

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Estimating the Mean and Variance of the Subsystem Reliability

Let the following notation be introduced:

ai = the ith parameter of the failure time density function.

&i = estimate of the ith parameter of the failure time density function.

fi(t) = failure time density function of the ith subsystem.

t , = mission time of the ith subsystem. Ti = prior operating time of the ith subsystem,

sometimes referred to as the subsystem age.

Ri(ti) = reliability of the ith subsystem during the mission time ti.

The subsystem reliability fii(ti) is by definition a conditional probability, that is, it is the probability of surviving the mission time ti from Ti to Ti + ti given that it has survived until Ti. The reliability is estimated using

Ri(ti) = Ri(Ti + ti)

Ri( Ti)

For exponential failure times it can be shown that Ri(ti) reduces to:

which when evaluated yields the well known reliability equation :

Ri(ti) = exp (- ti/liz) [31

where m is an estimate of the mean-time-between-failures.

Equation [ I ] can be used to obtain an estimate of the mean of the subsystem reliability for any failure time density function. Another important estimate, although often overlooked, is the variance of the reliability. The variance can be approximated without knowing the exact distribution of the mean of the subsystem reliability. Knowing the mean and variance of the subsystem reliability, the mean and variance of the reliability of the complete system can be obtained.

Let the failure time density function have j parameters that are estimated from test data and denote them by :

Let G(&) be a function of these parameters, that is, for our case G(&) is ~ i ( t i ) . The variance of G(&) can be

approximated by expanding G(&) into a Taylor series about

The Taylor series expansion of G(&) about a as given in 151 is:

akl+kz . . . +kiG(a) (hl - a l ) k ~ G(&) = G(a) + C

dc~l~lda2~2. . . dajkj kl!

[41

Ignoring derivatives of order greater than one yields:

Rearranging Eq. [5], squaring both sides and taking the expected value yields :

Since

VAR [G(&)] = E{[G(&) - G(a)I2}. [71

Equation [6] yields :

VAR [G(&)] = E [LaF (a1 - al)

For a single parameter distribution Eq. [S] with G(&) replaced by &(ti) becomes

For a two-parameter distribution we have

dRi(ti) ' VAB [Ri(ti)] = [ aal ] VAR ( & I )

where

COV ( & I , &2) = E(bl - & I ) E(&z - az).

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For the purposes of this article five failure time density functions are considered: exponential, gamma, Weibull, normal, and lognormal. Table 1 contains the mean of the subsystem reliability and the partial de- rivative for calculating the variance for exponential failure times. Equation [9] is used to obtain the variance. Table 2 contains the equations for estimating the mean and partial derivatives of IZi(ti) as a function of R ~ ( T ~ + ti) and R ~ ( T ~ ) . Table 3 contains the means and partial derivatives for Ri(Ti + ti) and B i ( ~ i ) for the remaining four failure time density functions. Using Table 3, Table 2, and Eq. [ lo ] , the mean and variance of ~ i ( t i ) can be estimated.

Table 1 : Mean and partial derivatives for evaluating the variance for exponential failure times

Density Function aB,(ti) Parameters R,(t,) a,, Exponential

& z = m exp (-t t,/m) (tt/h2)i%(tC)

The derivation of the results given in Table 2 is presented for the Weibull distribution. The estimates for the other distributions are derived in a similar manner.

Table 2: Mean and partial derivatives for evaluating the variance of Bi(ti)

Estimators for the Weibull Distribution

The Weibull failure density is

fi(ti) = @tie1 exp (- bits ti > 0 @ > O [Ill p" > 0.

For Weibull failure rates the probability that a subsystem will survive to time Ti + ti given that it has survived until time Ti is

I\

Ri(ti) = &(Ti + ti) fii( Ti)

where

b i (T i + t i ) = S" Ti+,i @tib1 exp (- &t,!)dti

= exp [- &(Ti + ti)$] [I21

and

R ~ ( T ~ ) = exp (- c r ~ ? ) . [I31

Therefore,

8i ( t i ) = exp [- &[(Ti + ti)$ - T!]) . [I41

The results given in Table 3 for the Weibull were obtained by evaluating the partial derivatives of R ~ ( T ~ + t i ) as given by Eq. [12] and R ~ ( T ~ ) as given by Eq. [13] with respect to & and 4. These results will not be repeated here.

Mean and Variance of the Reliability for Series and Parallel Subsystems

Once the mean and variance of the reliability of eaoh subsystem is known, the system reliability can be obtained for any mixture of series and parallel subsystems. The equations that are used to analyze series subsystems and parallel subsystems are presented. These equations will be used to obtain the system reliability later in this article.

When deriving the equations for the reliability of series and parallel subsystems it is convenient to use the expected value operator. In this respect the following notation will be used:

R = system reliability. Ri = reliability of the ith subsystem.

E(Rm) = the mth moment of the system reliability.

E(Rim) = the mth moment of the subsystem reliability.

VAR(R) = variance of the system reliability. VAR(Ri) = variance of the ith subsystem relia-

bility.

For this article the-values of m = 1 and m = 2 are used. The above notation will also be used with the reliability R replaced by the unreliability Q.

The first and second moments of the system reliability for N series subsystems is

Using the expression

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yields is obtained using Eq. [15]. The variance is evaluated using Eqs. [15] and [18] and the expression

The VAR(Ri) and E(Ri) are evaluated for each sub- The analysis for N parallel subsystems is similar to system using Eqs. [8] and [I], respectively. the series case except that the unreliability is used in-

The mean of the reliability for N series subsystem stead of the reliability. The first and second moments

I Table 3: Mean and partial derivatives for evaluating the variance of %(ti)

Density Function

Parameters E i ( ~ i + ti)

Gamma 1 a1 = a

b z = $ ti2-' exp (- ti/&, A

tP-1 exp (- ti/@)&

Weibull A

$1 = a 4

exp [ - S ( T ~ + ti)? $2 = p

Normal 1

exp (- @T,? I\ - (T i + ti) exp [- a T i - ti)']

log ( T ~ + ~ ~ ) - G \ Lognormal $1 = i; 1 I log ( T i +t;)-,i2

exp (- 1/2zz)dz 1 - exp (- 1/9z2)dz --- r\ A

24% [-i( a >'I Density a f i i ( ~ i ) a2,(Ti + ti) a B i ( ~ d Function

Parameters da^l 822 a 2 2

Gamma [*(a) - 1% ( W ~ I .

61 = @ 11 - Ri(Ti11 -

a = p ITi (log ti)fi(ti)dti

Weibull 81 = 2 a, = B

I\

- Tip exp (- 3 ~ ~ 3 - $(Ti + ti)'log ( T i + ti) exp [- 8 ( T i + t i)q

- d ~ i p l o ~ ~i exp (- @ ~ i ?

1 log ( T i + ti) - ji' 1 log T i - $2 Lognormal & = p 1 ,5wa a ~ 4 % B

@z = 8

-

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of the system unreliability for parallel subsystem are

where

Using the expressions I

1 yields

I ~ The mean of the reliability for N parallel subsystems is obtained using Eq. [23] and

The variance is obtained using Eqs. [20] and [23] and the expression

Stochastic Networks in Reliability Analysis

The procedure used to analyze the subsystem reliability and the reliability of series and parallel subsystems has been presented. We now proceed with the analysis of a system consisting of a mixture of series and parallel subsystems. This is where the block diagram and/or stochastic network is required. As previously mentioned the stochastic network provided the advantage of ease in programming. The subsystems are represented by branches on the network with the nodes designating the relationship between the subsystems.

Any reliability block diagram can be represented with a stochastic network by using two logical rela- tions on the input side and a deterministic output. Let us consider the block diagram of Fig. 1. The stochastic network for this diagram begins with a deterministic output node and an AND node is place after each series subsystem. The resulting network is shown in Fig. 2. Obviously the network can be extended to any number of series subsystems.

Figure 3 is a block diagram for two parallel subsystems. The stochastic network for such a system begins with a deterministic output node and ends with an Inclusive-or node. Each subsystem is represented by one branch from the beginning to the ending node. The resulting network is shown in Fig. 4.

Reliability block diagrams for series and parallel


Fig. 1. Block diagram for three series subsystems

Fig. 2. Stochastic network for three series subsystems

Fig. 3. Block diagram for three parallel subsystems

R2 Fig. 4. Stochastic network for three parallel subsystems

Fig. 5 . Block diagram for series and parallel subsystem

Fig. 6. Stochastic network for series and parallel subsystem

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systems similar to those presented in Figs. 1 and 3 work. The program is coded in Fortran IV and the can be represented by a stochastic network using the present version contains the density functions for AND and Inclusive-or nodes with deterministic out- time-to-failure given in Tables 1 and 3. In order to puts. This same technique can be extended to mixtures use the program the analyst needs to supply the fol- of series and parallel subsystems as illustrated in Figs. lowing data: 5 and 6. An example problem will be presented which 1. Start node and end node for each branch. consists of several series and parallel subsystems.

2. The number of different types of failures represented by the branch. An example would be where chance and wearout failures are considered. This represents two types of failures. Another example is where three different failure modes are considered for the subsystem. This represents three types of failures. The data listed under 3) and 4) below must be given for each failure type.

3. The time-to-failure density function, its parameters and their variances and covariance.

4. The prior operating time of the subsystem, except for chance failures, and the mission time for the subsystem.

When more than one branch is incident on a node, the program designates the node as an Inclusive-or node. Otherwise the node is identified as an AND node.

The first step in the analysis is to compute the first and second moments of the reliability for each branch. This is done by evaluating the moments of the reliability that corresponds to each failure type. The equations given in Table 1, 2 and 3 along with Eqs. [I] and [17] are used for this computation. The mo-

Fig. 8. Stochastic network for the esample problem ments of the reliability for the branch is the product of each of the resulting moments for each failure type.

A n Example Problem

Let us consider the reliability block diagram given in Fig. 7. Although this block diagram is considered typical of many systems i t is not an actual system. I t was selected primarily to illustrate the technique and the general nature of the problems that can be solved using the computer program.

The first step in using the network approach is to construct a stochastic network of the system as given in Fig. 8. The traditional block diagram is presented for the sake of clarity. For this example, it is assumed that chance failures are described by the exponential density function and wearout failures follow the normal density function.

The data for exponential failure rates are given in Table 4 and the data for wearout failures are given in Table 5. A selected portion of the results is presented in Table 6. From the results given in Table 6, i t can be seen that the first and second moments for the system reliability are 0.792 and 0.627 respectively. The resulting variance of the system reliability is 0.000022.

The solution to this example was obtained by using the computer program to analyze the stochastic net-

Table 4: Data for exponential failure rates

Start End Operating Mean time Variance node node time to failure VAR(m)

t t m

1 2 4.0 90.0 30.0 2 3 4.0 95.0 32.0 3 4 5.0 88.0 26.0 4 7 11.0 60.0 18.0 3 5 12.0 60.0 20.0 3 5 12 0 60.0 20.0 5 7 13 0 65.0 25.0 3 6 12.0 60.0 20.0 3 6 12.0 60.0 20.0 3 6 12.0 60.0 20.0 6 7 10.0 70.0 25.0 6 7 10.0 70.0 25.0 7 8 12.0 75.0 22.0 7 8 12.0 75.0 22.0 8 9 4.0 85.0 29.0

8.0 lo 12.0

60.0 16.0 10 13 55.0 16.0 9 12 12 0 55.0 15.0

12 13 10 0 55.0 17.0 9 11 10 0 65.0 20.0

11 13 9.0 58.0 20.0 13 14 4.0 100.0 25.0

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- Table 5: Data for normal failure rates

Start End Subsystem Operating Mean Standard node node age time deviation Variance Variance

Ti ti F a P B h

1 2 25.0 4 .0 130.0 30.0 2.00 0.020 2 3 25.0 4 .0 125.0 30.0 2.00 0.020 3 4 20.0 5.0 120.0 30.0 2.55 0.050 4 7 20.0 11.0 80.0 20.0 2.40 0.009 3 5 20.0 12.0 85.0 20.0 2.00 0.010 3 5 20.0 12.0 85.0 20.0 2.00 0.008 5 7 20.0 13.0 90.0 22.0 2.00 0.015 3 6 12.0 12.0 60.0 18.0 1.50 0.015 3 6 12.0 12.0 60.0 18.0 1.50 0.015 3 6 12.0 12.0 60.0 18.0 1.50 0.015 6 7 20.0 15.0 90.0 25.0 2.01 0.020 6 7 20.0 15.0 90.0 25.0 2.01 0.020 7 8 15.0 12.0 100.0 25.0 2.00 0.020 7 8 15.0 12.0 100.0 25.0 2.00 0.020 8 9 25.0 4.0 130.0 30.0 2.00 0.020 9 10 10.0 8 .0 80.0 18.0 2.00 0.020

10 13 15.0 12.0 70.0 16.0 1.50 0.007 9 14 10.0 12.0 70.0 14.0 1.50 0.008

12 13 20.0 10.0 70.0 20.0 1 .OO 0.006 9 11 15.0 10.0 90.0 25.0 2.50 0.010

11 13 25.0 9 .0 75.0 20.0 2.50 0.010 13 14 20.0 4 . 0 140.0 25.0 2.50 0.050 -

The computer program then analyzes each of the parallel subsystems in the appropriate order using the equations discussed for parallel subsystems. The network is continuously reduced until the first and second moments of the reliability of the complete system is obtained. The variance of the system reliability is obtained using Eq. [19]. A simplified flow chart of the computer program is given in Fig. 9.

The program has the provision for an intermediate output option. The first and second moments for each parallel structure are obtainable as intermediate output if desired. As previously mentioned, a sample of this is given in Table 6.

Analysis Using the Block Diagram

Previously i t was mentioned that the computer program can be used to analyze the block diagram, with additional information added, without the transformation to the stochastic network. The additional information required is an identifying number before and after each subsystem in the block diagram. Only one identifying number is required between any two subsystems which requires that a set procedure be estab- lished for the location of the number when diverging and converging paths are encountered. The rule for diverging paths is to place the number a t the first point of divergence after a subsystem. For instance, in Fig. 7 an identifying number is placed a t the point of divergence immediately after subsystem 2. Another identifying number is not required before any of its successor subsystems.

The rule for converging paths is to place the number a t the last point of convergence. Again referring to

Start

Ca l l Input

Rout~ne

Calculate

Vo,,once. and Output Rervl ts

4 Identmfy Parollel Networks

Calsulntc

Moments

Ca l l

Input

Fig. 9. Simplified flow chart of computer program


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U

Fig. 10. Block diagram for the example problem with ident,ifying numbers added

Fig. 7, an identifying number is placed a t the point where the paths from subsystems 7, 4, 11, and 12 con- verge. When converging paths and diverging paths both occur between subsystems it is arbitrary as to whether the identifying number is placed a t the last point of convergence or the first point of divergence.

The identifying numbers that are located on the block diagram actually correspond to node numbers on the stochastic network. However, it does eliminate the need for constructing the stochastic network which would be undesirable if block diagrams are already available. The block diagram given in Fig. 7 is presented again in Fig. 10 with the appropriate identifying numbers. The correspondence between the identifying numbers on Fig. 10 and the node numbers in Fig. 8 is apparent.

There is actually no difference in the data that is required for the program to analyze the block diagram. Instead of providing the start node and end node for each branch, the analyst supplies the identifying number before and the identifying number after each subsystem.

Conclusions

The technique presented in this article differs from the traditional approach to reliability analysis. A computer program was developed to obtain the mean

and variance of the subsystem reliability by replacing the block diagram by a stochastic network. The main features of the program are 1) the input data to the program can be supplied from either the stochastic network or from an appropriately modified block diagram, 2 ) i t will analyze any mixture of series and parallel subsystems, 3) several different failure time density functions are built into the program with the capability for adding additional ones, 4) multiple failure types such as chance failures, wearout failure, different modes, etc., can be handled for a subsystem, and 5) it is written in Fortran IV which means that it can be used on most computers.

Table 6: Selected portions of the results for the example

Start node End node First Second moment moment

3 5 0.9661 10 0.933378 3 6 0.992376 0.984810 6 7 0.979555 0.959532 3 7 0.998707 0.997415 7 8 0.977781 0.956059 9 13 0.971608 0.944027 1 14 0.797136 0.626631

VAR (R) = 0.000022


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Additional research efforts in applying stochastic networks to reliability analysis are currently in progress. Of particular interest are systems containing subsystems which are neither in series nor parallel.

References ( I ) Lloyd, D. K., and M. Lipow, Reliability: Management,

Methods, and Mathematics, Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1964).

(2) Pritsker, A. A. B., and W. W. Happ, "GERT: Graphicaf Evaluation and Review Technique, Part I-Fundamentals, Journal of Industrial Engineering, XVII, 5, May (1966).

(3) Pritsker, A. A. B., and G. E. Whitehouse, "GERT: Graph- ical Evaluation and Review, Part 11-Probabilistic and Industrial Engineering Applications," Journal of Industrial Engineering, XVII, 6, June (1966).

(4) Whitehouse, G. E., and A. A. B. Pritsker, "GERT: Part 111-Further Statistical Results; Counters, Itenewal Times, and Correlations," A I I E Transactions, I, 1, March (1969).

( .5) Protter, M. H., and C. B. Morrey, College Calculus with, Analytic Geometry, Addison-Wesley Publishing Co., Reading, Massach~isetts (1967).

(6) Whitehouse, G. E., "GERT, 4, Useful Technique for Analyzing Reliability Problems, Teehnometrics, 12, 1, February (1970).

Dr. Byers is an assistant professor of computer science at the University of Missouri-Rolla. He was formerly an operations research analyst for NASA. He earned a BSME degree and a PhD in industrial engineering from the University of Arkanras and a MSE degree from the University of Alabama in Hunts- ville. He is a member of ORSA.

Dr. Skeith is an associate professor of industrial engineering at the University of Arkansas. Formerly on the faculty of Arizona State University, he has worked as an I E for the Pittsburgh Plate Glass Company. He earned BSIE and MSIE degrees at Oklahoma State University and received a PhD in industrial engineering from Arizona State University. He is a member of ORSA, TIMS, ASQC, and ASEE. He is a member of the Editorial Board of A I I E Transactions.

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use of stochastic networks in reliability analysis

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