Confidence intervals for steady state availability of a system with exponential operating time and lognormal repair time

Applied Mathematics and Computation 137 (2003) 499–509 www.elsevier.com/locate/amc

Confidence intervals for steady state availability of a system with exponential operating time and lognormal repair time Malwane M.A. Ananda Department of Mathematical Sciences, University of Nevada, 4505 Maryland Parkway, Box 454020, Las Vegas, NV 89154-4020, USA

Abstract Long-run availability of a system assuming lognormal repair times and exponential failure times is considered. With many repairable systems, the time to repair is well characterized by a lognormal distribution. However, due to the difficulties associated with the lognormal distribution, repair times are often modeled using the lognormal distribution assuming that the variance of the lognormal distribution is known or by using some other distribution such as the exponential distribution instead of the lognormal distribution. In this paper using the generalized p-value approach, we propose confidence intervals and exact tests for steady state availability of systems using the twoparameter lognormal distribution for repair time and exponential distribution for operating time. Here, both parameters of the lognormal distribution are assumed to be unknown. A couple of examples are given to illustrate the proposed procedures. A simulation study is given to demonstrate the performance of the proposed procedure. Results are extended for the long-run availability of a system consisting of several independent parallel sub-systems. Ó 2002 Published by Elsevier Science Inc. Keywords: Steady state availability; Lognormal repair times; Confidence intervals

1. Introduction The steady state availability is an important measure in determining the long-term performance of a system which is to be operated continuously. E-mail address: [email protected] (M.M.A. Ananda). 0096-3003/02/$ - see front matter Ó 2002 Published by Elsevier Science Inc. PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 1 5 5 - 8

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Therefore, it is of interest to construct confidence intervals and perform hypotheses testing on the steady state availability of the system. These types of statistical problems occur in areas such as reliability engineering and environmental engineering when one is interested in determining the availability of a repairable system. The steady state availability A of a system is defined by ly A¼ ; ð1Þ ly þ lx where ly is the mean time between failures of the system and lx is the mean time to repair the system. Here the random variables X and Y represent the time to repair the system and the time between failures of the system. It is well known that the repair time of many repairable systems is well characterized by a lognormal distribution [10,11,13]. However, due to the difficulties associated with the lognormal distribution, repair times are often modeled using the lognormal distribution assuming that the variance of the lognormal distribution is known or using some other statistical distribution such as the exponential distribution instead of the lognormal distribution. In particular, when a system consists of several sub-systems, exponential distribution is used for mathematical tractability. Using exponential failure and repair times, Thompson [14] gave confidence intervals and tests of hypotheses for the system availability. Using a lognormal distribution with known variance (i.e. ln X  N ðl; r2 Þ; where r2 is known) for the repair times and an exponential distribution for the failure times, Gray and Lewis [9] provided exact confidence intervals for the steady state availability. The procedure involved numerically evaluating two dimensional integrals. They provided some table values which are required to construct these confidence intervals. Later several papers [5,6,10,12] discussed the inference on A using the lognormal and other related distributions. However, they all assumed that the variance parameter of the lognormal distribution is known and used the tables provided in [9] to construct their confidence intervals. When r2 is unknown, it is not clear how to extend this procedure. When r2 is unknown, one can use this procedure as an ad hoc procedure by guessing r2 value (use an estimate for r2 as a known value) though statistical properties of such an ad hoc procedure are unknown. In fact, simulation results from [10] shows that when sample sizes are 10 and 10 (10 repair times and 10 failure times) the empirical coverage for a 90% confidence interval can vary from 99% to 59% as r2 vary from 0.5 to 3.0 when the actual value of r2 is at 1.0. In this paper, we look at the confidence intervals and testing procedures for A when the repair time is lognormally distributed with an unknown mean and variance and the failure time is exponentially distributed with an unknown mean. Using the generalized p-value approach introduced by Tsui and Weerahandi [15], we construct an exact test for testing the long-run avail-

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ability. Using the generalized confidence interval concept introduced in [16], we construct a generalized confidence interval for the availability and a generalized lower confidence limit (LCL) for the availability. This is an exact test in significance testing. However, under the null hypothesis, the probability distribution of the p-value may not be uniform, and therefore the actual size of the test may not be equal to the intended size. As a result, the generalized confidence intervals which is essentially induced by these generalized p-values may not possess the intended coverage. Therefore, we carry out a simulation to estimate the exact coverage of these generalized confidence intervals. The simulation study shows that the exact coverages are nearly equal to the intended coverages. Furthermore, the results are extended for the long-run availability of a system consisting of several independent parallel sub-systems. When repair times and failure times are exponentially distributed, Ananda [2] gave similar procedures for a system consisting of several independent parallel sub-systems and they showed that these generalized procedures outperform the other known procedures. In many statistical problems involving nuisance parameters, conventional statistical methods do not provide exact solutions. As a result, even with small sample sizes practitioners often resort to asymptotic methods which are known to perform very poorly with small sample sizes. Generalized p-value approach is a recently developed method which is based on exact probability statements rather than on asymptotic approximations. As a result, the performance of these generalized methods are better than the performance of the approximate procedures. According to a number of simulation studies (cf. [1,3,8,18]), when compared, tests and confidence intervals obtained using the generalized approach have been found to outperform the approximate procedures both in size and power. The generalized procedure have successfully been applied to many areas in statistics including anova, regression, mixed models and growth curve models ([4,19,20,7], respectively). For a complete coverage of these generalized tests and confidence intervals, the reader is referred to Weerahandi [17].

2. Estimation and testing procedures with lognormal repair times Suppose the repair time X has a lognormal distribution with parameters l and r2 ; and the operating time Y has an exponential distribution with mean h; i.e., "  2 # 1=2 ð2pÞ 1 ln x  l 2 f ðx; l; r Þ ¼ exp  ; x > 0; ð2Þ 2 r xr 1 f ðy; hÞ ¼ ey=h ; h

y > 0;

ð3Þ

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where parameters l, r, and h are unknown. Then the steady state availability of the system is given by A¼

ly h ; ¼ ly þ lx h þ exp ðl þ r2 =2Þ

ð4Þ

where lx and ly are the means of X and Y , respectively. We are interested in testing the availability hypothesis H0 : A 6 A0

ð5Þ

vs: Ha : A > A0 ;

and constructing confidence intervals for the steady state availability A, based on the repair times X1 ; X2 ; . . . ; Xn and the operating times Y1 ; Y2 ; . . . ; Ym of the system. When r2 is known, 100(1  a)% confidence interval for A is given by [9]    1    1

1 þ exp r2 =2 bxg =ð2my Þ ; 1 þ exp r2 =2 axg =ð2my Þ ; ð6Þ Qn 1=n is the geometrical mean and the values a and b are where xg ¼ i¼1 xi coming from a table provided in [9]. Calculation of these constants involves numerically evaluating two dimensional integrals. Along with some simulation results, when r2 is known, Gray and Schucany [10] extended these results for testing the availability hypothesis given in (5). To describe the procedures when r2 , l, and h are unknown, let U¼

n X

U i =n;

S2 ¼

r¼1

n X 

2 Ui  U =ðn  1Þ;

r¼1

C¼

m X

Yi ;

ð7Þ

r¼1

where Ui ¼ ln ðXi Þ, i ¼ 1; 2; . . . ; n: Then   U  N l; r2 =n ; ðn  1ÞS 2 =r2  v2n1 ;

2C=h  v22m :

ð8Þ

Suppose Z¼

U l pffiffiffi ; r= n

V ¼

ðn  1ÞS 2 ; r2

W ¼

2C ; h

ð9Þ

then Z  N ð0; 1Þ, V  v2n1 , and W  v22m : First, let us consider the problem of constructing confidence intervals for A. 2.1. Generalized confidence intervals for A Define the random variable R¼

2cW 1 pffiffiffiffiffiffiffiffiffiffiffi ; 2cW 1 þ exp u  sZ n  1 n1=2 V 1=2 þ ðn  1Þs2 V 1 =2 

ð10Þ

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where Z, V, and W are independent standard normal, chi-square with degrees of freedom n  1 and chi-square with degrees of freedom n respectively. Here u, s2 , and c are the observed values of U , S 2 , and C given in Eq. (7). Then 100ð1  aÞ% generalized upper confidence interval for A is given by ðc0 ; 1Þ, where l0 in the ath quantile of the random variable R, i.e. PrðR 6 l0 Þ ¼ a. Furthermore, equal tail 100ð1  aÞ% confidence interval for A is given by ðl1 ; l2 Þ, where l1 is the (a=2)th quantile and the l2 is the (1  a=2)th quantile of the random variable R. The values l0 , l1 , and l2 can be evaluated using Monte Carlo Simulations. This can be done by generating a large number of random numbers from Z, V, and W and then evaluating R and looking at the empirical distribution of R. The proof of these generalized confidence intervals are given in Appendix A. 2.2. Generalized p-values for testing H0 The hypothesis H0 given in (5) can be tested using the generalized p-value 0 p ¼ Pr @

2cW 1

1



6 A0 A: 2cW 1 þ exp u  Zsðn  1Þ1=2 n1=2 V 1=2 þ ðn  1Þs2 V 1 =2 ð11Þ

Again, this p-value can easily be evaluated by generating a large number of random numbers from random variables Z, V and W. The derivation of this testing procedure is given in Appendix A. This p-value is an exact probability of a well defined extreme region of the sample space and measures the evidence in favor of the null hypothesis. This is an exact test in significance testing. In fixed level testing, one can use this pvalue by rejecting the null hypothesis, if the generalized p-value is less than the desired nominal level a.

3. Examples and simulation results In this section, we give two examples to demonstrate the advantages of the proposed procedures. The first example shows the calculated 95% confidence intervals for a simulated data set. The second example shows the results of a simulation study to compare the actual and the intended coverages. 3.1. Example 1 We consider a system having lognormal repair times and exponential failure times with the following parameters: l ¼ 1:0, r ¼ 1:0, h ¼ 100. Ten repair

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times and 10 failure times were generated from these distributions and their results are given below: Repair times: 3.69, 1.22, 0.43, 3.14, 4.59, 2.96, 12.11, 3.06, 1.45, 2.20 Failure times: 75.69, 46.50, 393.30, 476.17, 15.76, 340.92, 21.20, 14.06, 33.24, 2.83 The data were analyzed using the generalized method and the Gray–Lewis method (G–L method) with sample standard deviation in place of population standard deviation r: For the generalized method and the G–L method, 95% upper confidence interval for the availability of the system is given by (0.9339, 1) and (0.9530, 1) respectively. Although the G–L method gives a shorter confidence interval, its actual coverage is much lower than the intended coverage 95%. Indeed, our simulation study (see Table 1) shows that when l ¼ 1:0, r ¼ 1:0, h ¼ 100, actual coverages for the generalized method and the G–L method are: 0.9636 and 0.9078, respectively. 3.2. Example 2 In this example, by simulation, we have calculated the actual probability coverage for the generalized method and the G–L method. For this demon-

Table 1 Probability coverages for 95% upper confidence intervals Parameters

Sample sizes

Probability coverage for

l

r

h

n

m

Generalized method

G–L method

1.0 1.0 1.0 1.0 1.0 2.0 2.0 2.0 2.0 2.0 1.0 1.0 1.0 3.0 2.0 2.0 3.0 3.0 3.0

1.0 1.0 1.0 1.0 1.0 2.0 2.0 2.0 2.0 2.0 1.0 1.0 1.0 1.0 2.0 2.0 3.0 3.0 3.0

100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 1.0 50.0 200.0 20.0 50.0 200.0 20.0 50.0 200.0

5 10 10 15 20 5 10 10 15 20 10 10 10 10 10 10 10 10 10

5 10 15 10 20 5 10 15 10 20 10 10 10 10 10 10 10 10 10

0.9755 0.9636 0.9619 0.9631 0.9610 0.9612 0.9572 0.9550 0.9533 0.9528 0.9650 0.9671 0.9560 0.9650 0.9564 0.9543 0.9520 0.9560 0.9503

0.9022 0.9078 0.8983 0.9219 0.9187 0.7696 0.8014 0.7924 0.8144 0.8115 0.9090 0.9097 0.8320 0.9090 0.7975 0.7989 0.6990 0.7990 0.7009

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stration we used the upper confidence intervals instead of the two-sided confidence intervals, since the upper confidence interval is more important in availability analysis. The simulation results are based on 20,000 simulations. As in the first example, the G–L method is utilized as an ad hoc procedure by replacing the standard deviation r by the sample standard deviation (i.e., the sample standard deviation for Ui ¼ lnðXi Þ, i ¼ 1; 2; . . . ; n). In order to get more accurate calculations for the G–L method, rather than using interpolated values from tables given in [9], the values a and b needed for the Eq. (6) were calculated by simulation as well. The simulation results for different parameter values and sample sizes are given in Table 1. According to the simulation study, the actual probability coverage of the generalized method always stay very close to the intended coverage. However with the G–L method, the actual coverage can go as low as 70% when the intended coverage is set at 95%. In fact, this observation is in line with the simulation results given in [10]. By simulation they noticed that the actual coverage for a 90% confidence interval can vary from 99% to 59% as r2 varies from 0.5 to 3.0 when the actual value of r2 is at 1.0. 4. Availability of a system with several parallel sub-systems In this section, we extend the results for a system with several parallel subsystems. Suppose that there are k parallel sub-systems and each sub-system has its own failure time and operating time distributions. For the ith sub-system, suppose that the repair time has a lognormal distribution with parameters li and r2i and, the operating time has an exponential distribution with a mean hi . The steady state availability of the whole system is given by A ¼ 1  P ðall components are downÞ k Y ¼1 P ðthe ith component is downÞ i¼1

¼1

k  Y i¼1



ð12Þ



 exp li þ r2i =2 : hi þ exp ðli þ r2i =2Þ

Furthermore, suppose that from the ith sub-system, ni repair times and mi operating times are available. Let xi1 ; xi2 ; . . . ; xini and yi1 ; yi2 ; . . . ; yimi be their repair times and operating times. For each i ¼ 1; 2; . . . ; k, suppose Ui ¼

ni X

Uir =ni ;

Si2 ¼

r¼1

Zi ¼

U i  li pffiffiffiffi ; r i = ni

ni X 

2 Uir  U i =ðni  1Þ;

r¼1

Vi ¼

ðni  1ÞSi2 ; r2i

Ci ¼

mi X r¼1

Wi ¼

2Ci ; hi

Yir ;

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where Uir ¼ lnðXir Þ, r ¼ 1; 2; . . . ; ni . Define the random variable 

2 3 pffiffiffiffiffiffiffiffiffiffiffiffi 1=2 1=2 k exp ui  si Zi ni  1 ni Vi þ ðni  1Þs2i Vi 1 =2 Y 4 

5; R¼ pffiffiffiffiffiffiffiffiffiffiffiffi 1=2 1=2 1 2 1 n W þ exp u  s Z  1 n V þ ð n  1 Þs V =2 2c i¼1 i i i i i i i i i i i ð13Þ where for each i ¼ 1; 2; . . . k; Zi , Vi , and Wi are independent standard normal, chi-square with degrees of freedom ni  1 and chi-square with degrees of freedom ni respectively. Here ui , s2i , and ci are the observed values of U i , Si2 , and Ci . Then 100ð1  aÞ% generalized upper confidence interval for A is given by ð1  l0 ; 1Þ, where l0 in the (1  a)th quantile of the random variable R, i.e. Pr ð R 6 l0 Þ ¼ 1  a. Furthermore, equal tail 100ð1  aÞ% confidence interval for the availability A in (12) is given by ð1  l1 ; 1  l2 Þ; where l1 is the (1  a=2)th quantile and the l2 is the (a=2)th quantile of the random variable R. The availability hypothesis H0 : A 6 A0 vs. Ha : A > A0 can be tested using the p-value 0

2 k Y p ¼ Pr @ 4 i¼1



3 1=2 1=2 1=2 exp ui  Zi si ðni  1Þ ni Vi þ ðni  1Þs2i Vi 1 =2 

5 1=2 1=2 þ ðni  1Þs2i Vi 1 =2 2ci Wi 1 þ exp ui  Zi si ðni  1Þ1=2 ni Vi 1

 1 þ A0 P 0A:

ð14Þ

As in Section 3, these confidence intervals and the p-value can easily be evaluated by generating a large number of random numbers from random variables Zi , Vi and Wi : The outline for the derivation of these procedures are given in Appendix A.

Appendix A In this section, we outline the proofs of the main results given in Sections 3 and 4. Proofs are based on the concepts behind generalized p-values and generalized confidence intervals and for more details the reader is referred to [17]. Proof for the generalized confidence interval: One can show that the random variable R given in (10) can be written as RðX; Y; x; y; l; r; hÞ ¼

chC 1

chC 1   : þ exp u  sS 1 U  l þ r2 s2 S 2 =2 

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The probability distribution of RðX; Y; x; y; l; r; hÞ is free of unknown parameters. The observed value of RðX; Y; x; y; l; r; hÞ is equal to Rðx; y; x; y; l; r; hÞ ¼ A and it does not depend on nuisance parameters and therefore RðX; Y; x; y; l; r; hÞ is a generalized pivotal quantity to construct confidence intervals for A: In order to construct a 100ð1  aÞ% LCL for A; find l0 such that 1  a ¼ Pr ðRðX; Y; x; y; l; r; hÞ P l0 Þ ¼ Pr

chC 1 chC 1 þexp ðusS 1 ðU lÞþr2 s2 S 2 =2Þ

 P l0 :

Then the 100ð1  aÞ% upper confidence interval for A is given by ðl0 ; 1Þ. Proof for the generalized p-value: To show that the p-value given in (11) can be used to test the hypothesis in (5), consider the potential test variable T ðX; Y; x; y; l; r; hÞ ¼

chC 1     A: chC 1 þ exp u  sS 1 U  l þ r2 s2 S 2 =2 

Then one can show that this test variable can be written as T ðX; Y; x; y; l; r; hÞ ¼

2cW 1 

A 2cW 1 þ exp u  Zsðn  1Þ1=2 n1=2 V 1=2 þ ðn  1Þs2 V 1 =2

and its observed value is equal to Tobs ¼ T ðx; y; x; y; l; r; hÞ ¼ 0: The probability distribution of T ðX; Y; x; y; l; r; hÞ is free of nuisance parameters. When x, y and nuisance parameters are fixed, the cumulative distribution function of T, Pr ðT 6 t; AÞ is a monotonically increasing function of A for any given t. Therefore T ðX; Y; x; y; l; r; hÞ is stochastically decreasing in A and T ðX; Y; x; y; l; r; hÞ is a test variable that can be used to test the hypothesis in (5) and the generalized p-value is given by p ¼ Pr ðT P Tobs = A ¼ A0 Þ; which can be written in the form of (11). Proof for the procedures in Section 4: Let Zi ¼

U i  li pffiffiffiffi ; ri = ni

Vi ¼

ðni  1ÞSi2 ; r2i

Wi ¼

2Ci : hi

Then Zi  N ð0; 1Þ, Vi  v2ni 1 , and Wi  v22mi and they are independent. Consider the potential pivotal quantity

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RðX; Y; x; y; l; r; hÞ " #     k Y exp ui  si Si1 U i  li þ r2i s2i Si2 =2   :   ¼ 1 1 U i  li þ r2i s2i Si2 =2 i¼1 ci hi Ci þ exp ui  si Si The probability distribution of R is free of unknown parameters and the observed value of R is equal to A, hence R is a generalized pivotal quantity, this will yield the desired confidence intervals. The proof of (14) will follow by showing that T ðX; Y; x; y; l; r; hÞ " #     k Y exp ui  si Si1 U i  li þ r2i s2i Si2 =2   A   ¼ 1 1 U i  li þ r2i s2i Si2 =2 i¼1 ci hi Ci þ exp ui  si Si is a generalized test variable and following the same argument as in the previous proof.

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