- Email: [email protected]
On steady state availability of a system with lognormal repair time
Applied Mathematics and Computation 150 (2004) 409–416 www.elsevier.com/locate/amc
On steady state availability of a system with lognormal repair time Malwane M.A. Ananda
a,*
, Jinadasa Gamage
b
a
Department of Mathematical Sciences, University of Nevada, Box 455020 4505, Maryland Parkway, Las Vegas, NV 89154-4020, USA b Department of Mathematics, Illinois State University, Normal, IL 61761, USA
Abstract Assuming two-parameter lognormal distribution for repair times, statistical inference for the steady state availability of a system is considered. For the failure time distribution, weibull, gamma, and lognormal distributions were considered. Using the generalized p-value approach, we propose confidence intervals and exact tests for the steady state availability of a system. A couple of examples are given to illustrate the proposed procedures. Ó 2003 Published by Elsevier Ltd. Keywords: Steady state availability; Lognormal repair times; Confidence intervals
1. Introduction The steady state availability is a measure in determining the long term performance of a system which is to be operated continuously. The steady state availability A of a system is defined by A¼
ly ; ly þ lx
ð1Þ
where ly is the mean time between failures of the system and lx is the mean time to repair the system. The random variables X and Y represent the time to
*
Corresponding author. E-mail address: [email protected] (M.M.A. Ananda).
0096-3003/$ - see front matter Ó 2003 Published by Elsevier Ltd. doi:10.1016/S0096-3003(03)00281-9
410
M.M.A. Ananda, J. Gamage / Appl. Math. Comput. 150 (2004) 409–416
repair the system and the time between failures of the system. It is known that the repair time of many repairable systems is well characterized by a lognormal distribution [8,9,12]. Furthermore, it is well known [12] that the failure times of repairable systems are often modeled with weibull, gamma, and lognormal distributions. In availability studies, 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 which is essentially equivalent to using a one parameter lognormal distribution. In some other instance even when the repair times are known to be lognormally distributed, for mathematical tractability probability distributions such as the exponential distribution is used instead of the lognormal distribution. Gray and Lewis [7] provided exact confidence intervals for the steady state availability when the repair times are lognormally distributed with known variance (i.e. ln X N ðl; r2 Þ, where r2 is known) and the failure times are exponentially distributed. The procedure involved numerically evaluating two dimensional integrals and therefore, they provided some table values which are required to construct these confidence intervals. Masters and Lewis [10] provided exact confidence intervals for the availability under the assumption that the time between failures are distributed with a gamma random variable and repair times are distributed with a lognormal distribution with known variance. Masters et al. [11] provided exact intervals under the assumption that the time between failures and time to repair are weibull and lognormal (with known variance) random variables, respectively. Chandrasekhar et al. [5] provided confidence intervals when the failure times and the repair times both are lognormal with known variances. When r2 is unknown, the exact confidence intervals and testing procedures are not available in the literature. In this situation for some of the cases discussed above, Bayesian techniques were described in Martz and Waller [12]. When r2 is unknown, one can use the procedure given in Masters and Lewis [10], Masters et al. [11], and Chandrasekhar et al. [5] as an ad-hoc procedure by replacing r2 with an estimate of r2 , even though statistical properties of such an ad-hoc procedure are unknown. In fact, simulation results from Gary and Schucany [8] 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. Using the generalized p-value approach introduced in Tsui and Weerahandi [13], when repair times and failure times are exponentially distributed, Ananda [1] gave exact intervals and testing procedures for the availability of a system consisting of several independent parallel subsystems and they showed that these generalized procedures outperform the other known procedures. Ananda [2] extended these results and gave confidence intervals and testing procedures for the availability when the repair time is lognormally distributed with an
M.M.A. Ananda, J. Gamage / Appl. Math. Comput. 150 (2004) 409–416
411
unknown mean and variance and the failure time is exponentially distributed with an unknown mean. Using statistical simulation Ananda [2] showed that the exact probability coverage of their procedure is nearly equal to the intended coverage and their procedure outperform the other known two-parameter lognormal procedures. In this paper, we look at the exact testing procedures and confidence intervals for the availability A when the repair time is lognormally distributed with an unknown mean and variance. For the failure time distributions, we use gamma, weibull or lognormal distributions. For each case, using the generalized p-value approach, we construct an exact test for testing the long-run availability. Furthermore, for each case, we construct a generalized confidence interval for the availability and a generalized lower confidence limit (LCL) for the availability. Generalized p-value approach is a recently developed method which is based on an extended definition of p-values. The generalized tests and generalized confidence intervals are 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 (for example, [3,4,6,15]), 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 and for more details of these procedures see Weerahandi [14].
2. Estimation and testing procedures with lognormal repair times Suppose the repair time X has a lognormal distribution with parameters l and r2 ; " 2 # 1=2 ð2pÞ 1 ln x l f ðx; l; r2 Þ ¼ exp ; 2 r xr
x > 0;
ð2Þ
where parameters l and r are unknown parameters. Through out of this paper, we use this two-parameter lognormal model for the repair times. Suppose that repair times X1 ; X2 ; . . . ; Xn are available from this distribution. We are interested in testing the availability hypothesis H0 : A 6 A0
vs: Ha : A > A0
ð3Þ
and constructing confidence intervals for the steady state availability A, based on the observed repair times and the observed failure times Y1 ; Y2 ; . . . ; Ym of the
412
M.M.A. Ananda, J. Gamage / Appl. Math. Comput. 150 (2004) 409–416
system. These failure times are assumed to follow the underline probability distribution considered in each subsection. Suppose U¼
n X
Ui =n;
S2 ¼
r¼1
n X 2 ðUi U Þ =ðn 1Þ; r¼1
Z¼
U l pffiffiffi ; r= n
2
V ¼
ðn 1ÞS ; r2
ð4Þ
where Ui ¼ lnðXi Þ, i ¼ 1; 2; . . . ; n. Then Z N ð0; 1Þ, V v2n1 . 2.1. Statistical inference with weibull failure times First let us consider the case where the time between failures follow a weibull distribution. Suppose the failure time Y has a weibull distribution f ðy; k; cÞ ¼ cky c1 eky
c
ð5Þ
y > 0;
were c is a known parameter and k is an unknown parameter. Then EðY Þ ¼ Cðc1 þ 1Þk1=c and the steady state availability of the system is given by A¼
Cðc1 þ 1Þk1=c Cðc1 þ 1Þk1=c þ expðl þ r2 =2Þ
:
ð6Þ
Masters et al. [11] provided an approximate test and confidence interval when the parameters r and c are known and, l and k are unknown. In this section, using the generalized p-value approach, we will provide an exact test for the availability when the parameter c is known and l; r, and k are unknown. If c is unknown, statistical inference regarding the availability gets more complicated and needs further research. Let Y1 ; Y2 ; . . . ; Ym be the observed failure times from the failure time distribution given in (5). Let C¼
m X
Yic
and
W ¼ 2kC:
ð7Þ
r¼1
Then W v22m . The hypothesis H0 given in (3) can be tested using the p-value p ¼ Pr
Cðc1 þ 1Þ u Zsðn 1Þ1=2 n1=2 V 1=2 þ ðn 1Þs2 V 1 =2Þ Cðc1 þ 1Þ þ ð0:5W =cÞ1=c expð
! 6 A0 :
ð8Þ
M.M.A. Ananda, J. Gamage / Appl. Math. Comput. 150 (2004) 409–416
413
This p-value can be evaluated by generating a large number of random numbers from random variables Z, V and W . In fixed level testing, one can use this p-value by rejecting the null hypothesis, if the generalized p-value is less than the desired nominal level a. The proof of this testing procedure can be obtained by utilizing the generalized test variable T ¼
Cðc1 þ 1Þ Cðc1 þ 1Þ þ ðkC=cÞ
1=c
expð u sS 1 ðU lÞ þ r2 s2 S 2 =2Þ
A:
ð9Þ
Here u, s2 , and c are the observed values of U , S 2 , and C given in Eqs. (4) and (7). The observed value of this test variable is equal to zero and the probability distribution of T ðX; Y; x; y; l; r; hÞ is free of nuisance parameters. For any given t, when x, y and nuisance parameters are fixed, the cumulative distribution function of T , PrðT 6 t; AÞ is a monotonically increasing function of A, therefore T ðX; Y; x; y; l; r; hÞ is a test variable and this will yield the p-value given in (8). The confidence intervals on A induced by this p-value can be obtained by defining the random variable R¼
Cðc1 þ 1Þ
; u Zsðn 1Þ1=2 n1=2 V 1=2 þ ðn 1Þs2 V 1 =2Þ Cðc1 þ 1Þ þ ð0:5W =cÞ1=c expð ð10Þ
where Z, V , and W are independent standard normal, chi-square with degrees of freedom n 1 and chi-square with degrees of freedom 2m respectively. 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=2th quantile and the l2 is the 1 a=2th 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. For further details of these procedures or the proofs, the reader is referred to Weerahandi [14] and Ananda [2]. 2.2. Statistical inference with gamma failure times Suppose the operating time Y has a gamma distribution with parameters k and k, f ðy; hÞ ¼
kk y k1 eky CðkÞ
y > 0:
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M.M.A. Ananda, J. Gamage / Appl. Math. Comput. 150 (2004) 409–416
Then the steady state availability of the system is given by A ¼ ½1 þ kk 1 1 expðl þ r2 =2Þ . These types of failure time distributions occur in availability studies [10], in particular when a system composed of p components, the entire system fails only after k of the components have failed. Masters and Lewis [10] gave confidence intervals when repair times have lognormal distribution (with known r2 ) and failure times have gamma distribution. One can show that when parameters k, l, r are unknown, the generalized pvalue to test the hypothesis in (3) is given by p ¼ PrðW ð2ckÞ
1
expð u Zsðn 1Þ
1=2 1=2
n
V 1=2
þ ðn 1Þs2 V 1 =2Þ P A1 0 1Þ:
ð11Þ
The proof of this is quite similar to the previous proof and as before this pvalue can be evaluated using simulation by generating a large number of random numbers from the random variables W ; Z, and V . Confidence intervals and lower confidence limit on A can be obtained in a similar manner. 2.3. Statistical inference with lognormal failure times Suppose the failure time Y has a lognormal distribution with unknown parameters a and b2 . Then ly ¼ expða þ b2 =2Þ and A ¼ ½1 þ expððl aÞ þ 1 ðr2 b2 Þ=2Þ . When the parameters a, b, r are known, Chandrasekhar et al. [5] provided statistical inference for the availability. When a, b, l, r are unknown, one can show that the generalized p-value to test the hypothesis in (11) is given by p ¼ Pr þ
Z1 s1 ðm 1Þ 1=2
m1=2 V1
1=2
1=2
ðm 1Þs21 Zsðn 1Þ 2V1 n1=2 V 1=2 !
ðn 1Þs2 P lnðA1 u1 u ; 0 1Þ þ 2V1
ð12Þ
1 , s2 , s21 are sample means and sample variance of repair times and where u, u failure times respectively. Here Z, V , Z1 and V1 are independent random variables having distributions Z N ð0; 1Þ, V v2n1 , Z1 N ð0; 1Þ, and V1 v2m1 . To state the generalized confidence interval, define the random variable " ( 1=2 Z1 s1 ðm 1Þ R ¼ 1 þ exp u u1 þ 1=2 m1=2 V1 )#1 1=2 ðm 1Þs21 Zsðn 1Þ ðn 1Þs2 þ : ð13Þ 2V1 n1=2 V 1=2 2V1
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415
Then 100ð1 aÞ% upper confidence interval for A is given by ðc0 ; 1Þ, where l0 in the ath quantile of the random variable R. Moreover, equal tail 100ð1 aÞ% confidence interval for A is given by ðl1 ; l2 Þ, where l1 is the a=2th quantile and the l2 is the 1 a=2th quantile of the random variable R.
3. Examples In this section, we shall now apply our procedures to two examples with simulated data. Example 1. This example deals with weibull failure times and lognormal repair times with following population parameter configurations: c ¼ 0:5, k ¼ 0:1, l ¼ 1:0, r ¼ 0:5. With these population parameters, actual availability of the system is 0.9848. Ten failure times and ten repair times were generated from these distributions and their values are: Repair times: 4.906, 1.365, 5.381, 5.135, 3.160, 2.788, 1.604, 3.553, 5.623, 1.458 Failure times: 97.189, 17.635, 25.482, 15.905, 322.542, 1606.661, 18.725, 164.138, 0.627, 10.069 For this data, 95% generalized upper confidence interval described in Section 2 is (0.9541, 1). Furthermore, suppose we need to test the availability hypothesis H0 : A 6 A0 ¼ 0:95 vs. Ha : A > A0 ¼ 0:95 using this data. To test this hypothesis, the generalized p-value given in Eq. (8) is 0.0371. If we are to test the hypothesis H0 : A 6 A0 ¼ 0:90 vs. Ha : A > A0 ¼ 0:90, the generalized p-value is given by 0.0017. Example 2. The second example deals with lognormal failure times and lognormal repair times with following population parameter configurations: a ¼ 5:0, b ¼ 0:1, l ¼ 1:0, r ¼ 0:5. The actual availability of the system is 0.9798. The data generated from these distributions are: Repair times: 7.588, 4.812, 3.636, 2.570, 1.256, 1.227, 1.833, 3.228, 2.842, 1.347 Failure times: 165.177, 123.557, 154.328, 156.439, 153.304, 138.563, 148.950, 133.561, 140.769, 162.638 Assuming that all of the above parameters are unknown, 95% generalized upper confidence interval calculated from this data is (0.9663, 1). If we are to test the availability hypothesis H0 : A 6 A0 ¼ 0:95 vs. Ha : A > A0 ¼ 0:95 using this data, the generalized p-value given in (12) is 0.0083. The generalized p-value for the hypothesis H0 : A 6 A0 ¼ 0:95 vs. Ha : A > A0 ¼ 0:95 is 0.00088.
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M.M.A. Ananda, J. Gamage / Appl. Math. Comput. 150 (2004) 409–416
References [1] M.M.A. Ananda, Estimation and testing of availability of a parallel system with exponential failure and repair times, Journal of Statistical Planning and Inference 77 (1999) 237–246. [2] M.M.A. Ananda, Confidence intervals for steady state availability of a system with exponential operating time and lognormal repair time, submitted for publication. [3] M.M.A. Ananda, S. Weerahandi, Testing the difference of two exponential means using generalized p-values, Communications in Statistics––Simulation and Computation 25 (2) (1996) 521–532. [4] M.M.A. Ananda, S. Weerahandi, Two-way ANOVA with unequal cell frequencies and unequal variances, Statistica Sinica 7 (1997) 631–646. [5] P. Chandrasekhar, R. Natarajan, H.S. Sujatha, Confidence limits for steady state availability of systems, Microelectronics and Reliability 34 (8) (1994) 1365–1367. [6] J. Gamage, S. Weerahandi, Size performance of some tests in one-way ANOVA, Communications in Statistics––Simulation and Computations 27 (3) (1998) 625–640. [7] H.L. Gray, T.O. Lewis, A confidence interval for the availability ratio, Technometrics 9 (1967) 465–471. [8] H.L. Gray, W.R. Schucany, Lower confidence limits for availability assuming lognormally distributed repair times, IEEE Transactions on Reliability R-18 (1969) 157–162. [9] Maintainability Design Criteria Handbook for Designers of Shipboard Electronic Equipment, Bureau of Ships, Navships 94324, 1965. [10] B.N. Masters, T.O. Lewis, A note on the confidence interval for the availability ratio, Microelectronics and Reliability 27 (3) (1987) 487–492. [11] B.N. Masters, T.O. Lewis, W.J. Kolarik, A confidence interval for the availability ratio for systems with weibull operating time and lognormal repair time, Microelectronics and Reliability 32 (1/2) (1992) 89–99. [12] H.F. Martz, R.A. Waller, Bayesian Reliability Analysis, John Wiley, 1982. [13] K. Tsui, S. Weerahandi, Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters, Journal of the American Statistical Association 84 (1989) 602–607. [14] S. Weerahandi, Exact Statistical Methods for Data Analysis, Springer-Verlag, New York, 1995. [15] S. Weerahandi, R.A. Johnson, Testing reliability in a stress-strength model when X and Y are normally distributed, Technometrics 34 (1992) 83–91.
On steady state availability of a system with lognormal repair time Malwane M.A. Ananda
a,*
, Jinadasa Gamage
b
a
Department of Mathematical Sciences, University of Nevada, Box 455020 4505, Maryland Parkway, Las Vegas, NV 89154-4020, USA b Department of Mathematics, Illinois State University, Normal, IL 61761, USA
Abstract Assuming two-parameter lognormal distribution for repair times, statistical inference for the steady state availability of a system is considered. For the failure time distribution, weibull, gamma, and lognormal distributions were considered. Using the generalized p-value approach, we propose confidence intervals and exact tests for the steady state availability of a system. A couple of examples are given to illustrate the proposed procedures. Ó 2003 Published by Elsevier Ltd. Keywords: Steady state availability; Lognormal repair times; Confidence intervals
1. Introduction The steady state availability is a measure in determining the long term performance of a system which is to be operated continuously. The steady state availability A of a system is defined by A¼
ly ; ly þ lx
ð1Þ
where ly is the mean time between failures of the system and lx is the mean time to repair the system. The random variables X and Y represent the time to
*
Corresponding author. E-mail address: [email protected] (M.M.A. Ananda).
0096-3003/$ - see front matter Ó 2003 Published by Elsevier Ltd. doi:10.1016/S0096-3003(03)00281-9
410
M.M.A. Ananda, J. Gamage / Appl. Math. Comput. 150 (2004) 409–416
repair the system and the time between failures of the system. It is known that the repair time of many repairable systems is well characterized by a lognormal distribution [8,9,12]. Furthermore, it is well known [12] that the failure times of repairable systems are often modeled with weibull, gamma, and lognormal distributions. In availability studies, 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 which is essentially equivalent to using a one parameter lognormal distribution. In some other instance even when the repair times are known to be lognormally distributed, for mathematical tractability probability distributions such as the exponential distribution is used instead of the lognormal distribution. Gray and Lewis [7] provided exact confidence intervals for the steady state availability when the repair times are lognormally distributed with known variance (i.e. ln X N ðl; r2 Þ, where r2 is known) and the failure times are exponentially distributed. The procedure involved numerically evaluating two dimensional integrals and therefore, they provided some table values which are required to construct these confidence intervals. Masters and Lewis [10] provided exact confidence intervals for the availability under the assumption that the time between failures are distributed with a gamma random variable and repair times are distributed with a lognormal distribution with known variance. Masters et al. [11] provided exact intervals under the assumption that the time between failures and time to repair are weibull and lognormal (with known variance) random variables, respectively. Chandrasekhar et al. [5] provided confidence intervals when the failure times and the repair times both are lognormal with known variances. When r2 is unknown, the exact confidence intervals and testing procedures are not available in the literature. In this situation for some of the cases discussed above, Bayesian techniques were described in Martz and Waller [12]. When r2 is unknown, one can use the procedure given in Masters and Lewis [10], Masters et al. [11], and Chandrasekhar et al. [5] as an ad-hoc procedure by replacing r2 with an estimate of r2 , even though statistical properties of such an ad-hoc procedure are unknown. In fact, simulation results from Gary and Schucany [8] 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. Using the generalized p-value approach introduced in Tsui and Weerahandi [13], when repair times and failure times are exponentially distributed, Ananda [1] gave exact intervals and testing procedures for the availability of a system consisting of several independent parallel subsystems and they showed that these generalized procedures outperform the other known procedures. Ananda [2] extended these results and gave confidence intervals and testing procedures for the availability when the repair time is lognormally distributed with an
M.M.A. Ananda, J. Gamage / Appl. Math. Comput. 150 (2004) 409–416
411
unknown mean and variance and the failure time is exponentially distributed with an unknown mean. Using statistical simulation Ananda [2] showed that the exact probability coverage of their procedure is nearly equal to the intended coverage and their procedure outperform the other known two-parameter lognormal procedures. In this paper, we look at the exact testing procedures and confidence intervals for the availability A when the repair time is lognormally distributed with an unknown mean and variance. For the failure time distributions, we use gamma, weibull or lognormal distributions. For each case, using the generalized p-value approach, we construct an exact test for testing the long-run availability. Furthermore, for each case, we construct a generalized confidence interval for the availability and a generalized lower confidence limit (LCL) for the availability. Generalized p-value approach is a recently developed method which is based on an extended definition of p-values. The generalized tests and generalized confidence intervals are 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 (for example, [3,4,6,15]), 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 and for more details of these procedures see Weerahandi [14].
2. Estimation and testing procedures with lognormal repair times Suppose the repair time X has a lognormal distribution with parameters l and r2 ; " 2 # 1=2 ð2pÞ 1 ln x l f ðx; l; r2 Þ ¼ exp ; 2 r xr
x > 0;
ð2Þ
where parameters l and r are unknown parameters. Through out of this paper, we use this two-parameter lognormal model for the repair times. Suppose that repair times X1 ; X2 ; . . . ; Xn are available from this distribution. We are interested in testing the availability hypothesis H0 : A 6 A0
vs: Ha : A > A0
ð3Þ
and constructing confidence intervals for the steady state availability A, based on the observed repair times and the observed failure times Y1 ; Y2 ; . . . ; Ym of the
412
M.M.A. Ananda, J. Gamage / Appl. Math. Comput. 150 (2004) 409–416
system. These failure times are assumed to follow the underline probability distribution considered in each subsection. Suppose U¼
n X
Ui =n;
S2 ¼
r¼1
n X 2 ðUi U Þ =ðn 1Þ; r¼1
Z¼
U l pffiffiffi ; r= n
2
V ¼
ðn 1ÞS ; r2
ð4Þ
where Ui ¼ lnðXi Þ, i ¼ 1; 2; . . . ; n. Then Z N ð0; 1Þ, V v2n1 . 2.1. Statistical inference with weibull failure times First let us consider the case where the time between failures follow a weibull distribution. Suppose the failure time Y has a weibull distribution f ðy; k; cÞ ¼ cky c1 eky
c
ð5Þ
y > 0;
were c is a known parameter and k is an unknown parameter. Then EðY Þ ¼ Cðc1 þ 1Þk1=c and the steady state availability of the system is given by A¼
Cðc1 þ 1Þk1=c Cðc1 þ 1Þk1=c þ expðl þ r2 =2Þ
:
ð6Þ
Masters et al. [11] provided an approximate test and confidence interval when the parameters r and c are known and, l and k are unknown. In this section, using the generalized p-value approach, we will provide an exact test for the availability when the parameter c is known and l; r, and k are unknown. If c is unknown, statistical inference regarding the availability gets more complicated and needs further research. Let Y1 ; Y2 ; . . . ; Ym be the observed failure times from the failure time distribution given in (5). Let C¼
m X
Yic
and
W ¼ 2kC:
ð7Þ
r¼1
Then W v22m . The hypothesis H0 given in (3) can be tested using the p-value p ¼ Pr
Cðc1 þ 1Þ u Zsðn 1Þ1=2 n1=2 V 1=2 þ ðn 1Þs2 V 1 =2Þ Cðc1 þ 1Þ þ ð0:5W =cÞ1=c expð
! 6 A0 :
ð8Þ
M.M.A. Ananda, J. Gamage / Appl. Math. Comput. 150 (2004) 409–416
413
This p-value can be evaluated by generating a large number of random numbers from random variables Z, V and W . In fixed level testing, one can use this p-value by rejecting the null hypothesis, if the generalized p-value is less than the desired nominal level a. The proof of this testing procedure can be obtained by utilizing the generalized test variable T ¼
Cðc1 þ 1Þ Cðc1 þ 1Þ þ ðkC=cÞ
1=c
expð u sS 1 ðU lÞ þ r2 s2 S 2 =2Þ
A:
ð9Þ
Here u, s2 , and c are the observed values of U , S 2 , and C given in Eqs. (4) and (7). The observed value of this test variable is equal to zero and the probability distribution of T ðX; Y; x; y; l; r; hÞ is free of nuisance parameters. For any given t, when x, y and nuisance parameters are fixed, the cumulative distribution function of T , PrðT 6 t; AÞ is a monotonically increasing function of A, therefore T ðX; Y; x; y; l; r; hÞ is a test variable and this will yield the p-value given in (8). The confidence intervals on A induced by this p-value can be obtained by defining the random variable R¼
Cðc1 þ 1Þ
; u Zsðn 1Þ1=2 n1=2 V 1=2 þ ðn 1Þs2 V 1 =2Þ Cðc1 þ 1Þ þ ð0:5W =cÞ1=c expð ð10Þ
where Z, V , and W are independent standard normal, chi-square with degrees of freedom n 1 and chi-square with degrees of freedom 2m respectively. 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=2th quantile and the l2 is the 1 a=2th 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. For further details of these procedures or the proofs, the reader is referred to Weerahandi [14] and Ananda [2]. 2.2. Statistical inference with gamma failure times Suppose the operating time Y has a gamma distribution with parameters k and k, f ðy; hÞ ¼
kk y k1 eky CðkÞ
y > 0:
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Then the steady state availability of the system is given by A ¼ ½1 þ kk 1 1 expðl þ r2 =2Þ . These types of failure time distributions occur in availability studies [10], in particular when a system composed of p components, the entire system fails only after k of the components have failed. Masters and Lewis [10] gave confidence intervals when repair times have lognormal distribution (with known r2 ) and failure times have gamma distribution. One can show that when parameters k, l, r are unknown, the generalized pvalue to test the hypothesis in (3) is given by p ¼ PrðW ð2ckÞ
1
expð u Zsðn 1Þ
1=2 1=2
n
V 1=2
þ ðn 1Þs2 V 1 =2Þ P A1 0 1Þ:
ð11Þ
The proof of this is quite similar to the previous proof and as before this pvalue can be evaluated using simulation by generating a large number of random numbers from the random variables W ; Z, and V . Confidence intervals and lower confidence limit on A can be obtained in a similar manner. 2.3. Statistical inference with lognormal failure times Suppose the failure time Y has a lognormal distribution with unknown parameters a and b2 . Then ly ¼ expða þ b2 =2Þ and A ¼ ½1 þ expððl aÞ þ 1 ðr2 b2 Þ=2Þ . When the parameters a, b, r are known, Chandrasekhar et al. [5] provided statistical inference for the availability. When a, b, l, r are unknown, one can show that the generalized p-value to test the hypothesis in (11) is given by p ¼ Pr þ
Z1 s1 ðm 1Þ 1=2
m1=2 V1
1=2
1=2
ðm 1Þs21 Zsðn 1Þ 2V1 n1=2 V 1=2 !
ðn 1Þs2 P lnðA1 u1 u ; 0 1Þ þ 2V1
ð12Þ
1 , s2 , s21 are sample means and sample variance of repair times and where u, u failure times respectively. Here Z, V , Z1 and V1 are independent random variables having distributions Z N ð0; 1Þ, V v2n1 , Z1 N ð0; 1Þ, and V1 v2m1 . To state the generalized confidence interval, define the random variable " ( 1=2 Z1 s1 ðm 1Þ R ¼ 1 þ exp u u1 þ 1=2 m1=2 V1 )#1 1=2 ðm 1Þs21 Zsðn 1Þ ðn 1Þs2 þ : ð13Þ 2V1 n1=2 V 1=2 2V1
M.M.A. Ananda, J. Gamage / Appl. Math. Comput. 150 (2004) 409–416
415
Then 100ð1 aÞ% upper confidence interval for A is given by ðc0 ; 1Þ, where l0 in the ath quantile of the random variable R. Moreover, equal tail 100ð1 aÞ% confidence interval for A is given by ðl1 ; l2 Þ, where l1 is the a=2th quantile and the l2 is the 1 a=2th quantile of the random variable R.
3. Examples In this section, we shall now apply our procedures to two examples with simulated data. Example 1. This example deals with weibull failure times and lognormal repair times with following population parameter configurations: c ¼ 0:5, k ¼ 0:1, l ¼ 1:0, r ¼ 0:5. With these population parameters, actual availability of the system is 0.9848. Ten failure times and ten repair times were generated from these distributions and their values are: Repair times: 4.906, 1.365, 5.381, 5.135, 3.160, 2.788, 1.604, 3.553, 5.623, 1.458 Failure times: 97.189, 17.635, 25.482, 15.905, 322.542, 1606.661, 18.725, 164.138, 0.627, 10.069 For this data, 95% generalized upper confidence interval described in Section 2 is (0.9541, 1). Furthermore, suppose we need to test the availability hypothesis H0 : A 6 A0 ¼ 0:95 vs. Ha : A > A0 ¼ 0:95 using this data. To test this hypothesis, the generalized p-value given in Eq. (8) is 0.0371. If we are to test the hypothesis H0 : A 6 A0 ¼ 0:90 vs. Ha : A > A0 ¼ 0:90, the generalized p-value is given by 0.0017. Example 2. The second example deals with lognormal failure times and lognormal repair times with following population parameter configurations: a ¼ 5:0, b ¼ 0:1, l ¼ 1:0, r ¼ 0:5. The actual availability of the system is 0.9798. The data generated from these distributions are: Repair times: 7.588, 4.812, 3.636, 2.570, 1.256, 1.227, 1.833, 3.228, 2.842, 1.347 Failure times: 165.177, 123.557, 154.328, 156.439, 153.304, 138.563, 148.950, 133.561, 140.769, 162.638 Assuming that all of the above parameters are unknown, 95% generalized upper confidence interval calculated from this data is (0.9663, 1). If we are to test the availability hypothesis H0 : A 6 A0 ¼ 0:95 vs. Ha : A > A0 ¼ 0:95 using this data, the generalized p-value given in (12) is 0.0083. The generalized p-value for the hypothesis H0 : A 6 A0 ¼ 0:95 vs. Ha : A > A0 ¼ 0:95 is 0.00088.
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