- Email: [email protected]
Comparative analysis of availability for a redundant repairable system
Applied Mathematics and Computation 188 (2007) 332–338 www.elsevier.com/locate/amc
Comparative analysis of availability for a redundant repairable system Jau-Chuan Ke *, Yunn-Kuang Chu Department of Statistics, National Taichung Institute of Technology, Taichung 404, Taiwan, ROC
Abstract We consider the steady-state availability, denoted A, of a repairable system consisted of one operating unit and one spare. This paper we are interested in computational comparisons of confidence intervals for A based on four feasible b of A, defined as the ratio of the sample mean lifetime of the operating and efficient approaches. The natural estimator A unit to the sum of the sample mean lifetime of the operating unit and the sample mean downtime of system, is strongly consistent and asymptotically normal. The interval estimations for A are constructed through four bootstrap approaches; standard bootstrap confidence interval, percentile bootstrap confidence interval, bias-corrected percentile bootstrap confidence interval as well as bias-corrected and accelerated confidence interval. Finally, by calculating the coverage percentage and the average length of intervals, some numerical simulation comparisons are conducted in order to illustrate the b on various interval estimations. performances of A 2006 Elsevier Inc. All rights reserved. Keywords: Availability; Bootstrap approach; Consistent; Simulation; Slutsky’s theorem
1. Introduction Redundancy plays an important role in improving the reliability (or availability) of engineering systems. Standby redundant repairable systems have been studied extensively in the past (Kumar and Agarwal [1], Birolini [2], Yearout et al. [3], and detailed bibliographies are found in Osaki and Nakagawa [4] and Srinivasan and Subramanian [5]. Due to modelling complexity, a great deal of authors focused on the study of two-unit redundant systems under different assumptions (see [6–13]). However, their works are based on parametric lifetime and repair time distributions. On the other hand, Sen and Bhattacharjee [14] and Sen [15,16] proposed nonparametric estimators of availability under provisions of spare and repair (see Table 1). With regarding estimation of availability for a redundant repairable system with an operating unit and spare, the key problem is how to estimate precisely the mean E(D) of system downtimes. All estimation approaches in Table 1 proposed by Sen and Bhattacharjee [14–16] are just theoretically perfect, but have *
Corresponding author. E-mail addresses: [email protected] (J.-C. Ke), [email protected] (Y.-K. Chu).
0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.123
J.-C. Ke, Y.-K. Chu / Applied Mathematics and Computation 188 (2007) 332–338
333
Table 1 Some special approaches about estimating availability of the redundant repairable system with one operating unit and one spare Authors
Estimation approaches
Theoretical properties
Application in practical system
Case 1. Sen and Bhattacharjee [14] Case 2. Sen [15] Case 3. Sen [16]
• Jackknified natural estimator • Progressive censoring schemes Time-sequential point estimation Jackknified U-statistics point estimator • CAN point estimator based on sample means of data
CANa CAN CAN CAN
Troublesome Troublesome Troublesome Troublesome
CAN
The CAN estimates and the four bootstrap confidence intervals can be easily and efficiently implemented
• Four bootstrap interval estimation approaches: SB, PB, BCPB, and BCa
Feasible and efficient
Case 4. Ke and Chu (the approach for this paper is new)
a
and and and and
inefficient inefficient inefficient inefficient
CAN denotes consistent and asymptotically normal.
limited application in practical redundant repairable system. This motivates us to develop more useful estimation procedure for estimating availability of the system (referred to Case 4 in Table 1). In this paper, we consider one operating-unit system supported by one spare and a repair facility, when the operating unit fails, it is immediately replaced by the spare while it is being dispatched to the repair facility. Upon repair, it is sent to the spare box. The system fails when an operating unit fails and no spare is there to replace it, which occurs when the repairing time (Y) is greater than the lifetime (X) of the operating unit. This paper is aimed to investigate the accuracy of various interval estimations of availability for a redundant repairable system via four bootstrap approaches. Firstly in Section 2, we deduce the consistent and asymptotically normal (CAN) estimator of availability. Next, we use the bootstrap approaches to construct confidence intervals for availability in Section 3. We sketch the four bootstrap estimations of availability, and the four types of bootstrap confidence interval for availability are presented in the order of the standard bootstrap (SB), the percentile bootstrap (PB), the bias-corrected percentile bootstrap (BCPB), and the biascorrected and accelerated (BCa). Subsequently, Section 4 is devoted to evaluating performances of the four bootstrap confidence intervals for availability. Assuming three various types of lifetime distribution and two kinds of repairing time distribution, we make some numerical simulation studies. According to simulation results, we assess and compare the accuracy of these four bootstrap interval estimations by virtue of coverage percentages and average widths of intervals. Finally, we make some conclusions and recommendations for interval estimation of availability of a redundant repairable system based on the four bootstrap approaches. 2. Estimation of availability Consider a redundant repairable system described in Section 1. Let X represent the lifetime of the operating unit in the system and Y denote the repairing time when the operating unit fails. Then the system downtime D is equal to (Y X) I (Y > X), where I (Y > X) is the indicator function of event {Y > X}. Therefore, Sen and Bhattacharjee [14] wrote the system availability as A¼
EðX Þ ; EðX Þ þ EðDÞ
ð1Þ
where E(X) and E(D) are expected values of X and Y, respectively. Assume that X1, X2, . . . , Xn is a random sample of X and Y1, Y2, . . . , Yn is a random sample of Y. Define Di = (Yi Xi)I(Yi Xi) for i = 1, 2, . . . , n. Let X and D represent the sample means of Xs and Ds, respectively. According to the strong law of large numbers (cf. [17]), X and D are strongly consistent estimators of E(X) and E(D), respectively. Hence a strongly consistent estimator of the system availability A is b¼ A
X : X þD
ð2Þ
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b is asymptotically normally distributed by using the central limit theSen and Bhattacharjee [14] showed that A b obtained by [14] is complex and orem and Slutsky’s theorem. But the theoretic variance of CAN estimator A not easy to be computed for practical use. In real systems, the distributions of X and Y are seldom known, the b is more difficult to be derived and calculated. To estimate A efficiently, we will develop interval variance of A estimations by means of four bootstrap approaches. 3. Bootstrap estimation of availability Efron [18,19], the greatest statistician in the field of nonparametric resampling approach, originally developed and proposed the bootstrap, which is a resampling technique that can be effectively applied to estimate the sampling distribution of any statistic. Specifically, one can utilize the bootstrap approaches to approximate the sampling distribution of a statistic defined by a random sample from a population with unknown probability distribution. And due to the popularity of PC and statistical software, today the bootstrap becomes the most powerful nonparametric estimation procedure. Especially on the topic about interval estimation, Efron and Gong [20], Efron and Tibshirani [21], and Efron [22] presented four bootstrap confidence intervals: the standard bootstrap (SB) confidence interval, the percentile bootstrap (PB) confidence interval, the bias-corrected percentile bootstrap (BCPB) confidence interval, and the bias-corrected and accelerated (BCa) confidence interval. In order to understand how the bootstrap confidence interval is developed, let x1, x2, . . . , xn be a sample of n observations taken from the population X, and y1, y2, . . . , yn be a sample of n observations drawn from the population Y. Set di = (yi xi)I(yi > xi) for i = 1, 2, . . . , n. Then according to the bootstrap procedure, a simple random sample x1 ; x2 ; . . . ; xn can be taken from the empirical distribution of x1, x2, . . . , xn, called a bootstrap sample from x1, x2, . . . , xn. Similarly, we can draw a bootstrap sample ðd 1 ; d 2 ; . . . ; d n Þ from the empirical distribution of d1, d2, . . . , dn. Hence in terms of Eq. (2), a CAN estimate of system availability A can be calculated from original samples as b ¼ x ; A ð3Þ x þ d where x and d are the sample means of x1, x2, . . . , xn and y1, y2, . . . , yn , respectively. And another estimate of A computed from bootstrap samples is x ; ð4Þ þ d b is called a where x and d are the sample means of x1 ; x2 ; . . . ; xn and y 1 ; y 2 ; . . . ; y n , respectively. Also A bootstrap estimate of A. The above resampling process can be repeated multiple times, for example, B times. b ; A b ; . . . ; A b Þ can be computed from the bootstrap resamples. Averaging the B The B bootstrap estimates ð A 1 2 B bootstrap estimates, we obtain that b ¼ A
x
B X bB ¼ 1 b ; A A B j¼1 j
ð5Þ
b can be estimated by is the bootstrap estimate of A. And the standard deviation of the CAN estimator A ( )1=2 B 1 X b BÞ ¼ b A b B Þ2 sdð A ðA : ð6Þ B 1 j¼1 j For necessary backgrounds on bootstrap techniques, the interested readers can refer to Efron [18,19,22], Efron and Gong [20], Efron and Tibshirani [21], Gunter [23,24], Mooney and Duval [25] or Young [26]. Next, we describe four types of bootstrap confidence intervals for A as follows. 3.1. Standard bootstrap confidence interval Because the central limit theorem implies that D b AÞ=sdð A b BÞ ! ðA N ð0; 1Þ;
ð7Þ
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335
D b a where ! denotes convergence in distribution. Based upon the above asymptotic normality of A, 100(1 2a)% SB confidence interval for A is
b B Þ; A b þ za sdð A b B ÞÞ; b za sdð A ðA
ð8Þ
where za is the upper ath quantile of the standard normal distribution. 3.2. Percentile bootstrap confidence interval b ; A b ; . . . ; A b is called the bootstrap distribution of A, b and let A b ð1Þ; A b ð2Þ; . . . ; A b ðBÞ be order statistic of A 1 2 B b ; . . . ; A b . Then utilizing the 100ath and 100(1a)th percentage point of the bootstrap distribution, a b ; A A 1 2 B 100(1 2a)% PB confidence interval for A is obtained as b ð½Bð1 aÞÞÞ; b ð½BaÞ; A ðA
ð9Þ
where [x] denotes the greatest integer less than or equal to x. 3.3. Bias-corrected percentile bootstrap confidence interval b ; A b ; . . . ; A b may be biased. Consequently, the third approach is designed to The bootstrap distribution A 1 2 B P b < AÞ=B, b correct this potential bias of the bootstrap distribution. Set p0 ¼ Bj¼1 Ið A where I(Æ) is the indicator j 1 1 function. Define ^z0 ¼ U ðp0 Þ, where U denotes the inverse function of the standard normal distribution U. Then let a1 ¼ Uð2^z0 za Þ, and a2 ¼ Uð2^z0 þ za Þ. It follows that a 100(1 2 a)% BCPB confidence interval for A is given by b ð½Ba2 ÞÞ: b ð½Ba1 Þ; A ðA
ð10Þ
3.4. Bias-corrected and accelerated confidence interval Except for correcting the potential bias of the bootstrap distribution, we can accelerate convergence of the e ðiÞ and Ye ðiÞ denote the original samples with the ith observations xi and yi bootstrap distribution. Let X e i be the CAN estimate of A calculated by using X e ðiÞ and Ye ðiÞ. Define deleted, also let A e¼1 A n and
n X
ei A
i¼1
,8 " #3=2 9 < X = n eA e i Þ3 eA e i Þ2 ^ a¼ ; ðA ðA 6 : i¼1 ; i¼1 n X
^z0 and ^ a are named bias-correction and acceleration, respectively. Thus a 100(1 2a)% BCa confidence interval for A is constructed by b ð½Ba2 ÞÞ; b ð½Ba1 Þ; A ðA
ð11Þ
að^z0 za ÞÞ and a2 ¼ Uð^z0 þ ð^z0 þ za Þ=½1 ^að^z0 þ za ÞÞ. where a1 ¼ Uð^z0 þ ð^z0 za Þ=½1 ^ 4. Simulation study One principal goal of bootstrap approaches is to establish good confidence intervals. Efron and Tibshirani [21] indicated that ‘‘good’’ means that the bootstrap confidence intervals should have relatively accurate coverage performances and dependably short average interval lengths in all situations. Numerical simulation studies were carried out to evaluate performances of the four bootstrap approaches presented in Section 3. The four confidence intervals are assessed in terms of their coverage accuracy and
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average lengths. In order to reach this goal, we not only set three different failure time distributions for the lifetime X, but also assume two various continuous distributions for the repairing time Y. For concreteness, we let X have a gamma lifetime distribution G(a, b) with scale and shape parameters a and b, respectively. Three different levels (5, 2), (10, 1), and (20, 0.5) are assigned to (a, b). For convenience, the distribution of repairing time Y is assumed to be exponential with mean 8 (i.e., EXP(8)) and uniform on interval (2, 14) (i.e., U(2, 14)), respectively. For each pair of distributions (X, Y), a random sample (x1, y1), (x2, y2), . . . , (xn, yn) of size n(=15, 30, 60) is b is calculated. Next, B = 1000 bootstrap resamples generated from (X, Y). Using Eq. (3), the CAN estimate A fðx1 ; y 1 Þ; ðx2 ; y 2 Þ; . . . ; ðxn ; y n Þg are drawn from the original sample {(x1, y1), (x2, y2), . . . , (xn, yn)}. By virtue of b ; . . . ; A b are calculated from the bootstrap resamples. And utilizing (5) and b ; A (4), B bootstrap estimates A 1 2 B b is computed as sdð A b B Þ. Finally, applying the four bootstrap meth(6), the estimated standard deviation of A ods (Eqs. (8)–(11)) described in Section 3, we obtain the four bootstrap confidence intervals with confidence level 90%. Subsequently the above simulation process is replicated N=1000 times. We not only average the N simub but also compute coverage percentages and average lengths of the four bootstrap confidence lated estimates A, intervals. Matlab 7.0.4 code is easily implemented to accomplish all simulations. All simulation results are b is demonstrated by Table displayed in Tables 2 and 3. The consistency property of the CAN estimator A 2, while performances of the four bootstrap confidence intervals for system availability A can be examined in terms of Table 3. For this redundant repairable system with one operating unit and one spare, the mean of N simulated CAN b is shown in Table 2 for various sample size n and six different combinations of lifetime distribution estimate A and repairing time distribution. Based upon the strong law of large numbers (cf. [17,p. 196]), the true value of b with sufficiently large sample size. system availability A can be estimated by the simulated sample value of A b Thus the true value of A is produced from the average of N simulated A with sample size n = 107. We find that the approximated system availability approaches to the true value of A with six decimal places when n P 107. b converges to the true value of system availability A as the Table 2 implies that the average of N simulated A sample size n becomes large enough, for any specified lifetime and repairing time distributions, as we expect. b is a consistent estimator of system availability A. According to simulation analysis, we show that A Because the frequency of coverage for an accurate confidence interval is binomially distributed with N = 1000 and p = 0.90. Therefore, a 99% confidence range for the coverage probability is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:90 2:576 0:90 0:10=1000 ¼ 0:90 0:0244; that is, from 0.876 to 0.924. Hence, one could be 99% confident that a ‘‘true 90% confidence interval’’ would have a proportion of coverage between 0.876 and 0.924. All simulation results of coverage percentages and average lengths for the four bootstrap confidence intervals are displayed in Table 3. Examining these simulation results, we find that coverage percentages for the four bootstrap confidence intervals increase with the sample size n. The BCa approach always has the largest coverage percentages as well as the highest chances that fall into the 99% confidence range (0.876, 0.924). In addition, the SB approach always has the smallest coverage percentages as well as the lowest chance that fall into the 99% confidence range. When the sample size n is sufficiently large (n P 60), almost all coverage per-
Table 2 b Simulation analysis for consistency of A Distribution of X
G(5, 2) G(10, 1) G(20, 0.5) G(5, 2) G(10, 1) G(20, 0.5)
Distribution of Y
EXP(8) EXP(8) EXP(8) U(2, 14) U(2, 14) U(2, 14)
The true value of A
0.763546 0.736553 0.697354 0.832997 0.783819 0.728217
b The average of 1000 simulated A n = 15
n = 30
n = 60
0.769033 0.729428 0.692224 0.829010 0.771585 0.703688
0.768637 0.740861 0.694576 0.832421 0.781088 0.719426
0.767478 0.733981 0.695719 0.832954 0.781320 0.723276
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Table 3 Simulation results of coverage percentages and average lengths for the four 90% bootstrap confidence intervals Distribution of X
Distribution of Y
Bootstrap approaches
Coverage percentages n = 15
G(5, 2)
EXP(8)
G(10, 1)
EXP(8)
G(20, 0.5)
EXP(8)
G(5, 2)
U(2, 14)
G(10, 1)
U(2, 14)
G(20, 0.5)
U(2, 14)
a *
SB PB BCPB BCa SB PB BCPB BCa SB PB BCPB BCa SB PB BCPB BCa SB PB BCPB BCa SB PB BCPB BCa
0.791 0.819 0.838 0.849 0.803 0.849 0.856 0.875 0.793 0.848 0.859 0.868 0.850 0.861 0.873 0.887* 0.837 0.863 0.872 0.895* 0.827 0.839 0.849 0.861
n = 30 0.832 0.850 0.858 0.868 0.836 0.856 0.871 0.876* 0.871 0.884* 0.888* 0.898* 0.861 0.870 0.880* 0.897* 0.880* 0.884* 0.887* 0.893* 0.859 0.870 0.878* 0.878*
Average lengths n = 60 0.862 0.879* 0.882* 0.892* 0.876* 0.890* 0.884* 0.885* 0.873 0.886* 0.893* 0.898* 0.884* 0.885* 0.890* 0.898* 0.896* 0.894* 0.892* 0.900* 0.885* 0.885* 0.886* 0.891*
n = 15 0.3011 0.3004a 0.3049a 0.3161a 0.3399 0.3393a 0.3421 0.3502 0.3901 0.3888a 0.3891 0.3916 0.2238 0.2229a 0.2257 0.2319 0.2856 0.2838a 0.2846 0.2866 0.3573 0.3550 0.3518 0.3487a
n = 30 a
0.2281 0.2286 0.2299 0.2360 0.2514a 0.2516 0.2527 0.2569 0.2942 0.2941 0.2937a 0.2949 0.1609 0.1607a 0.1614 0.1635 0.2026 0.2023 0.2022a 0.2028 0.2605 0.2598 0.2570 0.2549a
n = 60 0.1703a 0.1708 0.1712 0.1741 0.1880a 0.1883 0.1885 0.1903 0.2146 0.2147 0.2145a 0.2149 0.1153a 0.1154 0.1155 0.1162 0.1458 0.1458 0.1456a 0.1458 0.1838 0.1836 0.1823 0.1816a
Indicates the shortest average length among the four bootstrap confidence intervals. Indicates a proportion that is not significantly different (at a = 0.01) from the expected value of 0.90.
centages for the four bootstrap intervals reach the 99% confidence range. Based on coverage percentages, the BCa approach has the best performances among the four bootstrap approaches. On the other hand, the four bootstrap approaches are scrutinized by average lengths of the confidence intervals. We can see from Table 3 that all average lengths for the four bootstrap approaches decrease with sample size n. Particularly the shortest average length among the four bootstrap confidence intervals is produced from the PB approach when the sample size is small (n 6 15). Whereas the differences among average lengths of the four bootstrap confidence intervals could be negligible when the sample size n is large enough (n P 30). In a word, the BCa approach among the four bootstrap approaches can construct the best confidence interval of availability A for a redunb dant repairable system, by applying the CAN estimator A. 5. Conclusions This paper provided four feasible and efficient interval estimations of availability A for a redundant repairb as well as four different bootstrap approaches SB, PB, BCPB and BCa, are able system. The CAN estimator A applied to construct confidence intervals for system availability A. The coverage percentages and average lengths are adopted to understand, compare, and assess performances of the resulted bootstrap confidence intervals. The simulation results imply that the BCa approach has the best performance on coverage percentages, while the SB approach is the poorest performer. While based on the criterion of average interval length, the PB approach is the most superior actor when the sample size n is small (n 6 15); but all the four bootstrap approaches perform almost equally well when the sample size n is sufficiently large (n P 30). Consequently, the BCa approach is the best choice made by practitioners who want to construct efficient confidence intervals of availability A for such a repairable system with sufficient data (n P 30) of lifetime X and repairing time Y.
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The bootstrap estimation approaches used by this paper would be successfully, efficiently, and easily applied to practical redundant systems. Further research may consider the investigations of other system characteristics of some redundant repairable systems by using the bootstrap estimation approaches. References [1] A. Kumar, M. Agarwal, A review of standby systems, IEEE Transactions on Reliability, R-29 (1980) 290–294. [2] A. Birolini, The Use of Stochastic Processes in Modeling Reliability Problems, Lecture Notes in Economics and Mathematical Systems, vol. 252, Springer-Verlag, Berlin, 1985. [3] R.D. Yearout, P. Reddy, D.L. Grosh, Standby redundancy in reliability—A review, IEEE Transactions on Reliability, R-35 (1986) 285–292. [4] S. Osaki, T. Nakagawa, Bibliography for reliability and availability of stochastic systems, IEEE Transactions on Reliability R-25 (1976) 284–287. [5] S.K. Srinivasan, R. Subramanian, Probabilistic Analysis of Redundant Systems, Lecture Notes in Economics and Mathematical Systems, 175, Springer-Verlag, Berlin, 1980. [6] L.R. Goel, P. Shrivastava, Profit analysis of a two-unit redundant system with provision for test and correlated failures and repairs, Microelectronics and Reliability 31 (1991) 827–833. [7] A.T. de Almeida, F.M. Campello de Souza, Decision theory in maintenance strategy for a 2-unit redundant standby system, IEEE Transactions on Reliability 42 (3) (1993) 401–407. [8] M. Gururajan, B. Srinivasan, A complex two-unit system with random breakdown of repair facility, Microelectronics and Reliability 35 (2) (1995) 299–302. [9] R. Subramanian, V. Anantharaman, Reliability analysis of a complex standby redundant system, Reliability Engineering and System Safety 48 (1995) 57–70. [10] S.P. Rajamanickam, B. Chandrasekar, Reliability measures for two-unit systems with a dependent structure for failure and repair times, Microelectronics and Reliability 37 (5) (1997) 829–833. [11] R. Billinton, J. Pan, Optimal maintenance scheduling in a two identical component parallel redundant system, Reliability Engineering and System Safety 59 (1998) 309–316. [12] V.S.S. Yadavalli, M. Botha, A. Bekker, Asymptotic confidence limits for the steady-state availability of a two-unit parallel system with preparation time for the repair facility, Asia–Pacific Journal of Operational Research 19 (2002) 249–256. [13] J.H. Seo, J.S. Jang, D.S. Bai, Lifetime and reliability estimation of repairable redundant system subject to periodic alternation, Reliability Engineering and System Safety 80 (2003) 197–204. [14] P.K. Sen, M.C. Bhattacharjee, Nonparametric estimators of availability under provisions of spare and repair I, in: A.P. Bausu (Ed.), Reliability and Quality Control, North Holland, Amsterdam, 1986, pp. 281–296. [15] P.K. Sen, Nonparametric estimators of availability under provisions of spare and repair, II, in: A.P. Bausu (Ed.), Reliability and Quality Control, Amsterdam, North Holland, 1986, pp. 297–308. [16] P.K. Sen, Statistical analysis of some reliability models: parametrics, semi-parametrics and nonparametrics, Journal of Statistical Planning and Inference 43 (1995) 41–66. [17] G.G. Roussas, A Course in Mathematical Statistics, second ed., Academic Press, 1997. [18] B. Efron, Bootstrap methods; another look at the jackknife, Annals of Statistics 7 (1979) 1–26. [19] B. Efron, The jackknife, the bootstrap, and other resampling plans, SIAM Monograph #38, 1982. [20] B. Efron, G. Gong, A leisurely look at the boot strap, the jackknife and cross-validation, American Statistician 37 (1983) 36–48. [21] B. Efron, R.J. Tibshirani, Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy, Statistical Science 1 (1986) 54–77. [22] B. Efron, Better bootstrap confidence intervals, Journal of the American Statistical Association 82 (1987) 171–200. [23] B. Gunter, Bootstrapping: How to make something from almost nothing and get statistically valid answers, Part I, Quality Progress (1991) 97–103. [24] B. Gunter, Bootstrapping: How to make something from almost nothing and get statistically valid answers, Part III, Quality Progress (1992) 119–122. [25] Z. Mooney, R.D. Duval, Bootstrapping: A Nonparametric Approach to Statistical Inference, Sage Publications, Newbury Park, CA, 1993. [26] A. Young, Bootstrap: more than a stab in the dark? Statistical Science 9 (1994) 382–415.
Comparative analysis of availability for a redundant repairable system Jau-Chuan Ke *, Yunn-Kuang Chu Department of Statistics, National Taichung Institute of Technology, Taichung 404, Taiwan, ROC
Abstract We consider the steady-state availability, denoted A, of a repairable system consisted of one operating unit and one spare. This paper we are interested in computational comparisons of confidence intervals for A based on four feasible b of A, defined as the ratio of the sample mean lifetime of the operating and efficient approaches. The natural estimator A unit to the sum of the sample mean lifetime of the operating unit and the sample mean downtime of system, is strongly consistent and asymptotically normal. The interval estimations for A are constructed through four bootstrap approaches; standard bootstrap confidence interval, percentile bootstrap confidence interval, bias-corrected percentile bootstrap confidence interval as well as bias-corrected and accelerated confidence interval. Finally, by calculating the coverage percentage and the average length of intervals, some numerical simulation comparisons are conducted in order to illustrate the b on various interval estimations. performances of A 2006 Elsevier Inc. All rights reserved. Keywords: Availability; Bootstrap approach; Consistent; Simulation; Slutsky’s theorem
1. Introduction Redundancy plays an important role in improving the reliability (or availability) of engineering systems. Standby redundant repairable systems have been studied extensively in the past (Kumar and Agarwal [1], Birolini [2], Yearout et al. [3], and detailed bibliographies are found in Osaki and Nakagawa [4] and Srinivasan and Subramanian [5]. Due to modelling complexity, a great deal of authors focused on the study of two-unit redundant systems under different assumptions (see [6–13]). However, their works are based on parametric lifetime and repair time distributions. On the other hand, Sen and Bhattacharjee [14] and Sen [15,16] proposed nonparametric estimators of availability under provisions of spare and repair (see Table 1). With regarding estimation of availability for a redundant repairable system with an operating unit and spare, the key problem is how to estimate precisely the mean E(D) of system downtimes. All estimation approaches in Table 1 proposed by Sen and Bhattacharjee [14–16] are just theoretically perfect, but have *
Corresponding author. E-mail addresses: [email protected] (J.-C. Ke), [email protected] (Y.-K. Chu).
0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.123
J.-C. Ke, Y.-K. Chu / Applied Mathematics and Computation 188 (2007) 332–338
333
Table 1 Some special approaches about estimating availability of the redundant repairable system with one operating unit and one spare Authors
Estimation approaches
Theoretical properties
Application in practical system
Case 1. Sen and Bhattacharjee [14] Case 2. Sen [15] Case 3. Sen [16]
• Jackknified natural estimator • Progressive censoring schemes Time-sequential point estimation Jackknified U-statistics point estimator • CAN point estimator based on sample means of data
CANa CAN CAN CAN
Troublesome Troublesome Troublesome Troublesome
CAN
The CAN estimates and the four bootstrap confidence intervals can be easily and efficiently implemented
• Four bootstrap interval estimation approaches: SB, PB, BCPB, and BCa
Feasible and efficient
Case 4. Ke and Chu (the approach for this paper is new)
a
and and and and
inefficient inefficient inefficient inefficient
CAN denotes consistent and asymptotically normal.
limited application in practical redundant repairable system. This motivates us to develop more useful estimation procedure for estimating availability of the system (referred to Case 4 in Table 1). In this paper, we consider one operating-unit system supported by one spare and a repair facility, when the operating unit fails, it is immediately replaced by the spare while it is being dispatched to the repair facility. Upon repair, it is sent to the spare box. The system fails when an operating unit fails and no spare is there to replace it, which occurs when the repairing time (Y) is greater than the lifetime (X) of the operating unit. This paper is aimed to investigate the accuracy of various interval estimations of availability for a redundant repairable system via four bootstrap approaches. Firstly in Section 2, we deduce the consistent and asymptotically normal (CAN) estimator of availability. Next, we use the bootstrap approaches to construct confidence intervals for availability in Section 3. We sketch the four bootstrap estimations of availability, and the four types of bootstrap confidence interval for availability are presented in the order of the standard bootstrap (SB), the percentile bootstrap (PB), the bias-corrected percentile bootstrap (BCPB), and the biascorrected and accelerated (BCa). Subsequently, Section 4 is devoted to evaluating performances of the four bootstrap confidence intervals for availability. Assuming three various types of lifetime distribution and two kinds of repairing time distribution, we make some numerical simulation studies. According to simulation results, we assess and compare the accuracy of these four bootstrap interval estimations by virtue of coverage percentages and average widths of intervals. Finally, we make some conclusions and recommendations for interval estimation of availability of a redundant repairable system based on the four bootstrap approaches. 2. Estimation of availability Consider a redundant repairable system described in Section 1. Let X represent the lifetime of the operating unit in the system and Y denote the repairing time when the operating unit fails. Then the system downtime D is equal to (Y X) I (Y > X), where I (Y > X) is the indicator function of event {Y > X}. Therefore, Sen and Bhattacharjee [14] wrote the system availability as A¼
EðX Þ ; EðX Þ þ EðDÞ
ð1Þ
where E(X) and E(D) are expected values of X and Y, respectively. Assume that X1, X2, . . . , Xn is a random sample of X and Y1, Y2, . . . , Yn is a random sample of Y. Define Di = (Yi Xi)I(Yi Xi) for i = 1, 2, . . . , n. Let X and D represent the sample means of Xs and Ds, respectively. According to the strong law of large numbers (cf. [17]), X and D are strongly consistent estimators of E(X) and E(D), respectively. Hence a strongly consistent estimator of the system availability A is b¼ A
X : X þD
ð2Þ
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b is asymptotically normally distributed by using the central limit theSen and Bhattacharjee [14] showed that A b obtained by [14] is complex and orem and Slutsky’s theorem. But the theoretic variance of CAN estimator A not easy to be computed for practical use. In real systems, the distributions of X and Y are seldom known, the b is more difficult to be derived and calculated. To estimate A efficiently, we will develop interval variance of A estimations by means of four bootstrap approaches. 3. Bootstrap estimation of availability Efron [18,19], the greatest statistician in the field of nonparametric resampling approach, originally developed and proposed the bootstrap, which is a resampling technique that can be effectively applied to estimate the sampling distribution of any statistic. Specifically, one can utilize the bootstrap approaches to approximate the sampling distribution of a statistic defined by a random sample from a population with unknown probability distribution. And due to the popularity of PC and statistical software, today the bootstrap becomes the most powerful nonparametric estimation procedure. Especially on the topic about interval estimation, Efron and Gong [20], Efron and Tibshirani [21], and Efron [22] presented four bootstrap confidence intervals: the standard bootstrap (SB) confidence interval, the percentile bootstrap (PB) confidence interval, the bias-corrected percentile bootstrap (BCPB) confidence interval, and the bias-corrected and accelerated (BCa) confidence interval. In order to understand how the bootstrap confidence interval is developed, let x1, x2, . . . , xn be a sample of n observations taken from the population X, and y1, y2, . . . , yn be a sample of n observations drawn from the population Y. Set di = (yi xi)I(yi > xi) for i = 1, 2, . . . , n. Then according to the bootstrap procedure, a simple random sample x1 ; x2 ; . . . ; xn can be taken from the empirical distribution of x1, x2, . . . , xn, called a bootstrap sample from x1, x2, . . . , xn. Similarly, we can draw a bootstrap sample ðd 1 ; d 2 ; . . . ; d n Þ from the empirical distribution of d1, d2, . . . , dn. Hence in terms of Eq. (2), a CAN estimate of system availability A can be calculated from original samples as b ¼ x ; A ð3Þ x þ d where x and d are the sample means of x1, x2, . . . , xn and y1, y2, . . . , yn , respectively. And another estimate of A computed from bootstrap samples is x ; ð4Þ þ d b is called a where x and d are the sample means of x1 ; x2 ; . . . ; xn and y 1 ; y 2 ; . . . ; y n , respectively. Also A bootstrap estimate of A. The above resampling process can be repeated multiple times, for example, B times. b ; A b ; . . . ; A b Þ can be computed from the bootstrap resamples. Averaging the B The B bootstrap estimates ð A 1 2 B bootstrap estimates, we obtain that b ¼ A
x
B X bB ¼ 1 b ; A A B j¼1 j
ð5Þ
b can be estimated by is the bootstrap estimate of A. And the standard deviation of the CAN estimator A ( )1=2 B 1 X b BÞ ¼ b A b B Þ2 sdð A ðA : ð6Þ B 1 j¼1 j For necessary backgrounds on bootstrap techniques, the interested readers can refer to Efron [18,19,22], Efron and Gong [20], Efron and Tibshirani [21], Gunter [23,24], Mooney and Duval [25] or Young [26]. Next, we describe four types of bootstrap confidence intervals for A as follows. 3.1. Standard bootstrap confidence interval Because the central limit theorem implies that D b AÞ=sdð A b BÞ ! ðA N ð0; 1Þ;
ð7Þ
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D b a where ! denotes convergence in distribution. Based upon the above asymptotic normality of A, 100(1 2a)% SB confidence interval for A is
b B Þ; A b þ za sdð A b B ÞÞ; b za sdð A ðA
ð8Þ
where za is the upper ath quantile of the standard normal distribution. 3.2. Percentile bootstrap confidence interval b ; A b ; . . . ; A b is called the bootstrap distribution of A, b and let A b ð1Þ; A b ð2Þ; . . . ; A b ðBÞ be order statistic of A 1 2 B b ; . . . ; A b . Then utilizing the 100ath and 100(1a)th percentage point of the bootstrap distribution, a b ; A A 1 2 B 100(1 2a)% PB confidence interval for A is obtained as b ð½Bð1 aÞÞÞ; b ð½BaÞ; A ðA
ð9Þ
where [x] denotes the greatest integer less than or equal to x. 3.3. Bias-corrected percentile bootstrap confidence interval b ; A b ; . . . ; A b may be biased. Consequently, the third approach is designed to The bootstrap distribution A 1 2 B P b < AÞ=B, b correct this potential bias of the bootstrap distribution. Set p0 ¼ Bj¼1 Ið A where I(Æ) is the indicator j 1 1 function. Define ^z0 ¼ U ðp0 Þ, where U denotes the inverse function of the standard normal distribution U. Then let a1 ¼ Uð2^z0 za Þ, and a2 ¼ Uð2^z0 þ za Þ. It follows that a 100(1 2 a)% BCPB confidence interval for A is given by b ð½Ba2 ÞÞ: b ð½Ba1 Þ; A ðA
ð10Þ
3.4. Bias-corrected and accelerated confidence interval Except for correcting the potential bias of the bootstrap distribution, we can accelerate convergence of the e ðiÞ and Ye ðiÞ denote the original samples with the ith observations xi and yi bootstrap distribution. Let X e i be the CAN estimate of A calculated by using X e ðiÞ and Ye ðiÞ. Define deleted, also let A e¼1 A n and
n X
ei A
i¼1
,8 " #3=2 9 < X = n eA e i Þ3 eA e i Þ2 ^ a¼ ; ðA ðA 6 : i¼1 ; i¼1 n X
^z0 and ^ a are named bias-correction and acceleration, respectively. Thus a 100(1 2a)% BCa confidence interval for A is constructed by b ð½Ba2 ÞÞ; b ð½Ba1 Þ; A ðA
ð11Þ
að^z0 za ÞÞ and a2 ¼ Uð^z0 þ ð^z0 þ za Þ=½1 ^að^z0 þ za ÞÞ. where a1 ¼ Uð^z0 þ ð^z0 za Þ=½1 ^ 4. Simulation study One principal goal of bootstrap approaches is to establish good confidence intervals. Efron and Tibshirani [21] indicated that ‘‘good’’ means that the bootstrap confidence intervals should have relatively accurate coverage performances and dependably short average interval lengths in all situations. Numerical simulation studies were carried out to evaluate performances of the four bootstrap approaches presented in Section 3. The four confidence intervals are assessed in terms of their coverage accuracy and
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average lengths. In order to reach this goal, we not only set three different failure time distributions for the lifetime X, but also assume two various continuous distributions for the repairing time Y. For concreteness, we let X have a gamma lifetime distribution G(a, b) with scale and shape parameters a and b, respectively. Three different levels (5, 2), (10, 1), and (20, 0.5) are assigned to (a, b). For convenience, the distribution of repairing time Y is assumed to be exponential with mean 8 (i.e., EXP(8)) and uniform on interval (2, 14) (i.e., U(2, 14)), respectively. For each pair of distributions (X, Y), a random sample (x1, y1), (x2, y2), . . . , (xn, yn) of size n(=15, 30, 60) is b is calculated. Next, B = 1000 bootstrap resamples generated from (X, Y). Using Eq. (3), the CAN estimate A fðx1 ; y 1 Þ; ðx2 ; y 2 Þ; . . . ; ðxn ; y n Þg are drawn from the original sample {(x1, y1), (x2, y2), . . . , (xn, yn)}. By virtue of b ; . . . ; A b are calculated from the bootstrap resamples. And utilizing (5) and b ; A (4), B bootstrap estimates A 1 2 B b is computed as sdð A b B Þ. Finally, applying the four bootstrap meth(6), the estimated standard deviation of A ods (Eqs. (8)–(11)) described in Section 3, we obtain the four bootstrap confidence intervals with confidence level 90%. Subsequently the above simulation process is replicated N=1000 times. We not only average the N simub but also compute coverage percentages and average lengths of the four bootstrap confidence lated estimates A, intervals. Matlab 7.0.4 code is easily implemented to accomplish all simulations. All simulation results are b is demonstrated by Table displayed in Tables 2 and 3. The consistency property of the CAN estimator A 2, while performances of the four bootstrap confidence intervals for system availability A can be examined in terms of Table 3. For this redundant repairable system with one operating unit and one spare, the mean of N simulated CAN b is shown in Table 2 for various sample size n and six different combinations of lifetime distribution estimate A and repairing time distribution. Based upon the strong law of large numbers (cf. [17,p. 196]), the true value of b with sufficiently large sample size. system availability A can be estimated by the simulated sample value of A b Thus the true value of A is produced from the average of N simulated A with sample size n = 107. We find that the approximated system availability approaches to the true value of A with six decimal places when n P 107. b converges to the true value of system availability A as the Table 2 implies that the average of N simulated A sample size n becomes large enough, for any specified lifetime and repairing time distributions, as we expect. b is a consistent estimator of system availability A. According to simulation analysis, we show that A Because the frequency of coverage for an accurate confidence interval is binomially distributed with N = 1000 and p = 0.90. Therefore, a 99% confidence range for the coverage probability is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:90 2:576 0:90 0:10=1000 ¼ 0:90 0:0244; that is, from 0.876 to 0.924. Hence, one could be 99% confident that a ‘‘true 90% confidence interval’’ would have a proportion of coverage between 0.876 and 0.924. All simulation results of coverage percentages and average lengths for the four bootstrap confidence intervals are displayed in Table 3. Examining these simulation results, we find that coverage percentages for the four bootstrap confidence intervals increase with the sample size n. The BCa approach always has the largest coverage percentages as well as the highest chances that fall into the 99% confidence range (0.876, 0.924). In addition, the SB approach always has the smallest coverage percentages as well as the lowest chance that fall into the 99% confidence range. When the sample size n is sufficiently large (n P 60), almost all coverage per-
Table 2 b Simulation analysis for consistency of A Distribution of X
G(5, 2) G(10, 1) G(20, 0.5) G(5, 2) G(10, 1) G(20, 0.5)
Distribution of Y
EXP(8) EXP(8) EXP(8) U(2, 14) U(2, 14) U(2, 14)
The true value of A
0.763546 0.736553 0.697354 0.832997 0.783819 0.728217
b The average of 1000 simulated A n = 15
n = 30
n = 60
0.769033 0.729428 0.692224 0.829010 0.771585 0.703688
0.768637 0.740861 0.694576 0.832421 0.781088 0.719426
0.767478 0.733981 0.695719 0.832954 0.781320 0.723276
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Table 3 Simulation results of coverage percentages and average lengths for the four 90% bootstrap confidence intervals Distribution of X
Distribution of Y
Bootstrap approaches
Coverage percentages n = 15
G(5, 2)
EXP(8)
G(10, 1)
EXP(8)
G(20, 0.5)
EXP(8)
G(5, 2)
U(2, 14)
G(10, 1)
U(2, 14)
G(20, 0.5)
U(2, 14)
a *
SB PB BCPB BCa SB PB BCPB BCa SB PB BCPB BCa SB PB BCPB BCa SB PB BCPB BCa SB PB BCPB BCa
0.791 0.819 0.838 0.849 0.803 0.849 0.856 0.875 0.793 0.848 0.859 0.868 0.850 0.861 0.873 0.887* 0.837 0.863 0.872 0.895* 0.827 0.839 0.849 0.861
n = 30 0.832 0.850 0.858 0.868 0.836 0.856 0.871 0.876* 0.871 0.884* 0.888* 0.898* 0.861 0.870 0.880* 0.897* 0.880* 0.884* 0.887* 0.893* 0.859 0.870 0.878* 0.878*
Average lengths n = 60 0.862 0.879* 0.882* 0.892* 0.876* 0.890* 0.884* 0.885* 0.873 0.886* 0.893* 0.898* 0.884* 0.885* 0.890* 0.898* 0.896* 0.894* 0.892* 0.900* 0.885* 0.885* 0.886* 0.891*
n = 15 0.3011 0.3004a 0.3049a 0.3161a 0.3399 0.3393a 0.3421 0.3502 0.3901 0.3888a 0.3891 0.3916 0.2238 0.2229a 0.2257 0.2319 0.2856 0.2838a 0.2846 0.2866 0.3573 0.3550 0.3518 0.3487a
n = 30 a
0.2281 0.2286 0.2299 0.2360 0.2514a 0.2516 0.2527 0.2569 0.2942 0.2941 0.2937a 0.2949 0.1609 0.1607a 0.1614 0.1635 0.2026 0.2023 0.2022a 0.2028 0.2605 0.2598 0.2570 0.2549a
n = 60 0.1703a 0.1708 0.1712 0.1741 0.1880a 0.1883 0.1885 0.1903 0.2146 0.2147 0.2145a 0.2149 0.1153a 0.1154 0.1155 0.1162 0.1458 0.1458 0.1456a 0.1458 0.1838 0.1836 0.1823 0.1816a
Indicates the shortest average length among the four bootstrap confidence intervals. Indicates a proportion that is not significantly different (at a = 0.01) from the expected value of 0.90.
centages for the four bootstrap intervals reach the 99% confidence range. Based on coverage percentages, the BCa approach has the best performances among the four bootstrap approaches. On the other hand, the four bootstrap approaches are scrutinized by average lengths of the confidence intervals. We can see from Table 3 that all average lengths for the four bootstrap approaches decrease with sample size n. Particularly the shortest average length among the four bootstrap confidence intervals is produced from the PB approach when the sample size is small (n 6 15). Whereas the differences among average lengths of the four bootstrap confidence intervals could be negligible when the sample size n is large enough (n P 30). In a word, the BCa approach among the four bootstrap approaches can construct the best confidence interval of availability A for a redunb dant repairable system, by applying the CAN estimator A. 5. Conclusions This paper provided four feasible and efficient interval estimations of availability A for a redundant repairb as well as four different bootstrap approaches SB, PB, BCPB and BCa, are able system. The CAN estimator A applied to construct confidence intervals for system availability A. The coverage percentages and average lengths are adopted to understand, compare, and assess performances of the resulted bootstrap confidence intervals. The simulation results imply that the BCa approach has the best performance on coverage percentages, while the SB approach is the poorest performer. While based on the criterion of average interval length, the PB approach is the most superior actor when the sample size n is small (n 6 15); but all the four bootstrap approaches perform almost equally well when the sample size n is sufficiently large (n P 30). Consequently, the BCa approach is the best choice made by practitioners who want to construct efficient confidence intervals of availability A for such a repairable system with sufficient data (n P 30) of lifetime X and repairing time Y.
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