American Journal of Applied Sciences 4 (11): 84-847, 007 ISSN 1546-99 007 Science Publications Asymptotic Confidence Limits for a Repairable System with Stbys Subject to Switching Failures 1 Jau-Chuan Ke Ssu-Lang Lee 1 Department of Statistics Department of Accounting, National Taichung Institute of Technology, Taichung 404, Taiwan, R.O.C Abstract: This paper studies system performance measures asymptotic confidence limits for the mean time to failure, steady-state availability, steady-state failure frequency of a repairable system which two primary units, two stby units, one repair facility when switching to stbys may fail. Keywords: Availability, Confidence, MTTF, Simulation, Switch failure INTRODUCTION Traditionally, most research about the reliability (availability) of a repairable system with cold or warm stbys assumes that the switchover from a stby to an operational unit is perfect. However, this might be unrealistic. Although a system with cold stbys has the advantage of a zero failure rate, there are also drawbacks, such as a higher probability of switching failure longer warm-up times. In this article, we study the reliability availability characteristics of a system with two primary units when switching failures may occur for cold or warm stbys. In other words, a stby unit with a lower failure rate might have a higher probability of switching failure. We not only investigate the impact of the switching failure to the reliability availability characteristics of the system but also present the behavior of asymptotic confidence limits for the system performance measures. Repairable systems are usually studied with reference to the evaluation of their performance measures in terms of reliability availability. Lewis [7] first introduced the concept of the stby switching failures in the reliability with stby system. Chung [] has ever provided the reliability of k active s cold stbys with multiple repair facilities multiple critical non-critical errors when the switching mechanism is subject to failure. He derived the reliability function in terms of LST of system state probabilities, which is very complicated is generally unsuitable for computational purposes. As for the analysis of two-unit redundant systems, different assumptions have been studied extensively in the past, Corresponding Author: a detailed bibliography is found in Srinivasan Subramanian [15]. However, many of the analysis always consider that the switchover from a stby to an operational unit is perfect (see Goel Shrivastava [4], Shi Li [1], Gururajan Srinivasan [5], Shi Liu [1], Rajamanickam Chrasekar [10], Sridharan Mohanavadivu [14], others). Confidence limits for availability reliability of the two-unit redundant systems were investigated by Abu-Salih et al. [1], Jie [6], Masters et al. [9]. Recently, Yadavalli et al. [16] examined asymptotic confidence limits for the steadystate availability of a two-unit parallel system with the introduction of preparation time for the service station. final section concludes. Jau-Chuan Ke, Department of Statistics, National Taichung Institute of Technology, No. 19, Sec., Sanmin Rd.,Taichung 404, Taiwan, R.O.C., Tel: +886-41960, Fax: +886-1961 84 [] Chrasekhar et al. derived a consistent asymptotically normal estimator an asymptotic confidence interval for the steady-state availability of a two-unit cold stby system in which the failure rate of the unit while online is a constant the repair time distribution is a two-stage Erlangian. This paper extends their statistical inference for system availability to encompass other useful performance measures in more realistic systems. The main objective of this paper is to study asymptotic confidence limits for the mean time to failure (MTTF), steady-state availability, failure frequency of a two-unit repairable system with stbys subject to switching failures. Problem formulation assumptions are given in Section. System reliability availability are developed in Sections 4, estimation confidence limits are developed in Sections 5 6, results are numerically illustrated in Section 7. Section 8 provides an example, the
Problem Formulation Notation: In this paper, we consider a system which consists of two identical primary units operating simultaneously in parallel, two stby units (which may be hot, warm, or cold), a reliable service station. The assumptions of the model are described as follows. Suppose that primary stby failures occur independently of the states of other units follow exponential distributions with parameters λ α ( where 0 α λ), respectively. In particular, a cold stby has α = 0 a hot stby has α = λ. When a primary unit fails, it is immediately replaced by a stby if one is available. It is assumed that the switchover time is instantaneous. However, the switch to a primary unit is imperfect; the switching failure probability q depends on the state of the stby unit decreases as α increases. In particular, a hot stby has q = 0. If a stby unit fails to switch to a primary unit, the next available stby unit attempts to switch. This process continues until switching is successful or all the stby units fail. When a stby unit switches over successfully, its failure characteristics become those of a primary unit. If a primary or a stby unit fails, it is immediately sent to the service station where service is performed on the first come first served (FCFS) convention. It is assumed that the service station can serve only one failed unit at a time that service is independent of the number of unit failures. In addition, the time to repair a failed unit is exponentially distributed with parameter µ. Once a unit is repaired, it instantly resumes stby status. In this research, system reliability availability characteristics are studied under the assumption that the system fails if the number of primary units is less than two; that is, three units fail. Therefore, the system fails if only if i <, where i denotes the number of primary units in the system. Such model has potential applications in both industrial military systems. For example, in an air plane with four engines, it may be possible to fly the plane if only two engines functioning. Am. J. Applied Sci., 4 (11): 84-847, 007 However, if less than two engines function, the plane will fail to fly (see Li Chen [8] ). Before further developing the model, we first present the notation used in later sections. λ failure rate of a primary unit α failure rate of a stby unit µ repair rate of a failed unit q switching failure probability of a stby unit to a primary unit P(i, j; t) probability that there are i primary units j stby units working in the system at time t, where i =, 1 j = 0, 1, P(i, j;0) initial probability of P(i, j; t) when t = 0 s Laplace transform variable ~ P ( i, j; s) Laplace transform of P(i, j; t) T time to failure of the system R T (t) reliability of the system at time MTTF mean time to system failure RELIABILITY ANALYSIS OF THE SYSTEM At time t = 0, the system commences operation with no failed units ( includes two primary units two stby units) an idle service station. That is, the initial conditions for this system are given by P (,;0) = 1, P(,1;0) = 0, P(,0;0) = 0, P(1,0;0) = 0. (1) The reliability availability characteristics with switching failure probabilities under exponential failure times exponential service times can be developed through the birth death process. Let P(i, j; t) denote the probability that there are i primary units j stby units in the system at time t, where i = 1,, j = 0, 1,, t 0. The differential equations governing the state probabilities of this system are: dp(, ; t) = [λ + α ] P(, ; t) + µ P(,1; t), dt () dp(,1; t) = [λ(1 + α ] P(, ; t) [λ + α + µ ] P(,1; t) + µ P(, 0; t), dt () dp(, 0; t) = λ (1 q P(, ; t) + [λ(1 + α] P(,1; t) [λ + µ ] P(, 0; t), dt (4) dp( 1,0; t) = λ q P(,; t) + λq P(,1; t) + λ P(, 0; t). dt (5) 85
Am. J. Applied Sci., 4 (11): 84-847, 007 Taking Laplace Transforms of both sides in ()-(5) using initial condition (1), these equations can be reduced to ~ sp ~ (,; s) 1 = [λ + α ] P ~ (, ;s) + µ P(,1;s), (6) sp ~ (,1; s) = [λ(1 + α ] P ~ (, ;s) [λ + α + µ ] P ~ (,1; s) + µ P ~ (, 0;s), (7) sp ~ (,0; s) = λ (1 q P ~ (, ;s) + [λ(1 + α] P ~ (,1; s) [λ + µ ] P ~ (, 0;s), (8) sp ~ (1,0; s) = λ q P ~ (, ;s) + λq P ~ (,1; s) + λ P ~ (, 0; s). (9) This system of linear equations can be solved to yield ~ 1 P (, ; s) = [ s + (4λ + α + µ ) s + 4λ + λµ + λα + λµ q + µ ], D (10) ~ 1 P (,1; s) = [( λ λq s + 4λ (1 + λµ (1 q ) + 4λα + αµ ], D (11) 1 P ~ (, 0; s) = [(λ q λq ) s + ( λ (λ 4λq( λ λq( µ + αq + µ q)], D (1) ~ λ P (1, 0; s) = [ q s + q(4λ + α + αq + µ q) s + ( λ (λ + µ q(4λ + α + µ q)], sd (1) where D = s + (6λ + α + µ ) s + [1λ ( λ + α ( α + µ ) + 4λµ (1 + q) + µ ] s + 4λ (λ ( λ + λµ q (4λ + α + µ q). After inversion, we obtain P (1,0; t), the probability that the system fails at time t. That is, P (1,0; t) is the probability that the system fails at or before time t. Thus the reliability of the system is R( t) = 1 P(1,0; t). Let T be the time to failure of the system; the Laplace transform of the failure density is then given by dr(t) dp(1,0; t) T (t) = =, dt dt ~ T ( s) = s P ~ (1,0; s) P(1,0;0), From the listed above equations, we have ~ 1 T ( s) = λ[ q s + q(4λ + α + αq + µ q) s + ( λ (λ + µ q(4λ + α + µ q)] D. (14) Instead of inverting this expression (14) to get the distribution of T, we will be content with obtaining the mean ~ time to failure (MTTF) using the derivative of T ( s ) with respect to s while s=0: MTTF λq[4λ + ( + q) α] 4λµ (1 + q q ) 1λ( λ ( α + µ ) α =. (15) λ[( λ (λ + µ q(4λ + α + µ q)] 4. AVAILABILITY ANALYSIS OF THE SYSTEM This section we will investigate the steady-state system availability frequency. A set of differential dp(, dt 0; t) 86 equations for the availability case can be established in a manner similar to that used for the reliability analysis in Section. The first two equations are the same as () (). However, the equations (4) (5) are rewritten in the following forms: = λ (1 q P(, ; t) + [λ(1 + α] P(,1; t) [λ + µ ] P(, 0; t) + µ P(1, 0; t), (16)
Am. J. Applied Sci., 4 (11): 84-847, 007 Table 1: (a). Asymptotic confidence interval limit of MTTF for λ = 0.1, α = 0.05, µ = 1. 0 various values of q n=0 n=50 n=100 q MTTF MT ˆ Lower TF Upper limit limit MT ˆ Lower TF Upper limit limit MT ˆ Lower TF Upper limit limit 0.00 11. 1.786 1.96 4.66 10.711 1.499 174.11 114.089 55.5577 17.610 0.01 105.657 116.5117 15.1114 17.911 96.688 1.4009 161.9667 106.741 5.6691 159.8151 0.50 15.065 17.50 10.54 4.6881 14.54 9.9184 19.199 14.7480 11.417 18.08 Table 1: (b). Asymptotic confidence interval limit of A ( ) for λ = 0.1, α = 0.05, µ = 1. 0 various values of q q A ( ) ˆ Lower A ( ) limit 0.00 0.01 0.50 n=0 n=50 n=100 Upper limit A ˆ( Lower ) limit Upper limit A ˆ( Lower ) limit Upper limit 0.98814 0.9880 0.977 1.00078 0.988087 0.9764 0.99974 0.988151 0.979668 0.9966 0.98764 0.98747 0.97019 1.0095 0.98747 0.9759 0.999455 0.9874 0.978498 0.996186 0.9101 0.9504 0.87680 0.9766 0.9405 0.886569 0.96041 0.91846 0.889607 0.94718 Table 1: (c). Asymptotic confidence interval limit of F ( ) for λ = 0.1, α = 0.05, µ = 1. 0 various values of q n=0 n=50 n=100 q F ( ) F ˆ ( ) Lower limit Upper limit F ˆ ( ) Lower limit Upper limit ˆ ( ) 0.00 0.01 0.50 F Lower limit Upper limit 0.010780 0.009681 0.00099 0.01906 0.01168 0.00894 0.0047 0.010686 0.00476 0.016645 0.011487 0.01080 0.000586 0.019975 0.01408 0.005 0.01490 0.011415 0.0056 0.017594 0.071808 0.061509 0.05761 0.08756 0.0749 0.050154 0.09805 0.0751 0.056550 0.09051 Table : (a). The biases the mean square errors of MTTF for λ = 0.1, α = 0.05, µ = 1. 0 various values of q n=0 n=50 n=100 n=00 q MTTF Bias MSE Bias MSE Bias MSE Bias MSE 0.00 11. 0.01 105.657 0.50 15.065 11.449 9.7996-9.60 1418.598 1.7506 894.758-0.67 411.180 11.1460 800.809-8.6819 1184.817 1.764 75.1419-0.041 41.001.4550 19.4019-0.5410 5.8149-0.17.9964 0.088 1.507 Table : (b). The biases the mean square errors of A ( ) for λ = 0.1, α = 0.05, µ = 1. 0 various values of q n=0 n=50 n=100 n=00 q A ( ) Bias MSE Bias MSE Bias MSE Bias MSE 0.00 0.01 0.50 0.98814 0.000060 0.000058-0.000056 0.00005 0.000009 0.000019-0.00000 0.000009 0.98764 0.000108 0.00006-0.000017 0.00008-0.0000 0.00000 0.00000 0.000010 0.9101 0.00401 0.0006 0.009 0.0008-0.00549 0.000 0.00090 0.000100 87
Am. J. Applied Sci., 4 (11): 84-847, 007 Table : (c) The biases the mean square errors of F ( ) for λ = 0.1, α = 0.05, µ = 1. 0 various values of q n=0 n=50 n=100 n=00 q F ( ) Bias MSE Bias MSE Bias MSE Bias MSE 0.00 0.010780-0.001099 0.00004 0.00090 0.00001-0.000094 0.000009 0.00004 0.000005 0.01 0.011487-0.00107 0.00006 0.00091 0.0000-0.00007 0.000010 0.00001 0.000005 0.50 0.071808-0.01099 0.00079 0.004 0.000157 0.00174 0.000078-0.0001 0.00005 Table : The coverage probability for λ = 0.1, α = 0.05, µ = 1. 0 various values of q MTTF ˆ A ˆ( ) F ˆ ( ) q n=0 n=50 n=100 n=0 n=50 n=100 n=0 n=50 n=100 0.00 0.914 0.97 0.946 0.94 0.896 0.97 0.97 0.949 0.915 0.01 0.918 0.99 0.945 0.94 0.897 0.98 0.97 0.950 0.916 0.50 0.96 0.948 0.957 0.95 0.9 0.947 0.95 0.960 0.948 dp( 1, 0; t) = λ q P(, ; t) + λq P(,1; t) + λ P(, 0; t) [ λ + µ ] P(1, 0; t) + µ P( 0, 0; t). (17) dt Moreover, we still need an equation for state (0, 0), which governs the system given by: dp(0, 0; t) = λ P(1, 0; t) µ P(0, 0; t). (18) dt In steady-state, let P( i, j) = lim P( i, j; t) hence the balance equations can be reduced: t Solving the above equations for P ( i, j) we get [ λ + α ] P(, ) = µ P(,1), (19) [ λ + α + µ ] P(,1) = [λ(1 + α ] P(, ) + µ P(, 0), (0) [ λ + µ ] P(, 0) = λ(1 q P(, ) + [λ(1 + α] P(,1) + µ P(1, 0), (1) [ λ + µ ] P(1, 0) = λq P(, ) + λq P(,1) + λ P(, 0) + µ P(0, 0), () µ P( 0, 0) = λ P(1, 0). () µ P(, ) = P(0, 0), (4) λ [( λ + α )(λ + α ) + µ q(4λ + α + µ q)] 88 4 µ ( λ P(,1) = P(0, 0), (5) λ [( λ + α )(λ + µ q(4λ + α + µ q)] µ [(λ ( λ + α ) + λµ q] P(, 0) = P(0, 0), (6) λ [( λ (λ + α ) + µ q(4λ + α + µ q)] µ P( 1, 0) = P(0, 0), (7) λ where P (0, 0) = λ [( λ + α )(λ + α ) + µ q(4λ + µ q + α )] 4 { µ + ( α + λ + λq + λq ) µ + [(λ + α )( λ + λq ( α + λ) + α q ] µ + 4λ[(λ + α )( λ + λq ( α + λ)] µ + 4λ [(λ + α )( λ + α )]}. (8) 1
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Am. J. Applied Sci., 4 (11): 84-847, 007 Since both states (0,0) (1,0) are system down states, the steady-state availability of the system is given by A ( ) = P(,) + P(,1) + P(,0) 4 µ + µ [( λ (λ + α + µ ) + λµ q] = P(0, 0). (9) λ [( λ (λ + µ q(4λ + α + µ q)] From (9), the steady-state unavailability is U ( ) = 1 A( ), the downtime in minutes per year is U ( ) 8760 60. Using the results by Shi Liu [1], the failure frequency of the system in steady-state is expressed as F( ) = λ q P(,) + λqp(,1) + λp(,0) 84
Am. J. Applied Sci., 4 (11): 84-847, 007 λµ [( µ q) + (λ ( λ + α + µ q)] = P(0, 0). (0) λ [( λ (λ + µ q(4λ + α + µ q)] Estimates For MTTF, Availability, And Failure Frequency: Let X, primary units with p.d.f. ( ) = λ λ x f x e, 0 < x <, λ > 0. Y, 1, Y, Y n be a sample of failure times for stby units with p.d.f. ( ) = α α y g y e, 0 < y <, α > 0, Z, 1, Z, Z n be a sample of repair times for failed units with p.d.f. ( ) = µ µ z h z e, 0 < z <, µ > 0. 1, X, X n be a sample of failure times for Let X Y represent the sample means of the times to failure for primary units stby units, respectively, Z represent the sample mean of the times to repair for failed units. Then E [ X ] = 1 λ, E [ Y ] = 1 α, E [ Z ] = 1 µ. It can be easily shown that X, Y, Z are the maximum likelihood estimates of 1 λ, 1 / α, 1 µ, respectively. Let MT ˆ TF be the estimator of MTTF. Inserting (15), we finally obtain ˆ X[ qyz (4Y + ( + q) X ) XY Z (1 + q q ) 6YZ ( X + Y ) X (( Y + Z ) + Z ) / ] MTTF =. (1) Z [( X + Y )( X + Y ) + qx Y (YZ + XZ + qxy / )] Furthermore, let ˆP (0,0) be an estimator of P (0,0) (the probability of all units failed including stby in the system). From (8), it yields Pˆ(0,0) = 4 X Y + [ X + Y (1 + q + q )] X Z[( X + Y )( X + Y ) Z YZ + Z [( X + Y )( X + Y )( X + qxy (4YZ + qxy + XZ )] + XZ + Z ) + qxy ( X + Y )( X + Z ) + q X Using the results (4)-(7) in the previous sections, we easily obtain the estimators of P (1,0 ), P (,0), P (,1), P (,) as follows ˆ X P (1,0) = Pˆ(0,0), Z 4. () ( X + Y )( X + Y ) + qxy Z Pˆ(,0) = Pˆ(0,0), Z ( X + Y )( X + Y ) + qxyz (XZ + 4YZ + qxy ) XY ( X + Y ) Pˆ(,1) = Pˆ(0,0), Z [Z ( X + Y )( X + Y ) + qxy (XZ + 4YZ + qxy )] ( XY ) Pˆ(,) = Pˆ(0,0). Z [Z ( X + Y )( X + Y ) + qxy (XZ + 4YZ + qxy )] From (9)-(0), we can obtain an estimator of A ( ) an estimator of F ( ) ( XY ) + Z[( X + Y )(YZ + XZ + XY ) + qxy ] Aˆ( ) = Pˆ(0,0), () Z [Z ( X + Y )( X + Y ) + qxy (XZ + 4YZ + qxy )] ˆ q( XY ) + Z ( X + Y )( YZ + XZ + qxy ) F( ) = Pˆ(0,0). (4) XZ [Z ( X + Y )( X + Y ) + qxy (XZ + 4YZ + qxy )] 844
Am. J. Applied Sci., 4 (11): 84-847, 007 Confidence Limits For MTTF, Availability, And Failure Frequency: From the discussion in the previous sections, we know that MT ˆ TF, A ˆ( ), F ˆ ( ) are real-valued functions in X, Y, Z which are also differentiable. Using the application of the multivariate central limit theorem (see Rao [11] ), it follows that n [( X, Y, Z ) ( θ1, θ, θ )] converges to N (0, ) in distribution as n, 1 1 1 where ( θ 1, θ, θ) = (,, ) the dispersion λ α µ matrix = [ σ i, j ] is given by = (,, diag θ1 θ θ ). Using the result by Rao [11] again, we have ˆ n[ MTTF MTTF ] converges to N (0, σ1 ( θ )) in distribution as n, with MTTF σ σ ii, θ i where θ = ). 1 ( θ ) = i= 1 ( θ1, θ, θ Let ( ˆ σ 1 θ ) be the estimator of σ 1 ( θ ) which is obtained by replacing θ with a consistent estimator ˆ θ = ( X, Y, Z ). Since σ ( ) is a continuous function of 1 θ θ, we know that σ ( ˆ) is a consistent estimator of 1 θ 1 θ σ ( ) (see Wackerly et al. [17] ). Thus σ 1 ( ˆ) θ σ1 ( θ ) as n. Using Slutsky's theorem, we have n[ MTTF ˆ MTTF ] converges to N (0,1) in σ 1 ( ˆ) θ distribution as n, which leads to n[ MTTF ˆ MTTF ] Pr[ z γ < < zγ ] = 1 σ ( ˆ) θ γ, 1 where z γ is determined from stard normal tables or statistical software packages. Hence, the asymptotic 100(1-γ )% confidence limits for MTTF are given by ˆ σ1 ( ˆ) θ MTTF ± zγ. (5) n Continuing in the same way for A ( ) F ( ) we obtain where σ ˆ σ ( ˆ) ( ) θ A ± zγ, (6) n ˆ σ ( ˆ) ( ) θ F ± zγ, (7) n σ A( ) ( θ ) = [ ] σ, ii i= 1 θi F( ) ( θ ) = [ ] σ. ii i= 1 θi Numerical Illustration: In this section we provide numerical results of the mean time to system failure, MTTF, steady-state availability, A ( ), failure frequency, F ( ), for different values of system parameters. The results of MTTF, A( ), F( ) are shown in Fig.1- for the following three cases, respectively. Case 1: We choose µ = 1. 0 α = 0. 01 vary the values of q λ. Case : We select λ = 0. 01 µ = 1. 0 vary the values of α q. Case : We choose q = 0. 1 α = 0. 05 vary the values of λ µ. Cases 1- are analyzed graphically to study the effects of various system parameters on MTTF, A ( ), MTTF. Fig.1 shows that (i) MTTF increases as either q or λ decreases, (ii) A ( ) decreases as either q or λ increases, (iii) F ( ) increases as either q or λ increases. From Fig., we observe that (i) MTTF increases as either α or q decreases, (ii) A ( ) decreases as either α or q increases, (iii) F ( ) increases as either α or q increases. Fig. depicts that (i) MTTF decreases as λ increases or µ decreases, (ii) A ( ) increases as λ decreases or µ increases, (iii) F ( ) increases as λ increases or µ decreases. Next, we perform asymptotic confidence limits for MTTF, A ( ), F ( ) for various values of system parameters. The following cases are analyzed to study the effects of various parameters on the estimating behavior of the system performance measures: results are respectively shown in Fig.4 Table 1. 845
Case 4: We set λ = 0.1, α = 0.01, µ = 1.0, consider q = 0.0, 0.01. Case 5: We set λ = 0.1, α = 0.05 µ = 1.0, examine q = 0.0, 0.01, 0.5. Asymptotic 95% upper lower confidence limits of MTTF, A ( ), F ( ) are shown in Fig.4. One observes from Fig.4 that (i) at a 0.05 significance level, MTTF, A ( ), F ( ) lie between the upper lower confidence limits for different values of q, (ii) for a given sample size, the confidence interval bs are almost the same when q varies from 0.0 to 0.01. The true values, estimates, upper lower 95% confidence limits of MTTF, A ( ), F ( ) are shown in Tables 1(a)-(c). Table 1 shows the effects on estimating behavior of MTTF, A ( ), F ( ) for different values of q. It is evident from the results that the estimators approach the true values of MTTF, A ( ), F ( ) as the sample size gets larger. It should also be noted that the confidence intervals narrow as the sample size grows. ROBUSTNESS OF CONSISTENT ASYMPTOTIC NORMAL (CAN) ESTIMATOR FOR MT ˆ TF, F ˆ ( ), AND A ˆ ( ) In order to see how good the normal approximation based on proposed above is, a simulation study of sensitivity is carried out to check on how accurate of this approximation is. Let Ω = ( X, Y, Z ), = 1,, be a sequence of independent identically distributed -dimensional rom vectors. From each single simulation (replication) Ω with sample size n, we can obtain MTTF ˆ j, F ˆ j ( ), A ˆ j ( ). The histograms of these MTTF ˆ j, F ˆ j ( ), A ˆ j ( ) are shown in Fig.5-7, respectively, based on N=1,000 replications. As expected the spread of the distributions will decrease with increasing sample size n by the law of large numbers (see De Groott [18] ). The simulation results also indicate that the asymptotical normality of MT ˆ TF, F ˆ ( ), A ˆ ( ) for n 100 are obviously. The biases mean square errors of MT ˆ TF, F ˆ ( ), A ˆ ( ) for different sample size of n are displayed in Tables. One observes from Tables that Am. J. Applied Sci., 4 (11): 84-847, 007 846 the biases mean square errors are smaller as n gets larger. In particular, the bias magnitude approaches to zero when n is large enough. Finally, Table shows the coverage probability respectively for MT ˆ TF, F ˆ ( ), A ˆ( ) at different sample size of n; it shows that the coverage probabilities are close to the nominated value of 0.95 when n is sufficient large. Since the number of confidence intervals, within which the true value of interested is contained, follows a binomial distribution with p=0.95 N=1,000, the 99% confidence interval for the coverage probability itself can also be constructed as 0.95 ±.575 0.95(1 0.95) /1000 = 0.95 ± 0.0177 (i.e. from 0.9 to 0.9678) In particular, when n 100 all these CAN estimators perform well on interval estimation based on simulation results in Table, which give reasonably good probability coverage for MT ˆ TF, F ˆ ( ), A ˆ( ). From the analysis listed above, we can say that the consistent estimators σ ( ˆ i θ ) ( i = 1,, ) are robust, confidence intervals based on the previous section with n 50 have moderately good performance. In contrast, the confidence intervals perform well adequately when n 100 AN EXAMPLE Consider a duplex system consists of two processors connected in parallel. Besides the two primary units, there are two stby units so that when a unit breaks down, a stby unit is immediately substituted thus the reliability of the system is improved. Units in operation or stby state are subject to breakdowns which occur by a Poisson process. When a unit is broken, it is repaired by an operator. System reliability characteristics are defined as previous section. During a sufficiently large amount of time t 0 (in order to obtain enough information), the managers collect three sets of thirty observations concerning failures: time to failure of primary units ( X i), to failure of stby units ( Y i ), to repair of the failed units ( Z i ). Simulated sample means are computed as follows: X =.9749, Y = 18.7010, Z = 0.8646. If 1 = 0. q, it follows from (1)-(4) that we obtain the estimated system performance measures as
Am. J. Applied Sci., 4 (11): 84-847, 007 M TTF ˆ = 9.4, Aˆ( ) = 0.87, Fˆ ( ) = 0.1458. Using (5)-(7), approximate 95% confidence intervals for MTTF, A( ), F( ) are respectively given by (.417, 15.0450), (0.6864, 0.9881), (0.057, 0.4). CONCLUSIONS In this paper, we study a two-unit system with stbys switching failures. We derive the explicit expressions for the system performance measures such as MTTF, steady-state availability, failure frequency. Some numerical illustrations are performed. The results indicate that the performances of this system are different from those of a system without switching failures. Confidence interval formulas for the MTTF, steady-state availability, failure frequency are developed for this redundant repairable system with stbys switching failures. We also provide the numerical simulations to examine the statistical behavior of varying the switching failure probability q, which gain some further insight on the system performance measures. REFERENCES 1 M. Abu-Salih, N. Anakerh M. Salahuddin Ahmed, 1999, Confidence limits for steady state availability, Pakistan Journal of Statistics 6()A: 189-196. P. Chrasekhar, R. Natarajan, V. S. S. Yadavalli, 004. A study on a two unit stby system with Erlangian repair time. Asia-Pacific Journal of Operation Research 1():71-77. W. K. Chung, 1995, Reliability of imperfect switching of cold stby systems with multiple non-critical critical errors, Microelectronics Reliability 5: 1479-148. 4 L. R. Goel P. Shrivastava, 1991, Profit analysis of a two-unit redundant system with provision for test correlated failures repairs, Microelectronics Reliability 1: 87-8. 5 M. Gururajan B. Srinivasan, 1995, A complex two-unit system with rom breakdown of repair facility, Microelectronics Reliability 5(): 99-0. 6 M. Jie, 1991, Interval estimation of availability of a series system, IEEE Transactions in Reliability R- 40(5): 541-546. 7 E.E. Lewis, 1996, Introduction to Reliability Engineering, nd Edition, John Wiley & Sons, New York. 8 X. Li J. Chen, 004, Aging properties of the residual life length of k-out-of-n systems with independent but non-identical components, Applied Stochastic Models in Business Industry 0:14-15. 9 B. N. Masters, T. O. Lewis W. J. Kolarik, 199, A confidence interval availability for systems with Weibull operating time lognormal repair time, Microelectronics Reliability : 89-99. 10 S. P Rajamanickam B. Chrasekar, 1997, Reliability measures for two-unit systems with a dependent structure for failure repair times, Microelectronics Reliability 7(5): 89-8. 11 C. R. Rao, 197, Linear Statistical Inference Its Applications, John Wiley & Sons, New York. 1 D.H. Shi W. Li, 199, Availability analysis of a two unit series system with shut-off rule firstfail, first-repaired, Acta Mathematicae Applicatae Sinica 1:88-91. 1 D. H. Shi L. Liu, 1996, Availability analysis of a two-unit series system with a priority shut-off rule, Naval Research Logistics 4:1009-104. 14 V. Sridharan P. Mohanavadivu, 1998, Some statistical characteristics of a repairable, stby, human \& machine system, IEEE Transactions on Reliability 47(4):41-45. 15 S. K. Srinivasan R. Subramanian, 1980, Probabilistic Analysis of Redundant Systems, Lecture Notes in Economics Mathematical Systems, 175. Springer Verlag, Berlin. 16 V. S. S. Yadavalli, M. Botha A.Bekker, 00, 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: 49-56. 17 D. D. Wackerly, 00, W. Mendenhall III R.L. Scheaffer, Mathematical Statistics with Applications, Duxbury, Thomson Learning. 18 M. H. de Groot, 1970, Optimal Statistical Decisions. New York: Wiley. 847