ESTIMATION OF SYSTEM RELIABILITY IN A TWO COMPONENT STRESS-STRENGTH MODELS DAVID D. HANAGAL Department of Statistics, University of Poona, Pune-411007, India. Abstract In this paper, we estimate the reliability of a parallel system with two components. We assume strengths of these components follow bivariate exponential(bve) distribution. These two components are subected to two independent stresses which are independent of the strengths of two components. If (X 1, X 2 ) are strengths of two components subected to two independent stresses Y 1 and Y 2, then the reliability of a parallel system or system reliability (R) is given by P [Y 1 < X 1, Y 2 < X 2 ]. We estimate R when (X 1, X 2 ) have different BVE models proposed by Marshall-Olkin(1967), Block-Basu(1974), Freund(1961) and Proschan-Sullo(1974). We assume the distribution of Y 1 and Y 2 as exponential or gamma. We obtain the asymptotic normal(an) distributions of these estimates. 1 Introduction Estimation of system reliability has been discussed by Bhattacharyya (1977), Bhattacharyya and Johnson(1974, 1975, 1977), Johnson(1988). Most of the authors considered the distribution of strengths are independent and indentically distributed random variables. In this paper, we estimate system reliability (R) = P [Y 1 < X 1, Y 2 < X 2 ] when (X 1, X 2 ) are strengths of two dependent components and (Y 1, Y 2 ) are two independent stresses. An example of a problem where system reliability (R) would be of interest is as follows. Two hands of a person are always subected to two independent and different stresses and R is the probability that a person can able to work 1
successfully with two hands. Ebrahimi(1982) obtained the estimate of system reliability (R) = P [Y 1 < X 1, Y 2 < X 2 ] when the strengths (X 1, X 2 ) have BVE of Marshall- Olkin(1967) and the stresses (Y 1, Y 2 ) have independent exponentials. We estimate system reliability P [Y 1 < X 1, Y 2 < X 2 ] when (X 1, X 2 ) follow BVE models proposed by Marshall- Olkin(1967), Block-Basu(1974), Freund(1961) and Proschan- Sullo(1974). We assume the distribution of Y 1 and Y 2 as exponential distribution in Section 2 and gamma distribution in Section 3. Marshall-Olkin(1967) proposed BVE with survival function given by F M (x 1, x 2 ) = P [X 1 > x 1, X 2 > x 2 ] = exp[ λ 1 x 1 λ 2 x 2 λ 3 Max(x 1, x 2 )], λ 1, λ 2, λ 3 > 0. Block-Basu(1974) proposed absolutely continuous BVE with survival function given by F B (x 1, x 2 ) = λ/(λ 1 + λ 2 )F M (x 1, x 2 ) λ 3 /(λ 1 + λ 2 )exp[ λmax(x 1, x 2 )] where F M (x 1, x 2 ) is the survival function of BVE of Marshall-Olkin(1967) and λ = λ 1 + λ 2 + λ 3. The above model is absolutely continuous part of BVE of Marshall- Olkin(1967). Freund(1961) proposed absolutely continuous BVE with oint p.d.f. given by θ 1 θ 2exp[ θ 2x 2 (θ 1 + θ 2 θ 2)x 1 ], x 1 < x 2 f(x 1, x 2 ) = θ 2 θ 1exp[ θ 1x 1 (θ 1 + θ 2 θ 1)x 2 ], x 2 < x 1. Proschan-Sullo(1974) proposed BVE which is the combination of both Freund(1961) and Marshall-Olkin(1967). Its p.d.f. is given by f(x 1, x 2 ) = θ 1 η 2exp[ η 2x 2 (θ η 2)x 1 ], x 1 < x 2 θ 2 η 1exp[ η 1x 1 (θ η 1)x 2 ], x 2 < x 1 θ 3 exp( θx), x 1 = x 2 = x 2
where θ = θ 1 + θ 2 + θ 3, η 1 = θ 1 + θ 3 and θ 2 = θ 2 + θ 3. When θ 3 = 0, the BVE Proschan-Sullo(1974) reduces to BVE of Freund(1961). Let (X 1i, X 2i ) and (Y 1i, Y 2i ), i=1,...,n be i.i.d. random sample of size n and n 1 be the number of observations with (Y 1i < X 1i, Y 2i < X 2i ) in the sample of size n. Then the distribution of n 1 is binomial(n,r). The natural estimate of R is n 1 /n which is AN[R, R(1 R)/n]. Here R is the function of the parameters of the distributions of (X 1, X 2, Y 1, Y 2 ). In Section 2, we assume Y i follow exponential with failure rate µ i, i=1,2 and in Section 3, we assume Y i follow Gamma with parameters (α i, µ i ), i=1,2 when (X 1, X 2 ) have different BVE models. We estimate R and obtain its asymptotic normal(an) distribution. 2 Independent Exponential Stresses In this Section, we assume the stresses (Y 1, Y 2 ) has distributions given by G i (y i ) = P [Y i < y i ] = 1 exp[ µ i y i ], i = 1, 2. The estimate of (µ 1, µ 2 ) based on MLEs are ˆµ i = n/ n =1 y i, i=1,2 and variances are V ar(ˆµ i ) = µ 2 i /n, i=1,2. The system reliability (R) is P [Y 1 < X 1, Y 2 < X 2 ] = 0 0 F (y 1, y 2 )dg 1 (y 1 )dg 2 (y 2 ) We first consider the estimate of R when (X 1, X 2 ) follow BVE of Marshall-Olkin(1967). 3
Hence µ 1 µ 2 /(λ + µ 1 + µ 2 )[1/(λ 1 + λ 3 + µ 1 ) + 1/(λ 2 + λ 3 + µ 2 )] where λ = λ 1 + λ 2 + λ 3. The estimate of R based on MLEs of (λ 1, λ 2, λ 3, µ 1, µ 2 ) is ˆ ˆµ 1ˆµ 2 /(ˆλ + ˆµ 1 + ˆµ 2 )[1/(ˆλ 1 + ˆλ 3 + ˆµ 1 ) + 1/(ˆλ 2 + ˆλ 3 + ˆµ 2 )], ˆλ = ˆλ1 + ˆλ 2 + ˆλ 3 The MLEs of (λ 1, λ 2, λ 3 ) can be obtained either by Newton-Raphson procedure or Fisher s method of scoring. [See Hanagal and Kale(1991b,1992)]. The elements of Fisherinformation matrix I(λ 1, λ 2, λ 3, µ 1, µ 2 ) are I 12 = 0, I 13 = λ 2 λ(λ 1 + λ 3 ) 2, I 23 = λ 1 λ(λ 2 + λ 3 ) 2, I 11 = 1/(λλ 1 ) + I 13, I 22 = 1/(λλ 2 ) + I 23, I 33 = 1/(λλ 3 ) + I 13 + I 23, I 44 = 1/µ 2 1, I 55 = 1/µ 2 2, I 45 = 0, I i = 0, i = 1, 2, 3; = 4, 5. The distribution of ˆR is AN[R, G 1 Λ 1 G 1 ] where G 1 = ( R/ λ 1, R/ λ 2, R/ λ 3, R/ µ 1, R/ µ 2 ) and Λ 1 = I 1 (λ 1, λ 2, λ 3, µ 1, µ 2 )/n and I 1 is the inverse of the Fisher information matrix. We next consider the estimate of R when(x 1, X 2 ) follow BVE of Block-Basu(1974). The system reliability (R) in this model is λµ 1 µ 2 (λ 1 + λ 2 )(λ + µ 1 + µ 2 ) i=1 1 λ 3 µ 1 µ 2 λ i + λ 3 + µ i (λ 1 + λ 2 )(λ + µ 1 + µ 2 ) i=1 1 λ + µ i The estimate of R based on MLEs of (λ 1, λ 2, λ 3, µ 1, µ 2 ) is ˆ ˆλˆµ 1ˆµ 2 (ˆλ 1 + ˆλ 2 )(ˆλ + ˆµ 1 + ˆµ 2 ) 1 ˆλ 3ˆµ 1ˆµ 2 1 i=1 ˆλ i + ˆλ 3 + ˆµ i (ˆλ 1 + ˆλ 2 )(ˆλ + ˆµ 1 + ˆµ 2 ) i=1 ˆλ + ˆµ i 4
The MLEs of (λ 1, λ 2, λ 3 ) can be obtained either by Newton-Raphson procedure or Fisher s method of scoring. [See Hanagal and Kale(1991a)]. The elements of Fisherinformation matrix I(λ 1, λ 2, λ 3, µ 1, µ 2 ) are I 12 = 1 λ 2 1 (λ 1 + λ 2 ) 2, I 13 = 1 λ 2 + λ 2 λ(λ 1 + λ 3 ) 2, I 23 = 1 λ 2 + λ 1 λ(λ 2 + λ 3 ) 2, I 11 = 1/(λλ 1 ) 1 λ 2 + I 12 + I 13, I 22 = 1/(λλ 2 ) 1 λ 2 + I 12 + I 23, I 33 = I 13 + I 23 1 λ 2, I 44 = 1/µ 2 1, I 55 = 1/µ 2 2, I 45 = 0, I i = 0, i = 1, 2, 3; = 4, 5. The distribution of ˆR is AN[R, G 2 Λ 2 G 2 ] where G 2 = ( R/ λ 1, R/ λ 2, R/ λ 3, R/ µ 1, R/ µ 2 ) and Λ 2 = I 1 (λ 1, λ 2, λ 3, µ 1, µ 2 )/n and I 1 is the inverse of the Fisher information matrix. We next consider the estimate of R when (X 1, X 2 ) follow BVE of Proschan-Sullo- (1974). The survival function of (X 1, X 2 ) is [(θ 2 + θ 3 η 2)exp( θx 2 ) + θ 1 exp{ (θ η 2)x 1 η 2x 2 }]/(θ η 2) x 1 x 2 F (x 1, x 2 ) = [(θ 1 + θ 3 η 1)exp( θx 1 ) + θ 2 exp{ (θ η 1)x 2 η 2x 1 }]/(θ η 1) x 2 x 1 provided η 1 θ η 2. When η 1 = θ = η 2, the survival function is [θ 1 (x 2 x 1 ) + 1]exp( θx 2 ) x 1 x 2 F (x 1, x 2 ) = [θ 2 (x 1 x 2 ) + 1]exp( θx 1 ) x 2 x 1. Now the system reliability R is given by 2i =1 µ 1 µ 2 (θ η )(θ+µ 1+µ 2 ) [ θ +θ 3 η θ+µ + θ i η +µ ] η 1 θ η 2 2i =1 µ 1 µ 2 (θ+µ )(θ+µ 1 +µ 2 ) [ θ i θ+µ + 1] η 1 = θ = η 2. The estimate of R based on MLEs of (θ 1, θ 2, θ 3, η 1, η 2, µ 1, µ 2 ) is ˆ 2i =1 ˆµ 1 ˆµ 2 (ˆθ ˆη )(ˆθ+ˆµ 1 +ˆµ 2 ) [ ˆθ +ˆθ 3 ˆη ˆθ+ˆµ + ˆθ i ˆη +ˆµ ] ˆη 1 ˆθ ˆη 2 2i =1 ˆµ 1 ˆµ 2 (ˆθ+ˆµ )(ˆθ+ˆµ 1 +ˆµ 2 ) [ ˆθi ˆθ+ˆµ + 1] ˆη 1 = ˆθ = ˆη 2. 5
The MLEs of (θ 1, θ 2, θ 3, η 1, η 2) obtained by Hanagal(1992). He also obtained asymptotic multivariate normal(amvn) distribution of MLEs. The MLEs of (θ 1, θ 2, θ 3, η 1, η 2) are ˆθ i = n m i / Min(x 1, x 2 ), i = 1, 2, 3; =1 ˆη i = n m i/ I i W, i i = 1, 2 =1 where m i = n =1 I i, i=1,2,3 be the number of observations with X 1 < X 2, X 1 > X 2 and X 1 = X 2 such that m 1 + m 2 + m 3 = n, I i, i=1,2,3 are the indicator functions of the respective regions and W = X 1 X 2. The MLEs of (θ 1, θ 2, θ 3 ) when η 1 = θ = η 2 are ˆθ i = n (2n m 3 )/[n Max(x 1i, x 2i )], i = 1, 2, 3. i=1 The asymptotic distribution of ˆR is AN(R, G 3Λ 3 G 3 ) where G 3 = ( R/ θ 1, R/ θ 2, R/ θ 3, R/ η 1, R/ η 2, R/ µ 1, R/ µ 2 ), η 1 θ η 2 = ( R/ θ 1, R/ θ 2, R/ θ 3, R/ µ 1, R/ µ 2 ), η 1 = θ = η 2 Λ 3 = 1 n diag(θθ 1, θθ 2, θθ 3, θη 2 1 /θ 2, θη 2 2 /θ 1, µ 2 1, µ 2 2), η 1 θ η 2 = 1 n I 1 (θ 1, θ 2, θ 3, µ 1, µ 2 ), η 1 = θ = η 2 and I 1 is the inverse of the Fisher information matrix I(θ 1, θ 2, θ 3, µ 1, µ 2 ). The elements of I(θ 1, θ 2, θ 3, µ 1, µ 2 ) are I ii = 1/(θθ i ) + (θ 1 + θ 2 )/θ 3, i = 1, 2, 3; I i = (θ 1 + θ 2 )/θ 3, i = 1, 2, 3; I ii = 1/µ 2 i, i = 4, 5; I i = 0, i = 1, 2, 3; = 4, 5. We next consider the estimate of R when (X 1, X 2 ) follow BVE of Freund(1961). When θ 3 = 0, the BVE of Proschan-Sullo(1974) reduces to the BVE of Freund(1961). 6
Now the system reliability (R) is given by 2i =1 µ 1 µ 2 (θ 1 +θ 2 θ )(θ 1+θ 2 +µ 1 +µ 2 ) [ θ θ θ 1 +θ 2 +µ + θ i θ +µ ] θ 1 θ 1 + θ 2 η 2 2i =1 µ 1 µ 2 (θ 1 +θ 2 +µ )(θ 1 +θ 2 +µ 1 +µ 2 ) [ θ i θ 1 +θ 2 +µ + 1] θ 1 = θ 1 + θ 2 = θ 2. The estimate of R based on MLEs of (θ 1, θ 2, θ 3, θ 1, θ 2, µ 1, µ 2 ) is ˆ 2i =1 ˆµ 1 ˆµ 2 [ (ˆθ 1 +ˆθ 2 ˆθ )(ˆθ 1 +ˆθ 2 +ˆµ 1 +ˆµ 2 ) 2i =1 ˆµ 1 ˆµ 2 [ (ˆθ 1 +ˆθ 2 +ˆµ )(ˆθ 1 +ˆθ 2 +ˆµ 1 +ˆµ 2 ) ˆθ ˆθ + ˆθ i ˆθ 1 +ˆθ 2 +ˆµ ˆθ +ˆµ ˆθi ] ˆθ 1 ˆθ 1 + ˆθ 2 ˆθ 2 ˆθ 1 +ˆθ 2 +ˆµ + 1] ˆθ 1 = ˆθ 1 + ˆθ 2 = ˆθ 2. The parameter θ 3 = 0 which leads to m 3 = 0 and hence n = m 1 + m 2. Substitute θ 3 = 0 and m 3 = 0 in the expressions of MLEs and variance-covariance matrix of BVE of Proschan-Sullo(1974), we get the corresponding expressions in BVE of Freund(1961). Here ˆR is AN(R, G 4Λ 4 G 4 ) where G 4 = ( R/ θ 1, R/ θ 2, R/ θ 1, R/ θ 2, R/ µ 1, R/ µ 2 ), θ 1 θ 1 + θ 2 θ 2 = ( R/ θ 1, R/ θ 2, R/ µ 1, R/ µ 2 ), θ 1 = θ 1 + θ 2 = θ 2 Λ 4 = 1 n diag[(θ 1 + θ 2 )θ 1, (θ 1 + θ 2 )θ 2, (θ 1 + θ 2 )θ 2 1 /θ 2, (θ 1 + θ 2 )θ 2 2 /θ 1, µ 2 1, µ 2 2], θ 1 θ 1 + θ 2 θ 2 = 1 n I 1 [(θ 1, θ 2, µ 1, µ 2 ], θ 1 = θ 1 + θ 2 = θ 2 3 Independent Gamma Stresses In this Section, we assume the p.d.f. of Y 1 and Y 2 as g i (y i ) = µ α i i exp[ µ i y i ]y α i 1 i /Γ(α i ), y i, µ i > 0, where α i is known integer. We assume Y 1 and Y 2 are independent and also independent of (X 1, X 2 ). So, The MLEs of µ 1 and µ 2 are independent of MLEs of the parameters of BVE of (X 1, X 2 ). 7
The MLEs of (µ 1, µ 2 ) are ˆµ i = n nα i / X i, i = 1, 2 =1 and their asymptotic variances are V (ˆµ i ) = µ 2 i /(nα i ), i = 1, 2. The system reliability (R) when (X 1, X 2 ) has BVE of Marshall- OLkin(1967) is Γ(α i + k)µ α i i µ α Γ(α i )Γ(k + 1)(µ i + µ + λ) α i+k (µ + λ + λ 3 ) α k The estimate of R based on MLEs is given by ˆ Γ(α i + k)ˆµ α i i ˆµ α Γ(α i )Γ(k + 1)(ˆµ i + ˆµ + ˆλ) α i+k (ˆµ + ˆλ + ˆλ 3 ) α k which is AN(R, G 5Λ 5 G 5 ) where G 5 = ( R/ λ 1, R/ λ 2, R/ λ 3, R/ µ 1, R/ µ 2 ) and Λ 5 = I 1 (λ 1, λ 2, λ 3, µ 1, µ 2 )/n and I 1 is the inverse of the Fisher information matrix. The system reliability when (X 1, X 2 ) has BVE of Bolck- Basu(1974) is Γ(α i + k)µ α i i µ α Γ(α i )Γ(k + 1)(µ i + µ + λ) α i+k (λ 1 + λ 2 ) [ λ (µ + λ + λ 3 ) α k λ 3 (µ + λ) ] α k The estimate of R based on MLEs is given by ˆ Γ(α i + k)ˆµ α i i ˆµ α Γ(α i )Γ(k + 1)(ˆµ i + ˆµ + ˆλ) α i+k (ˆλ 1 + ˆλ 2 ) [ ˆλ (ˆµ + ˆλ + ˆλ 3 ) α k ˆλ 3 (ˆµ + ˆλ) ] α k which is AN(R, G 6Λ 6 G 6 ) where G 6 = ( R/ λ 1, R/ λ 2, R/ λ 3, R/ µ 1, R/ µ 2 ) and Λ 6 = I 1 (λ 1, λ 2, λ 3, µ 1, µ 2 )/n and I 1 is the inverse of the Fisher information matrix. 8
The system reliability (R) when (X 1, X 2 ) has BVE of Proschan- Sullo(1974) is Γ(α i + k)µ α i i µ α Γ(α i )Γ(k + 1)(µ i + µ + θ) α i+k (θ η ) [ θ + θ 3 η (µ + θ) + θ i α k (µ + η ) ] α k provided η 1 θ η 2. When η 1 = θ = η 2, the system reliability (R) is α Γ(α i + k)µ α i i µ α θ i Γ(α i )Γ(k + 1)(µ i + µ + θ) α i+k (θ + µ i ) α k+1 Γ(α i + k)µ α i i µ α [µ i + µ + θ θ i (α i + k)] Γ(α i )Γ(k + 1)(µ i + µ + θ) α i+k+1 (θ + µ ). α k The estimate of R based on MLEs is ˆ Γ(α i + k)ˆµ α i i ˆµ α ˆθ Γ(α i )Γ(k + 1)(ˆµ i + ˆµ + ˆθ) α i+k (ˆθ ˆη ) [ + ˆθ 3 ˆη (ˆµ + ˆθ) + α k ˆθi (ˆµ + ˆη ) α k ] provided ˆη 1 ˆθ ˆη 2. When ˆη 1 = ˆθ = ˆη 2, the MLE of system reliability ( ˆR) is ˆ α + ˆµ α ˆθ i Γ(α i + k)ˆµ α i i Γ(α i )Γ(k + 1)(ˆµ i + ˆµ + ˆθ) α i+k (ˆθ + ˆµ i ) α k+1 Γ(α i + k)ˆµ α i i ˆµ α [ˆµ i + ˆµ + ˆθ ˆθ i (α i + k)] Γ(α i )Γ(k + 1)(ˆµ i + ˆµ + ˆθ) α i+k+1 (ˆθ + ˆµ ). α k The distribution of ˆR is AN[R, G 7Λ 7 G 7 ) where G 7 = ( R/ θ 1, R/ θ 2, R/ θ 3, R/ η 1, R/ η 2, R/ µ 1, R/ µ 2 ), η 1 θ η 2 = ( R/ θ 1, R/ θ 2, R/ θ 3, R/ µ 1, R/ µ 2 ), η 1 = θ = η 2 and Λ 7 is variance-covariance matrix of MLEs of the parameters. The system reliability (R) and its MLE when (X 1, X 2 ) follow BVE of Freund(1961) are given by the expressions of R and ˆR in the case of Proschan-Sullo(1974) model by substituting θ 3 = 0 and ˆθ 3 = 0. 9
References [1] Bhattacharyya, G.K. and Johnson, R.A.(1974). Estimation of reliability in a component stress-strtength model. J. Amer. Statist. Assn., 69, 966-70. [2] Bhattacharyya, G.K. and Johnson, R.A.(1975). Stress-strength models for system reliability. Proc. Symp. on Reliability and Fault-tree Analysis, SIAM, 509-32. [3] Bhattacharyya, G.K. and Johnson, R.A.(1977). Estimation of system reliability by nonparametric techniques. Bulletin of Mathematical Society of Greece. (Memorial Volume), 94-105. [4] Bhattacharyya, G.K.(1977). Reliability estimation from survivor count data in a stress-strength setting. IAPQR transactions, 2, 1-15. [5] Block, H.W. and Basu, A.P.(1974). A continous bivariate exponential extension. J. Amer. Statist. Assn., 69, 1031-37. [6] Ebrahimi, N.(1982). Estimation of reliability for a series stress-strength system. IEEE transactions on Reliability, R-31, 202-205. [7] Freund, J.E.(1961). A bivariate extension of the exponential distribution. J. Amer. Statist. Assn., 56, 971-77. [8] Hanagal, D.D.(1992). Some inference results in modified Freund s bivariate exponential distribution. Biometrical Journal, 34(6), 745-56. [9] Hanagal, D.D. and Kale, B.K.(1991a). Large sample tests of independence in an absolutely continuous bivariate exponential distribution. Communications in Statistics, Theory and Methods, 20(4), 1301-13. 10
[10] Hanagal, D.D. and Kale, B.K.(1991b). Large sample tests of λ 3 in the bivariate exponential distribution. Statistics and Probability Letters, 12(4), 311-13. [11] Hanagal, D.D. and Kale, B.K.(1992). Large sample tests for testing symmetry and independence in bivariate exponential models. Communications in Statistics, Theory and Methods, 21(9), 2625-43. [12] Johnson, R.A.(1988). Stress-strength models for reliability. Handbook of Statistics. Vol 7, Quality Control and Reliability, 27-54. [13] Marshall, A.W. and Olkin, I.(1967). A multivariate exponential distribution. J. Amer. Statist. Assn., 62, 30-44. [14] Proschan, F. and Sullo, P.(1974). Estimating the parameters of bivariate exponential distributions in several sampling situations. Reliability and Biometry, SIAM, 423-40. 11