Microsoft Word - 11-Wu Jianhua.doc Advances in Systems Science and Applications (2010), Vol.10, No.1 67-72 ISSN 1078-6236 International Institute for General Systems Studies, Inc. Characteristics of Item Replacement in Weibull Distribution Jianhua Wu, Dexin Tao and Hongxiang Li Wuhan University of Technology, Wuhan, China Abstract The aim of this paper is to study preventive replacement in order to increase system’s MTBF by replacing item following the Weibull Distribution. Here, we discuss the periodic preventive replacement and random preventive replacement as preventive replacement. According to item preventive replacement following Weibull Distribution, based on the MTBF evaluation of item to study the characteristics of item replacement. Keywords Weibull Distribution Preventive replacement MTBF 1. Introduction Outline and background of the preventive replacement theory Notation used in this paper. F(t): the failure distribution function of replacement item. dt/)t(dF)t(f);t(F1:)t(F =− G(t): the distribution function of the preventive replacement dt/)t(dG)t(g);t(G1:)t(G =− : first moment or MTBF under the preventive replacement : n moment μ: the replacement factor of preventive replacement distribution function G(t)=1-exp(-μt) T : time interval of preventive replacement Г(·): Gamma function; Г(·,·): in-complete Gamma function; δ(·): Derta function β: shape parameter of Weibull Distribution η: scale parameter of Weibull Distribution CV: coefficient of variation τ: reliability improvement rate Now let’s gather up the result of fundamental and general theory about preventive replacement (1) First moment or MTBF under preventive replacement MTBF or the first moment is given as: 0 0 ( ) ( ) 1 F(t)g(t)dt F t G t dt t ∞ ∞< >= − ∫ ∫ (1) (2) Second moment and variance under preventive replacement the Second moment is given as: 68 Wu: Characteristics of Item Replacement in Weibull Distribution 2 0 0 2 0 0 0 2 0 2{1 F( ) ( ) }{ tG( ) ( ) } (1 ( ) ( ) ) 2{ G( ) ( ) }{ tF( ) ( ) } (1 F( ) ( ) ) t g t dt t F t dt t F t g t dt t F t dt t g t dt t g t dt ∞ ∞ ∞ ∞ ∞ ∞ − < >= + − − ∫ ∫ ∫ ∫ ∫ ∫ (2) thus the varianceδ2 is easily got fromδ 2 =- 2 2. Periodic Preventive Replacement and Random Preventive Replacement The Periodic preventive replacement is the most general way in the preventive replacement, so that we study on this case, and then take up the random preventive replacement as extreme example 2.1 The Average p and the 2nd Moment p of the Periodic Preventive Replacement If theb periodic preventive replacement is done at time T, non- preventive replacement )t(G is as Figure 1, then 1 0 ( ) 0 t T G t theother ≤ ≤⎧ = ⎨ ⎩ (3) g(t)=δ(t-T) (4) Figure 1 Consequently, the molecule and the denominator of (Eq.1) is described as following ∫∫ ∫∫ ∞∞ ∞ =−δ= = 00 T 00 )T(Fdt)Tt()t(Fdt)t(g)t(F dt)t(Fdt)t(F)t(G and )T(F)T(F1dt)t(g)t(F1 0 =−=− ∫ ∞ Therefore the 1st moment p 0 ( ) ( ) T p F t dt t F T < > = ∫ (5) On the other hand, the 2nd moment p can be also obtained in the same way. 2 0 0 2 2 ( ) 2 ( ) ( ) ( ) ( ) T T p tF t dt T F T F t dt t F T F T < > = +∫ ∫ (6) So the variance of preventive replacement can be obtained through formula (Eq.5) and (Eq.6). Advances in Systems Science and Applications (2010), Vol.10, No.1 69 2.2 The 1st Moment and the 2nd Moment of the Random Replacement In the case of the random preventive replacement, as maintenance function we put G(t)=1-exp(-μt) into formula (Eq.1) and (Eq.2), then 1st moment R and 2nd moment R are obtained by 0 0 ( ) exp( ) 1 ( )exp( ) R F t t dt t F t t dt μ μ μ ∞ ∞ − < > = − − ∫ ∫ (7) 2 0 0 0 0 2 0 2 ( )exp( ) 1 ( ) exp( ) 2( ( )exp( ) )( ( ) exp( ) ) (1 ( ) exp( ) ) R tF t t dt t F t t dt F t t dt tF t t dt F t t dt μ μ μ μ μ μ μ ∞ ∞ ∞ ∞ ∞ − < > = + − − − − − − ∫ ∫ ∫ ∫ ∫ (8) 3. Applying to Weibull Distribution When applying these theoretical, we consider preventive replacement characteristic from average value increase of the failure interval. We suppose the Weibull Distribution as ( ) exp{ ( / ) }F t t βη= − (9) 3.1 The Periodic Preventive Replacement We insert it into (Eq.5), the average value is as follow: 0 exp{ ( / ) } 1 exp{ ( / ) } T p t dt t T β β η η − < > = − − ∫ (10) Moreover, it can be rearrange as [(1/ ), ( / ) ] [1, ( / ) ]p Tt T β β η β η β η Γ < > = Γ (11) On the other hand, the 2nd moment can be got by (Eq.6) 2 2 2 2 [(2 / ), ( / ) ] [1, ( / ) ] 2 exp{ ( / ) } [1/ , ( / ) ] [1, ( / ) ] p Tt t T T T T β β β β β η β η β η η η β η β η Γ < > = + Γ − Γ Γ (12) 3.2 The Random Preventive Replacement We apply the Weibull Distribution into (Eq.7) and (Eq.8) yield the following: Average value i.e. 0 0 exp( ) 1 exp( ) R t dt t t dt λ μ λ ∞ ∞ − < > = − − ∫ ∫ (13) 70 Wu: Characteristics of Item Replacement in Weibull Distribution 2 0 0 0 0 2 0 2 exp( ) 1 exp( ) 2 ( exp( ) )( exp( ) ) (1 exp( ) ) R t t dt t t dt t dt t t dt t dt λ μ λ μ λ λ μ λ ∞ ∞ ∞ ∞ ∞ ⋅ − < > = + − − − ⋅ − − − ∫ ∫ ∫ ∫ ∫ (14) Simplify, here (t/η)β +μt =λt is used 4. Consideration 4.1 Condition for Calculation First, for convenience and simple expression, we suppose average of the Weibull Distribution E(t)=ηГ(1/β+1)=1 and normalize the real time. If the periodic normalized time is T=0.1, the real time is 0.1ηГ(1/β+1). More once we define the replacement rate μof the preventive replacement distribution function G(t)=1-exp(- μt) in the random replacement Because reverse of the μ is average replacement intervals. Suppose T is 0.1, μ is 10, it show random maintenance had 0.1 intervals in average, the real time is 0.1ηГ(1/β+1).Moreover in order to compare the preventive replacement characteristics, we use the evolution reliability improvement rate in MTBF and coefficient of variation in dispersion. The reliability improvement rate is defined as following (1/ 1) 1 (1/ 1) t tη βτ η β < > Γ + = =< > − Γ + (15) 1 T t T Tt FF F − >< = −>< =τ And the coefficient of variation CV is defined as 2 2 2 2 2 var 1iance t t tCV average t t < > − < > < > = = = − < > < > (16) 4.2 Reliability improvement rate In the case of the periodic replacement, according to formula (Eq.15) andηГ(1/β+1)=1 Improvement rate of periodic replacement is [(1/ ), ( / ) ] 1 1 exp( ( / ) )p T T β β η β ητ η Γ < > = − − − (17) About random replacement use the same way. Improvement rate of random replacement is 0 0 exp( ) 1 1 exp( ) R t dt t dt λ τ μ λ ∞ ∞ − < > = − − − ∫ ∫ (18) 4.3 Coefficient of Variation The CV value of the periodic replacement is Advances in Systems Science and Applications (2010), Vol.10, No.1 71 2 2 { } 1 [2 / , ( ( )) ](1 exp( ( / )) ) [1/ , ( ( )) ] ( ) exp( ( ( )) ) [1/ , ( ( )) ] pcv a b T Ta T T Tb T β β β β β β β η β β = + − Γ Γ ⋅ − − = Γ Γ ⋅ Γ ⋅ − Γ ⋅ = Γ Γ ⋅ (19) Here, Г(·) is Г(1/β+1),η=1/Г(1/β+1) On the other hand, the CV value of the random replacement is 0 2 0 2 exp( ) 1 ( exp( ) ) R t t dt cv t dt λ λ ∞ ∞ ⋅ − < > = − − ∫ ∫ (20) 4.4 Numerical Results Table 1 Shows an example of calculated results in the periodic preventive replacement β= 2.5 β= 3.0 β = 3.5 β= 4.0 T τp CVp τp CVp τp CVp τp CVp 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 479 84 30 14 7 4 2 1 1 0 0.664 1.305 1.688 1.986 2.250 2.510 2.791 3.126 3.557 4.144 1571 195 57 23 11 6 3 2 1 0 0.973 1.605 2.000 2.305 2.575 2.844 3.147 3.526 4.052 4.837 5085 448 107 38 17 8 4 2 1 0 1.234 1.882 2.287 2.595 2.865 3.136 3.451 3.866 4.481 5.486 16344 1020 200 62 25 11 5 2 1 0 1.469 2.141 2.554 2.862 3.128 3.397 3.716 4.158 4.857 6.099 Comparing to above table, we can get two picture as following: Figure 2 Improvement rate of periodic replacement From the Figure 2 of above, we can see that: the interval T of exchange is smaller, reliability improvement rate τP is bigger. Turn over, the interval T of exchange is bigger, reliability improvement rate τP is smaller. This kind of trend has nothing to do with the size of β. The interval of exchange around one, the reliability improvement rate will become very small, the effect of preventive replacement will disappear. In a word, it is very important to exchange 72 Wu: Characteristics of Item Replacement in Weibull Distribution with the interval which is smaller to the average of item. The shape parameter β is bigger, the reliability improvement rate is bigger, the interval between breakdown and breakdown will become bigger. Figure 3 CV value of periodic replacement From the Figure 3 of above, we can see that: the interval T of exchange is bigger, the coefficient of variation CVP is bigger. Turn over, the interval T of exchange is smaller, the coefficient of variation CVP is smaller. This kind of trend has nothing to do with the size of β. The shape parameter β is bigger, the coefficient of variation is bigger. If we want to get the optimum value, we don’t only consider the reliability improvement rate but also consider the coefficient of variation CVP Synthesize: It will obtain the good effect when short the interval time of exchange. This is in accordance with our experience. 5. Discussion and Conclusion In this report, we obtain the results of theory about periodic preventive replacement and random preventive replacement and then introduced clearly the formula of average and variance in the both preventive replacement, more applied these result to the Weibull Distribution, and showed the example of replacement characteristics about preventive replacement. References [1] Barlow. R, Proschan. F, Hunter. L. “Mathematical Theory of Reliability”, John Wiley&Sons Inc., PP.61-63, 1965. [2] Weiss. G. H. On the Theory of Replacement of Machinery with a Random Failure Time. 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