A simple approximation for the renewal function with an increasing failure rate

ARTICLE IN PRESS Reliability Engineering and System Safety 95 (2010) 963–969

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A simple approximation for the renewal function with an increasing failure rate R. Jiang n Faculty of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha, Hunan 410114, China

a r t i c l e in fo

abstract

Article history: Received 23 May 2009 Received in revised form 13 February 2010 Accepted 14 April 2010 Available online 24 April 2010

This paper proposes a simple approximation for the renewal function of a failure distribution with an (equivalently) increasing failure rate. The approximation is a linear combination of the cumulative distribution and hazard functions, and the coefficients are functions of the shape parameter of the distribution. The approximation is applied to the Weibull, gamma and lognormal distributions, and it is shown that the approximation is accurate for t up to a certain value of larger than the characteristic life. The approximation is useful for maintenance policy analysis and optimization where the renewal function needs to be evaluated. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Renewal function Approximation Weibull distribution Gamma distribution Lognormal distribution

1. Introduction The renewal function (RF, M(t)) plays an important role in areas of reliability and maintenance (e.g., see [1–3]). However, it is not possible to obtain M(t) analytically for most of distribution families. Many different approaches have been developed in the literature to approximate or numerically compute RF. Refs [4,5] comprehensively review the approaches for computing RF. In some situations such as inventory planning, it requires that RF is accurate for sufficiently large t. For such situations, a numerical method (e.g., see [6]) or an approximation (e.g., see [7]) can be used to directly compute RF. In the other situations such as analysis and optimization of various replacement policies (e.g., see [1,8]), it requires that RF is accurate for a relatively small t (e.g., the interval from zero to a time close to the median life, see [9]) and a simple and closedform approximation is desired so as to simplify the optimization algorithm and solving process. The following requirements for such approximations are desired:

 Simplicity: Namely, it has a closed-form expression and can be directly used without a need of further numerical computation. Abbreviations: Cdf, Cumulative distribution function; Chf, Cumulative hazard function; RF, Renewal function; RHS, Right-hand side. n Tel.: + 86 731 2309122; fax: + 96 731 5258646. E-mail address: [email protected] 0951-8320/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2010.04.007

 Accuracy: It is accurate from an engineering perspective for 

the potentially possible range of the decision variable (e.g., a preventive replacement age). Applicability: This deals with two aspects. One is that the range that it is accurate should be as large as possible; and the other is that it is applicable for more distribution families rather than a specific distribution.

Though some efforts have been made to develop such approximations (e.g., see [10,11]), it appears that no approximation meets all the above three requirements. This may be due to the complexity of the shape of RF (e.g., see [12]). In this paper, we present such an approximation. It is suitable for a distribution with an increasing failure rate or equivalently increasing failure rate. For the notion of equivalently increasing failure rate, see [13]. Roughly speaking, a non-monotonic failure rate can be viewed as an equivalently increasing failure if its first change point, where the failure rate changes from increasing to decreasing, is sufficiently large (relative to the mean life). This constrain (i.e., (equivalently) increasing failure rate) is usually a necessary condition for a preventive replacement policy to have an optimal solution, and requires that a subset of the parameter space of a distribution is considered (e.g., the shape parameter 41 for the Weibull distribution). Considering the fact that the cumulative hazard function (Chf) [cumulative distribution function (Cdf)] is an upper [lower] limit of RF and is a good approximation to RF for small t, a linear combination of Cdf and Chf may be a good approximation for RF if the coefficients are appropriately determined. The paper studies this approximation

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and presents an approach to properly determine the coefficients. The approximation is applied to the Weibull, gamma and lognormal distributions and the accuracy is examined. It is shown that the approximation is accurate in an interval from zero to a certain time larger than the characteristic life. The paper is organized as follows. Section 2 presents the proposed approximation. Sections 3 through 5 apply the approximation to the Weibull, gamma and lognormal distributions, respectively, and obtain the empirical expressions of the coefficients. The paper is concluded with a brief summary in Section 6.

p,qA(0,1). From Eq. (9), we have the renewal density function given by

2. Approximation

we hope to ensure that the approximation is at least accurate in (0, tc). To achieve this, we introduce the following two constraints to determine p and q:

2.1. Cdf, Chf and RF Let F(t) and H(t) denote Cdf and Chf, respectively. Further, let r(t)¼ dH(t)/dt denote the failure rate function. Assume that F(t) has an (equivalently) increasing failure rate. F(t) and H(t) are linked by FðtÞ ¼ 1e

HðtÞ

or HðtÞ ¼ ln½1FðtÞ:

ð1Þ

The integral form of the renewal function associated with F(t) is given by Z t MðtxÞf ðxÞdx, ð2Þ MðtÞ ¼ FðtÞ þ 0

where f(t)¼dF(t)/dt is the probability density function. The renewal density function is given by Z t mðtxÞf ðxÞdx: ð3Þ mðtÞ ¼ dMðtÞ=dt ¼ f ðtÞ þ 0

When t is small, from Eq. (2) we have Z t MðtxÞf ðxÞdx  0, MðtÞ  FðtÞ:

(a) The optimal preventive replacement age is usually smaller than and somehow close to the life mean or median (t0.5), and (b) F(tc)¼1  e  1 ¼ 0.632140.5 and H(tc)¼1, where tc is the characteristic life (which equals the scale parameter for the Weibull distribution),

Ma ðtc Þ ¼ Mðtc Þ,

Mðtc Þ ¼ pð1e1 Þ þq,

ð5Þ

ð6Þ

ð7Þ

Eqs. (6) and (7) imply that H(t) is not only an upper limit of M(t) but also a good approximation of M(t) for small t. Specially, for the exponential distribution (both the gamma and Weibull distributions include it as a special case), we have ð8Þ

2.2. Proposed approximation According to the above discussion, we propose the following model to approximate RF: Ma ðtÞ ¼ pFðtÞ þqHðtÞ  MðtÞ,

e1 mðtc Þ=f ðtc Þ ¼ pe1 þ q:

, we ð12Þ

From Eq. (12) yields Mðtc Þe1 mðtc Þ=f ðtc Þ , q ¼ Mðtc Þpð1e1 Þ: 12e1 Thus, the approximation is given by Eqs. (9) and (13).

p¼

ð13Þ

2.3. Properties of the approximation 2.3.1. Relation between p, q and the scale parameter of a distribution Suppose that F(t) has a scale parameter s. Then we have f ðt; sÞ ¼ f ðz; 1Þ=s,

ð14Þ

F ðnÞ ðt; sÞ ¼ F ðnÞ ðt=s; 1Þ ¼ F ðnÞ ðz; 1Þ,

n ¼ 2,3,. . .

ð15Þ

(n)

When t is small, from Eqs. (1) and (5) we have

MðtÞ ¼ HðtÞ:

ð11Þ 1

where z¼t/s. From Eq. (14) it is easy to show ð4Þ

Namely, F(t) is not only a lower limit of M(t) but also a rough approximation of M(t) for small t. It is well-known that for a reparable item the expected number of minimal repairs over (0, t) is given by Chf, H(t). On the other hand, the expected number of perfect replacements over (0, t) is RF, M(t). Clearly, for a (or an equivalently) positive aging item and given time interval, the expected number of minimal repairs must be larger than the expected number of perfect replacements, i.e.,

MðtÞ  FðtÞ ¼ 1exp½HðtÞ  HðtÞ:

ma ðtc Þ ¼ mðtc Þ:

Using Eq. (11) in Eqs. (9) and (10) and noting F(tc) ¼1 e have

where F (.) is the n-fold convolution of F(.) with itself. As a result, we have

0

MðtÞ oHðtÞ:

ð10Þ

Considering the following facts:

Fðt; sÞ ¼ Fðt=s; 1Þ ¼ Fðz; 1Þ,

Ignoring the second term of the RHS of Eq. (2), we have MðtÞ 4FðtÞ:

ma ðtÞ ¼ f ðtÞ½p þq=RðtÞ  mðtÞ:

ð9Þ

where p and q are the model parameters to be determined. Empirically, if the failure rate is (equivalently) increasing, then

Mðt; sÞ ¼

1 X

F ðnÞ ðt; sÞ ¼

n¼1

1 X

F ðnÞ ðt=s; 1Þ ¼ Mðt=s; 1Þ,

mðt; sÞ ¼ mðz; 1Þ=s:

n¼1

ð16Þ Using Eqs. (14) and (16) to Eq. (13), we can conclude that p and q are independent of the scale parameter. In addition, we only need to study RF for s¼1 and RF for sa1 can be obtained from Eq. (16). 2.3.2. Relation between the approximation and exponential RF For the exponential distribution, we have M(tc)¼1, m(tc)¼1/Z and f(tc) ¼e  1/Z. Using these to Eq. (13) yields p¼0 and q¼1. Using these in Eq. (9) yields Ma(t) ¼H(t). Namely, the approximation is exact for the exponential distribution. 2.3.3. Relation between the approximation and Eq. (5) When F(tc)EM(tc), from Eq. (13) yields p E1 and qE0, and Ma(t)EF(t). As a result, the approximation reduces into Eq. (5) if F(tc)EM(tc). 2.3.4. Absolute error curve From Eqs. (9) and (11), the approximation is exact at t ¼0 and t¼tc; and Ma(t)-p +qH(t) for large t, which quickly deviates from M(t). As a result, the absolute error given by

eðtÞ ¼ 9MðtÞMa ðtÞ9

ð17Þ

generally has the shape shown in Fig. 1. It is noted that the relative error is not suitable for measuring the accuracy for both small and large t, we use the absolute error

ARTICLE IN PRESS

ε (t)

R. Jiang / Reliability Engineering and System Safety 95 (2010) 963–969

respectively. Without loss of generality, we let Z ¼1 and hence tc ¼1. According to Harter and Moore [14], in various specific applications of the Weibull distribution, b ranges from 0.09 to 6.3. To make the failure rate increasing, it requires b 41. As such, we only consider the range 1o b r6.5. Since Z ¼1 and f(1)¼ be  1, Eq. (13) becomes

ε εm

p¼

t

tc

to measure the accuracy. According to Fig. 1, the accuracy can be measured by the following two parameters:

 em ¼ max ðeðtÞÞ, and t A ð0,tc Þ

 te ¼ inf(t:e(t)¼ e), where e is a certain error level such as 0.01. Clearly, the larger ts is and the smaller em is, the more accurate the model is. 2.4. Approach to specify p and q for a specific distribution We consider a two-parameter distribution with a shape parameter and a scale parameter. To specify Eq. (14), we need to evaluate M(tc) and m(tc). This can be done by a numerical approach, e.g., the approach of Xie [6] or the like. To facilitate the implementation of the approximation, we can fit p and q into empirical relations of the shape parameter for a given distribution family. Thus, there is not a need for the user to evaluate M(tc) and m(tc). The specific procedure is as follows:

 Step 1: For a given distribution family and a set of given values



Mð1Þmð1Þ=b , 12e1

of the shape parameter (the scale parameter is always set as 1), evaluate M(tc) and m(tc) using a numerical method. Step 2: Evaluate the corresponding values of p and q using Eq. (13). Step 3: Fit the computational results for p and q into empirical relations of the shape parameter.

3. Approximation for the Weibull RF 3.1. Empirical relation for p

Z

ð18Þ

Here, b is the shape parameter and Z is the scale parameter. The mean, variance and coefficient of variation (cv) are given by mean ¼ ZGð1 þ1=bÞ, var ¼ Z2 Gð1 þ 2=bÞm2 , vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u Gð1 þ 2=bÞ 2 1 cv ¼ t 2 G ð1 þ 1=bÞ

ð20Þ

(a) M(1)-1 e  1 ¼F(1) as b increases, implying that the approximation gets close to Eq. (5) for a large b. Define a special point p(b0) ¼0.99. Then Eq. (5) can be thought to be accurate for b Z b0. Using the second-order polynomial interpolation, we obtained b0 ¼5.2007. (b) p +q can be either larger or smaller than but very close to 1. This implies that the approximation given by Eq. (9) can be simplified into a weighted arithmetic mean of F(t) and H(t). Thus, we only need to fit p or q as a function of b. (c) p [q] smoothly increases [decreases] from 0 [1] to 1 [0] as b increases, implying that p [q] versus b is similar to a distribution [reliability] function. Using the least squared error method, p versus b can be fitted as below "   # b1 b , b Z1 ð21Þ pðbÞ ¼ 1exp  a where a¼0.8731 and b¼0.9269. Fig. 2 illustrates the accuracy of Eq. (21). It is noted that p or q ¼1 p obtained from Eq. (21) is slightly different from the one obtained from Eq. (20) and hence generally e(tc)a0.

Table 1 M(1) and m(1) for the Weibull distribution.

b

An Excel spreadsheet program has been written to perform the first step with the step length Dt ¼tc/2000. The program has been verified and it gives very accurate RF.

The Weibull distribution is given by "   # t b , Z, b 4 0: FðtÞ ¼ 1exp 

q ¼ Mð1Þpð1e1 Þ:

For b ¼1(0.25)6.5, Table 1 shows values of M(1) and m(1). Using these in Eq. (20), we obtained the values of p and q shown in the 4th and 5th columns of Table 1. From the table, we have the following observations:

tε

Fig. 1. Absolute error curve.



965

ð19Þ

1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5 5.75 6 6.25 6.5

M(1)

m(1)

p

q

p+ q

1 0.908508 0.841574 0.791571 0.753691 0.724737 0.702507 0.685428 0.672329 0.662319 0.654705 0.64894 0.644599 0.641345 0.638917 0.637113 0.635778 0.634794 0.63407 0.633539 0.633151 0.632868 0.632662

1 1.043894 1.080426 1.114758 1.149557 1.186424 1.226509 1.270685 1.319558 1.373457 1.432441 1.496334 1.564788 1.637346 1.713496 1.792719 1.874522 1.958455 2.04412 2.13118 2.21935 2.308394 2.398124

0 0.2777 0.4590 0.5849 0.6771 0.7472 0.8019 0.8453 0.8798 0.9072 0.9288 0.9458 0.9590 0.9691 0.9769 0.9828 0.9873 0.9906 0.9931 0.9949 0.9963 0.9973 0.9980

1 0.7329 0.5514 0.4218 0.3257 0.2524 0.1956 0.1511 0.1162 0.0889 0.0676 0.0511 0.0384 0.0287 0.0214 0.0159 0.0117 0.0086 0.0063 0.0046 0.0034 0.0025 0.0018

1 1.0107 1.0104 1.0068 1.0028 0.9996 0.9975 0.9964 0.9960 0.9961 0.9964 0.9969 0.9974 0.9979 0.9983 0.9987 0.9990 0.9992 0.9994 0.9996 0.9997 0.9998 0.9998

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obtained from of the normal series approximation [7] and the point (te) where the maximum is achieved. It is clearly shown that the proposed approximation is more accurate in (0, te) for most of b’s values.

1 0.8 p 0.6 0.4

4. Approximation for the gamma RF

q 0.2

4.1. Empirical relations for p and q 0 1

2

3

4

5

6

7

β Fig. 2. Accuracy of (21).

b

1.25

1.5

1.75

2.0

2.5

3.0

em,  10  2

0.7576 1.9900 0.9059

0.6848 1.7252 0.8963

0.5114 1.6517 0.9099

0.3215 1.6510 0.9345

0.0418 1.7910 0.9863

0.2384 1.4907 0.9636

10.6367 0.5039 3.5

6.1909 0.6087 4.0

3.8883 0.6907 4.5

2.5560 0.7571 5.0

1.1651 0.8568 5.5

0.4976 0.9114 6.0

0.3368 1.4394 0.9721

0.3449 1.4297 0.9847

0.3037 1.4309 0.9934

0.2444 1.4364 0.9978

0.1853 1.4439 0.9995

0.1348 1.4529 0.9999

0.2750 1.4394

0.3279 1.3685

0.4587 1.3482

0.5816 1.3680

0.6857 1.3974

0.7732 1.4285

em,  10  2

te b

em,  10  2 te F(te) em,  10  2

te

ð22Þ

where G(.) is the complete gamma function, a is the shape parameter and d is the scale parameter. The mean, variance and cv are given, respectively, by pffiffiffi 2 mean ¼ ad, var ¼ ad , cv ¼ 1= a: ð23Þ

Table 2 Accuracy measures for the Weibull distribution.

te F(te)

The gamma distribution is given by Z t 1 xa1 ex=d dx, t Z0, FðtÞ ¼ a d GðaÞ 0

2

Without loss of generality, we let d ¼1. To determine the range of a, we look at cv’s of both the gamma and Weibull distributions. cv of the gamma distribution is given by Eq. (23), a function of a; and cv of the Weibull distribution is given by Eq. (19), a function of b. For a given value of b, we can find a value of a so that the gamma and Weibull distributions have the same cv. This yields " #1 Gð1 þ2=bÞ 1 a¼ 2 : ð24Þ G ð1 þ 1=bÞ In Eq. (24), letting b ¼(1, 6.3) yields a ¼(1, 29.1). Therefore, we consider the range 1 o a r30, which corresponds to b ¼(1, 6.4). For a set of given values of a, we obtained the values of p and q, and the results are shown in Table 3. From the table we have the following observations:

tε

1.5 1.4289

(a) M(tc)-1  e  1 as a increases, implying that the approximation gets close to Eq. (5) for a large a. Define a special point p(a0) ¼0.99. Then Eq. (5) can be thought to be accurate for a Z a0. Using the second-order polynomial interpolation, we obtained a0 ¼15.4500, which corresponds to cv¼0.2544. (b) p +q is slightly smaller than 1; and p +qE1 can be met only for large a. Thus, we need to fit both p and q as functions of a.

F(tε)

1 0.8962 0.5 1.48

4.19

0 1

2

3

4

5

6

β Fig. 3. Minimums of te and F(te) for the Weibull distribution.

3.2. Accuracy measures For a set of values of b, we find the values of te and em (see Fig. 1), and the results are shown in Table 2 and Fig. 3. As can be seen from the table and figure:

 em roughly decreases as b increases and is always smaller than e ¼0.01, and

 Using the second-order polynomial interpolation yields the minimum of te equals 1.4289; and the minimum of F(te) equals 0.8962. These show that the approximation is sufficiently accurate for analyzing and optimizing a preventive replacement policy with a failure probability of smaller than 0.90 and an optimal preventive replacement age of smaller than 1.43Z. To further illustrate the accuracy of the proposed approximation, Table 2 also shows the maximum absolute error (em) in (0, te)

Table 3 p and q for the gamma distribution.

a 1 1.5 2 3 4 5 6 7 8 10 12 14 16 18 20 22 24 26 28 30

tc

M(tc)

m(tc)

p

q

p+ q

1 1.5791 2.1462 3.2583 4.3520 5.4341 6.5080 7.5757 8.6386 10.7532 12.8566 14.9514 17.0395 19.1222 21.2003 23.2746 25.3455 27.4135 29.4788 31.5419

1 0.89194 0.826515 0.750821 0.70887 0.683083 0.666418 0.655373 0.647953 0.639517 0.635597 0.63376 0.632896 0.632488 0.632295 0.632203 0.63216 0.632139 0.632129 0.632125

1 0.65628 0.493164 0.334351 0.255985 0.209286 0.178525 0.156995 0.141281 0.12023 0.106942 0.097757 0.090921 0.085537 0.081114 0.077372 0.074136 0.071292 0.068763 0.066491

0 0.2500 0.3919 0.5610 0.6687 0.7479 0.8091 0.8566 0.8932 0.9421 0.9693 0.9840 0.9918 0.9958 0.9979 0.9989 0.9995 0.9997 0.9999 0.9999

1 0.7339 0.5788 0.3962 0.2862 0.2103 0.1550 0.1139 0.0833 0.0440 0.0229 0.0118 0.0060 0.0030 0.0015 0.0008 0.0004 0.0002 0.0001 0.0000

1 0.9839 0.9707 0.9572 0.9549 0.9582 0.9641 0.9705 0.9766 0.9861 0.9922 0.9958 0.9977 0.9988 0.9994 0.9997 0.9998 0.9999 1.0000 1.0000

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1

replacement age of smaller than 1.20tc. However, the minimums of te and F(te) are smaller than those for the Weibull distribution.

p

0.8

967

0.6

5. Approximation for the lognormal RF

0.4

5.1. Empirical relations for p and q 0.2 0

q 0

5

10

15 α

20

25

30

where F(.) is the standard normal distribution with mean zero and standard deviation 1. The mean, variance and cv are given, respectively, by pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ð27Þ mean ¼ em þ s =2 , var ¼ ðe2s es Þe2m , cv ¼ es2 1:

Table 4 Accuracy measures for the gamma distribution. 1.5

em,  10 te/tc F(te)

2

a em,  10  2 te/tc F(te)

0.5315 1.7820 0.8688 10 0.2910 1.2087 0.8340

2 0.3486 3.2687 0.9928 14 0.1873 1.2446 0.8859

3

4

0.3778 1.4840 0.8608 18 0.1129 1.2835 0.9284

6

0.3968 1.2988 0.8150 22 0.0658 1.3206 0.9582

8

0.3892 1.2070 0.7951 26 0.0376 1.3551 0.9771

0.3461 1.1979 0.8095 30 0.0213 1.3865 0.9881

3.5 3 2.5 2 tε/tc

1.5 1.20 1

F(tε)

0.5 0.795 6.16

0 0

5

7.92 10

15 α

ð26Þ

s

Fig. 4. Accuracy of (25).

a

The lognormal distribution is given by   lnðtÞm FðtÞ ¼ F

20

25

30

Fig. 5. Minimums of te and F(te) for the gamma distribution.

(c) p [q] smoothly increases [decreases] from 0 [1] to 1 [0] as a increases, implying that p [q] versus a is similar to a distribution [reliability] function. Using the least squared error method, empirical relations for p and q can be fitted as below "  # "  #   a1 0:7856 a1 0:7853 p ¼ 1exp  , q ¼ exp  : ð25Þ 2:5482 2:2173 The accuracy of the relations is illustrated by Fig. 4. 4.2. Accuracy measures For a set of values of a, we find the values of te and em, and the results are shown in Table 4 and Fig. 5. As can be seen from the table and figure:

 em is always smaller than e ¼0.01, and  Using the second-order polynomial interpolation yields the minimum of te equals 1.1979; and the minimum of F(te) equals 0.7950. These show that the approximation is sufficiently accurate for analyzing and optimizing a preventive replacement policy with a failure probability of smaller than 0.80 and an optimal preventive

It is noted h i lnðtÞm ¼ ln ðt=em Þ1=s :

ð28Þ

s

Comparing Eq. (28) with Eq. (18), em is similar to Z and hence is a scale parameter; and 1/s is similar to b and hence is a shape parameter. Without loss of generality, we let em ¼1, i.e., m ¼0. In this case, tc ¼ exp[sF  1(1 e  1)]¼1.4014s. According to Jiang et al. [13], the failure rate for this distribution is unimodal and it is equivalent to increasing when s is small. Similarly, for a given value of the Weibull shape parameter b, we can find a value of s so that the lognormal and Weibull distributions have the same cv. From Eqs. (19) and (27) yields ( " #)1=2 Gð1 þ 2=bÞ s ¼ ln 2 : ð29Þ G ð1þ 1=bÞ In Eq. (29), letting b ¼(1, 6.3) yields s ¼(0.8326, 0.1837). This implies that it requires s r0.8326 for the lognormal distribution to have cvr1, an equivalently positive aging condition. In this paper, we consider a slightly larger range: s r1. For s ¼0.2(0.05)1.0, we obtained the values of p and q shown in Table 5. From the table we have the following observations: (a) M(tc)-1  e  1 as s decreases, implying that the approximation gets close to Eq. (5) for a small s. Define a special point p(s0) ¼0.99. Then Eq. (5) can be thought to be accurate for s r s0. Using the second-order polynomial interpolation, we obtained s0 ¼0.2696, which corresponds to cv ¼0.2745. Table 5 p and q for the lognormal distribution.

s 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2

tc

M(tc)

m(tc)

p

q

p+ q

1.40140 1.37796 1.35490 1.33223 1.30994 1.28802 1.26647 1.24528 1.22444 1.20395 1.18381 1.16400 1.14453 1.12537 1.10654 1.08803 1.06982

0.91022 0.88853 0.86642 0.84393 0.82112 0.79806 0.77486 0.75168 0.72875 0.70645 0.68536 0.66640 0.65082 0.63992 0.63416 0.63234 0.63212

0.69618 0.71305 0.73037 0.74820 0.76661 0.78566 0.80545 0.82605 0.84752 0.87002 0.89422 0.92272 0.96321 1.03315 1.16260 1.38981 1.76143

 0.1595  0.0857  0.0113 0.0638 0.1396 0.2164 0.2945 0.3746 0.4577 0.5452 0.6384 0.7364 0.8339 0.9184 0.9742 0.9965 0.9999

1.0111 0.9427 0.8735 0.8036 0.7329 0.6613 0.5887 0.5149 0.4394 0.3618 0.2818 0.2009 0.1237 0.0594 0.0184 0.0024 0

0.8515 0.8570 0.8623 0.8674 0.8725 0.8777 0.8832 0.8895 0.8971 0.9070 0.9202 0.9373 0.9576 0.9778 0.9925 0.9989 1

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R. Jiang / Reliability Engineering and System Safety 95 (2010) 963–969

(b) We have

1.6

ð30Þ

1.4

(c) p + q is much smaller than 1 for a relatively large s; and p + qE1 can be met only for small s. Thus, we need to fit both p and q as functions of s for s 40.2.

1.2

p  1, q  0 for s r0:2

A B

F(tε)

0.8

C

0.6

Using the least squared error method, we have fitted the following empirical relations for s 40.2:   lnðs0:2Þ þ1:3327 p þ q ¼ 10:1358F , 0:4643   lnðs0:2Þ0:2146 q ¼ 2:9606F : ð31Þ 1:0704

A: (0.6647, 1.5203) B: (0.5852, 1.0666) C: (0.5935, 0.6728)

0.4 0.2 0 0.2

εm, %

tε/tc

1

0.473 0.3

0.4

0.5

0.6 σ

0.7

0.8

0.9

1

Fig. 7. Maximum of em and minimums of te and F(te) for the lognormal distribution.

The accuracy of the fitted relations is illustrated by Fig. 6. 5.2. Accuracy measures

Table 7 Summary of properties (e ¼ 0.01).

For s ¼0.2(0.05)1.0, we find the corresponding values of te and em, and the results are shown in Table 6 and Fig. 7. As can be seen from the table and figure:

 em 4 e ¼0.01 for a large s. Using the second-order polynomial interpolation, we have em ¼0.01 at s ¼0.4738. The vertical line s ¼0.4738 partitions the plot into two parts: the left [right] part corresponds to em o0.01 [em 40.01]. As a result, for e ¼0.01, the applicable range of the approximation is s r0.4738 with te/tc Z1.1539 and F(te)Z0.7387.

Model

q ¼ 1  p?

em r e

(te/tc)min

Fmin(te)

b0, a0, s0

Weibull Gamma Lognormal

Yes No No

bZ1

1.4289 1.1979 1.1539

0.8962 0.7950 0.7387

5.2007 15.4500 0.2696

aZ1 s r 0.4738

 em(s) has a maximum equal to 0.0152, which is achieved at s ¼0.6647 (see the point A of Fig. 7). If this maximum is acceptable, then the approximation is also applicable for the range sA(0.4738,1). In this case, we have te ¼ 0.01/tc Z1.0666 and F(te ¼ 0.01)Z0.6728 (see the points B and C of Fig. 7).

1 p+q

0.8

In summary, the approximation for the lognormal distribution is poorer than the ones for the Weibull and gamma distributions. This is probably because the lognormal distribution has a nonmonotonic failure rate.

0.6 q

0.4 0.2

6. Conclusions 0 0.2

0.3

0.4

0.5

0.6 s

0.7

0.8

0.9

1

Fig. 6. Accuracy of (31).

Table 6 Accuracy measures for the lognormal distribution.

s 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

em,  10  2

te

te/tc

F(te)

0.0004 0.1237 0.6562 0.9007 0.5769 0.8535 1.1720 1.3784 1.4840 1.5184 1.5095 1.4778 1.4360 1.3916 1.3487 1.3090 1.2734

1.4517 1.3854 1.4084 1.4225 1.4085 1.3734 1.3304 1.2986 1.3086 1.3788 1.4900 1.6171 1.7438 1.8598 1.9605 2.0449 2.1135

1.3570 1.2733 1.2728 1.2640 1.2306 1.1799 1.1238 1.0786 1.0687 1.1072 1.1765 1.2555 1.3312 1.3960 1.4470 1.4840 1.5081

0.9688 0.9039 0.8732 0.8430 0.8041 0.7596 0.7160 0.6826 0.6730 0.6894 0.7156 0.7392 0.7565 0.7673 0.7728 0.7743 0.7729

In this paper, we have developed a simple approximation for the RF of a (or an equivalently) positive aging distribution. The approximation has been applied to the Weibull, gamma and lognormal distributions, and the main properties are summarized in Table 7. The second column in the table indicates whether qE1p holds. If yes, the approximation reduces into the weighted arithmetic mean of F(t) and H(t). It is shown the relation holds for the Weibull distributions and not for the other two. The third column indicates the range of the shape parameter that meets em r0.01. The range is large for the Weibull and gamma distributions and relatively small for the lognormal distribution. The 4th and 5th columns give the minimums of te/tc and F(te). These values measure the applicability of the approximation. Large values are desired. It is shown that the approximation for the Weibull [lognormal] distribution has the best [worst] applicability. The last column gives the critical point where the approximation given by Eq. (5) (i.e., M(t) EF(t)) is accurate or the proposed approximation is close to Eq. (5). It is shown that Eq. (5) is only suitable for the situation where the item is strongly aging. In summary, the proposed approximation is simple and accurate for the interval from zero to a certain time larger than the characteristic life, and it can be used to analyze or/and optimize preventive replacement policies. We stress that Eq. (21) can be directly used for the Weibull distribution, Eq. (25) for the

ARTICLE IN PRESS R. Jiang / Reliability Engineering and System Safety 95 (2010) 963–969

gamma distribution and Eq. (31) for the lognormal distribution without a need to evaluate M(tc) and m(tc). Using a similar approach, the approximation can be specified for other distributions.

Acknowledgement The author would like to thank the editor and anonymous referees for their comments and suggestions for improvement on an earlier version of this paper. This research was supported by the National Natural Science Foundation (No. 70771015). References [1] Barlow RE, Proschan F. Mathematical theory of reliability. New York: John Wiley & Sons; 1965. [2] Sheikh AK, Younas M. Renewal models in reliability engineering. ASME 1985:93–103. [3] Bahrami GK, Price JWH, Mathew J. The constant-interval replacement model for preventive maintenance: a new perspective. International Journal of Quality and Reliability Management 2000;17(8):822–38.

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