Reliability equivalence factors in Gamma distribution

Applied Mathematics and Computation 187 (2007) 567–573

www.elsevier.com/locate/amc

Reliability equivalence factors in Gamma distribution

Yan Xia *, Guofen Zhang

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Abstract

In the article, the failure rates of the system’s components are functions about time t, with a life distribution of Gammadistribution. The reliability equivalence factors of a parallel system with n independent and identical components areobtained. Three different methods are used to improve the given system. Numerical examples are presented to interprethow one can utilize the obtained results.� 2006 Elsevier Inc. All rights reserved.

Keywords: Gamma distribution; Reliability equivalence factor; Hot duplication; Cold duplication; Reduction method

1. Introduction

In reliability analysis, sometimes different system designs should be compared based on a reliability char-acteristic such as the reliability function or mean time to failure in case of no repairs. The concept of reliabilityequivalence has been introduced in Rade [1]. Rade [2–5] and Sarhan [6,7] have applied such concept to varioussystems. Rade [4,5] has calculated the reliability equivalence factors for a single component and for two inde-pendent and identical component series and parallel systems. He assumed that the reliability function of thesystem can be improved by three different methods as follows: (1) improving the quality of one or several com-ponents by decreasing their failure rates; (2) adding some hot redundant components to the system; and (3)adding some cold redundant components to the system.

Rade [4,5] and Sarhan [6] have used the survival function as the performance measure of the reliabilitysystem. Rade [4,5] has calculated the reliability equivalence factors for a single component and for two inde-pendent and identical component series and parallel systems. Sarhan [6] has obtained the reliability equiva-lence factors of n independent and non-identical components series system. He used the survival functionand MTTF as characteristics to compare different system designs. Sarhan [8,9] has applied the concept of reli-ability equivalence on a parallel system consisting of n independent and identical components.

All articles mentioned above are about components of exponential distribution. We hope to discuss morelife distributions, Gamma distribution for example. Different from the constant failure rate of exponential

0096-3003/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.07.016

* Corresponding author.E-mail addresses: [email protected] (Y. Xia), [email protected] (G. Zhang).

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568 Y. Xia, G. Zhang / Applied Mathematics and Computation 187 (2007) 567–573

distribution, Gamma distribution has a failure rate of function of time t. So the reliability equivalence factorsshould be generalized accordingly.

In the current study, we shall use the survival function to calculate the reliability equivalence factors forparallel system, consisting of n i.i.d. components. These components are assumed to be independently andidentically Gamma distributed.

This paper is organized as follows. In Section 2, we present the parallel system. The survival functions of theoriginal and improved systems are presented in Section 3. The generalized reliability equivalence factors arederived in Section 4. The b-fractiles of the original and improved systems are introduced in Section 5. Numer-ical results and conclusion are introduced in Section 6.

2. n-Component parallel system

The system considered here consists of n independent and identical components connected in parallel. Thelife time of component i, say Ti, is a Gamma distribution with the following reliability function:

RiðtÞ ¼Z 1

kt

ua�1

CðaÞ e�u du t P 0; k > 0; a > 0; i ¼ 1; 2; . . . n: ð1Þ

The reliability function of the parallel system which consists of n-independent components, denoted R(t),can be obtained as follows:

RðtÞ ¼ 1�Yn

i¼1

½1� RiðtÞ�: ð2Þ

To simplify, we denote the distribution function of Gamma distribution as /ðkt; aÞ ¼R kt

0ua�1

CðaÞ e�u du.

Using Eqs. (1) and (2), the reliability function of the system can be written as

RðtÞ ¼ 1� /ðkt; aÞn: ð3Þ

3. The improved systems

The reliability of the system can be improved according to one of the following three different methods:

(1) Reducing the failure rates of r, 1 6 r 6 n, of the components by the same factor. This method will becalled the reduction method.

(2) Assuming hot duplications of m, 1 6 m 6 n, of the components. This means that each of m of the com-ponents is duplicated by a hot redundant standby component. This method will be called the hot dupli-cation method.

(3) Assuming cold duplications of m, 1 6 m 6 n, of the components. This means that each of m of the com-ponents is duplicated by a cold redundant standby component. This method will be called the cold dupli-cation method.

Then we shall make equivalence between the improved systems obtained according to the reduction methodand the hot and cold duplication methods based on the value of the reliability function. Next, we give the reli-ability functions of the improved systems obtained according the above-described methods.

3.1. The reduction method

Assuming that the system is improved by improving r, 1 6 r 6 n, of its components according the reductionmethod. That is, the failure rates of r components are reduced from k(t) to r(t)k(t), 0 < r(t) < 1. Here, we con-sider reducing the failure rate by reducing the scale parameter of r components by the factor q only. LetR(r),q(t) denote the reliability function of the system improved by reducing the scale parameter of r compo-nents by the factor q. We can obtain R(r),q(t) to be

Y. Xia, G. Zhang / Applied Mathematics and Computation 187 (2007) 567–573 569

RðrÞ;qðtÞ ¼ 1�Z qkt

0

ua�1

CðaÞ e�u du

� �r Z kt

0

ua�1

CðaÞ e�u du

� �n�r

¼ 1� /ðqkt; aÞr/ðkt; aÞn�r: ð4Þ

3.2. The hot duplication method

Let us assume that the system is improved by improving m, 1 6 m 6 n, of its components according the hotduplication method. Let RH

ðmÞðtÞ denote the reliability function of the system improved by improving m of itscomponents by hot duplication. We can obtain RH

ðmÞðtÞ to be

RHðmÞðtÞ ¼ 1�

Z kt

0

ua�1

CðaÞ e�u du

� �nþm

¼ 1� /ðkt; aÞnþm: ð5Þ

3.3. The cold duplication method

Let us consider now that the system is improved by improving m, 1 6 m 6 n, of its components accordingthe cold duplication method. Let RC

ðmÞðtÞ denote the reliability function of the system improved by improving m

of its components by cold duplication. We can obtain RCðmÞðtÞ as follows:

RCðmÞðtÞ ¼ 1� ½1� RC

1 ðtÞ�m½1� R1ðtÞ�n�m

; ð6Þ

where RC1 ðtÞ denotes to the reliability of a system’s component after it was improved according to cold dupli-

cation method. Following the technique given in Ref. [10], we can obtain RC1 ðtÞ to be

RC1 ðtÞ ¼

Z 1

kt

u2a�1

Cð2aÞ e�u du ¼ 1� /ðkt; 2aÞ: ð7Þ

So we have

RCðmÞðtÞ ¼ 1� /ðkt; 2aÞm/ðkt; aÞn�m

: ð8Þ

4. Reliability equivalence factors

Now we give the definition of reliability equivalence factor.Def. A reliability equivalence factor is a factor by which a characteristic of components of a system design

has to be multiplied in order to reach equality of a characteristic of this design and a different design regardedas a standard Ref. [7].

As above, the reliability equivalence factor is defined as the factor by which the failure rates of some of thesystem’s components should be reduced in order to reach equality of the reliability of another better system.

Different from the constant failure rate of exponential distribution, the failure rate of Gamma distributionis a function about time t. Accordingly, the reliability equivalence factors of Gamma distribution is a functionabout time t.

For convenience of calculation, while failure rate is reduced by factor r(t), we consider the scale parameterof Gamma distribution reduced from k to qk, only. From the failure rate of Gamma distributionkðtÞ ¼ 1R1

0ð1þu

tÞa�1e�ku du

, we know

rðtÞkðtÞ ¼ 1R10 ð1þ u

t Þa�1e�qku du

: ð9Þ

Obviously, r(t) will increase as q increases, and they fall in interval (0, 1) also. In what follows, we will presenthow to calculate q only, and we obtain r(t) by taking q in Eq. (9). Next, we present some of reliability equiv-alence factors of the improved parallel system studied here.


4.1. Hot reliability equivalence factor rHðmÞ;ðrÞðb; tÞ

The hot reliability equivalence factor, say rHðmÞ;ðrÞðb; tÞ, is defined as that factor by which the failure rates of r

of the system’s components should be reduced in order to reach the reliability of that system which improvedby improving m of the original system’s components according hot duplication method. As mentioned above,the failure rate reduced by rH

ðmÞ;ðrÞðb; tÞ is equal to the scale parameter reduced from k to qHðmÞ;ðrÞðbÞk. That is,

q ¼ qHðmÞ;ðrÞðbÞ is the solution of the following system of two equations:

RðrÞ;qðtÞ ¼ b; RHðmÞðtÞ ¼ b; b 2 ð0; 1Þ ð10Þ

equally

1� /ðqkt; aÞr/ðkt; aÞn�r ¼ b; 1� /ðkt; aÞnþm ¼ b: ð11Þ
From the second equation of Eq. (11), we have
/ðkt; aÞ ¼ ð1� bÞ1

mþn ð12Þ

put Eq. (12) in the first equation of Eq. (11), we have

/ðqkt; aÞ ¼ ð1� bÞmþr

rðmþnÞ: ð13Þ

Given b, m, n, r, kt and qkt can both be obtained from Gamma distribution table. So we have q ¼ qHðmÞ;ðrÞðbÞ.

Hot reliability equivalence factor

rHðmÞ;ðrÞðb; tÞ ¼

R10ð1þ u

t Þa�1e�ku duR1

0ð1þ u

t Þa�1e�qku du

; ð14Þ

where q ¼ qHðmÞ;ðrÞðbÞ.

4.2. Cold reliability equivalence factor rCðmÞ;ðrÞðb; tÞ

The cold reliability equivalence factor, say rCðmÞ;ðrÞðb; tÞ, is defined as that factor by which the failure rates of r

of the system’s components should be reduced in order to reach the reliability of that system which improvedby improving m of the original system’s components according cold duplication method. As mentioned above,the failure rate reduced by rC

ðmÞ;ðrÞðb; tÞ is equal to the scale parameter reduced from k to qCðmÞ;ðrÞðbÞk. That is,

q ¼ qCðmÞ;ðrÞðbÞ is the solution of the following system of two equations:

RðrÞ;qðtÞ ¼ b; RCðmÞðtÞ ¼ b; b 2 ð0; 1Þ ð15Þ

take Eqs. (4) and (8) in Eq. (15) respectively, we have

1� /ðqkt; aÞr/ðkt; aÞn�r ¼ b; ð16Þ1� /ðkt; 2aÞm/ðkt; aÞn�m ¼ b: ð17Þ

In what follows, we prove there is only one solution q from Eqs. (16) and (17). Let

F ðq; ktÞ ¼ 1� b� /ðqkt; aÞr/ðkt; aÞn�r; ð18Þ

GðktÞ ¼ 1� b� /ðkt; 2aÞm/ðkt; aÞn�m: ð19Þ

We will prove G(kt) = 0 has the only one solution in (0,1) first. For /(kt, a) is increased monotonely accord-ing kt, G(kt) is decreased monotonely, and G(0) = 1�b > 0, G(1) = �b < 0, so G(kt) = 0 has the only onesolution in (0,1).

Next, we prove F(q, kt) = 0 has the only one solution in q 2 (0,1), when kt fits G(kt) = 0. In fact, F(q, kt) isdecreased monotonely according q, and F(0, kt) = 1 � b > 0, we can only prove F(1, kt) = 1 � b � /(kt, a)n < 0.


The failure rate function of Gamma distribution is

f ða; ktÞ ¼ � d ln½1� /ðkt; aÞ�dt

ð20Þ

thus the distribution function of C(a, k) can be written as

/ðkt; aÞ ¼ 1� e�R t

0f ða;kuÞdu

; ð21Þ

so we know, /(kt, a) has the same monotone to f(a, kt) in a. From Ref. [10]

f ða; ktÞ ¼ 1R10ð1þ u

t Þa�1e�ku du

; ð22Þ

f(a, kt) is decreased monotonely in a, so the /(kt, a). Now we have

/ðkt; 2aÞ < /ðkt; aÞ; ð23Þ

G(kt) = 0 = 1 � b � /(kt, 2a)m/(kt, a)n�m > 1 � b � /(kt, a)n = F(1, kt).So F(q, kt) = 0 has the only one solution q ¼ qC

ðmÞ;ðrÞðbÞ when q 2 (0, 1). Cold reliability equivalence factor

rCðmÞ;ðrÞðb; tÞ ¼

R10 ð1þ u

t Þa�1e�ku duR1

0ð1þ u

t Þa�1e�qku du

; ð24Þ

where q ¼ qCðmÞ;ðrÞðbÞ.

5. b-Fractiles

The b-fractiles of the original and improved systems are presented in this section. Let L(a, b) be the b-frac-tile of the original system in distribution of C(a, 1), so

R1Lða;bÞ

ua�1

CðaÞ e�u ¼ b. Let LH

ðmÞða; bÞ and LCðmÞða; bÞ denote,

respectively, to the b-fractile of the improved system obtained by improving m of the system’s componentsaccording to hot and cold duplication methods.

The fractile L(a, b) can be found by solving the following equation with respect to L:

RLk

� �¼ b: ð25Þ

The fractile LHðmÞða; bÞ can be deduced by solving the following equation with respect to L:

RHðmÞ

Lk

� �¼ b: ð26Þ

Also, the fractile LCðmÞða; bÞ can be deduced by solving the following equation with respect to L:

RCðmÞ

Lk

� �¼ b: ð27Þ

As we know, three equations above never have closed solution in L. Thus, to find out L, we have to use thenumerical technique method.

6. Numerical results and conclusion

Some numerical results are given in this section to illustrate how to interpret the theoretical results previ-ously obtained. In this example, we assume a parallel system consisting of n = 3 independent and identicalcomponents.

Table 1a = 3, n = 3, m = 1, b-Fractiles and qH

ð1ÞðrÞ, qCð1ÞðrÞ

b L(3, b) LHð1Þð3;bÞ LC

ð1Þð3;bÞ qHð1Þð1Þ qH

ð1Þð2Þ qHð1Þð3Þ qC

ð1Þð1Þ qCð1Þð2Þ qC

ð1Þð3Þ

0.1 6.7971 7.1732 9.4292 0.8727 0.9260 0.9476 0.5772 0.6700 0.72090.2 5.7967 6.1837 8.1287 0.8480 0.9116 0.9374 0.5470 0.6548 0.71310.3 5.1562 5.5504 7.2883 0.8276 0.8997 0.9290 0.5244 0.6437 0.70750.4 4.6581 5.0576 6.6313 0.8084 0.8884 0.9210 0.5046 0.6338 0.70240.5 4.2300 4.6335 6.0656 0.7891 0.8771 0.9129 0.4858 0.6243 0.69740.6 3.8356 4.2421 5.5435 0.7685 0.8648 0.9042 0.4668 0.6144 0.69190.7 3.4480 3.8563 5.0300 0.7450 0.8507 0.8941 0.4463 0.6034 0.68550.8 3.0359 3.4444 4.4832 0.7156 0.8330 0.8814 0.4221 0.5897 0.67720.9 2.5312 2.9361 3.8114 0.6723 0.8063 0.8621 0.3883 0.5695 0.6641







ð2Þð3Þ

0.1 6.7971 7.4622 10.498 0.8062 0.8786 0.9109 0.5116 0.5986 0.64750.2 5.7967 6.4809 9.2155 0.7703 0.8562 0.8944 0.4717 0.5726 0.62900.3 5.1562 5.8532 8.3777 0.7411 0.8379 0.8809 0.4416 0.5532 0.61550.4 4.6580 5.3646 7.7159 0.7141 0.8207 0.8683 0.4154 0.5362 0.60370.5 4.2300 4.9440 7.1393 0.6872 0.8035 0.8556 0.3906 0.5201 0.59250.6 3.8356 4.5553 6.6011 0.6588 0.7852 0.8420 0.3657 0.5036 0.58110.7 3.4480 4.1718 6.0648 0.6271 0.7644 0.8265 0.3393 0.4858 0.56850.8 3.0359 3.7612 5.4857 0.5883 0.7386 0.8072 0.3088 0.4646 0.55340.9 2.5312 3.2525 4.7613 0.5325 0.7004 0.7782 0.2678 0.4347 0.5316







ð3Þð3Þ

0.1 6.7971 7.6968 11.1394 0.7632 0.8449 0.8831 0.4803 0.5632 0.61020.2 5.7967 6.7218 9.8820 0.7210 0.8173 0.8624 0.4370 0.5326 0.58660.3 5.1562 6.0985 9.0600 0.6870 0.7949 0.8455 0.4044 0.5096 0.56910.4 4.6580 5.6134 8.4091 0.6558 0.7741 0.8298 0.3760 0.4895 0.55390.5 4.2300 5.1958 7.8404 0.6251 0.7535 0.8141 0.3494 0.4705 0.53950.6 3.8356 4.8097 7.3073 0.5930 0.7316 0.7975 0.3228 0.4512 0.52490.7 3.4480 4.4282 6.7735 0.5575 0.7071 0.7786 0.2947 0.4304 0.50900.8 3.0359 4.0194 6.1931 0.5148 0.6769 0.7553 0.2626 0.4059 0.49020.9 2.5312 3.5116 5.4597 0.4543 0.6327 0.7208 0.2202 0.3720 0.4636


We have written programs to calculate the b-fractiles and the changed parameter factors when a = 3, m,r = 1, 2, 3, and a = 5, 10, m = 1, r = 1, 2, 3.

From Tables 1–3 and comparing the b-fractiles of the original and improved systems, it is conformable forall parameters that RC

ðmÞðtÞ > RHðmÞðtÞ > RðtÞ, which presents that the improved systems are better than the ori-

ginal system, and that the cold duplication method is better than the hot duplication method (see Tables 2 and3) is omitted.

Based on the results summarized in Tables 1, 4, and 5 we can conclude that:As a increases, the b-fractiles of the improved systems increase, too, but qH

ðmÞ;ðrÞ and qCðmÞ;ðrÞ have different

tendencies of change, where qHðmÞ;ðrÞ increases slowly, and accordingly the hot reliability equivalence factor

rHðmÞ;ðrÞ increases slowly, too; while qC

ðmÞ;ðrÞ decreases quickly, the same thing happens with the cold reliabilityequivalence factor rC

ðmÞ;ðrÞ. This presents the difference of three improving methods. In actual applications,we can choose among the three methods according to different instance.



b L(5,b) LHð1Þð5;bÞ LC




ð1Þð3Þ

0.1 9.7429 10.1817 14.2674 0.8949 0.9390 0.9569 0.5636 0.6402 0.68290.2 8.5616 9.0211 12.6240 0.8756 0.9279 0.9491 0.5386 0.6286 0.67820.3 7.7932 8.2675 11.5377 0.8598 0.9188 0.9426 0.5200 0.6205 0.67550.4 7.1875 7.6739 10.6748 0.8450 0.9103 0.9366 0.5038 0.6137 0.67330.5 6.6603 7.1575 9.9218 0.8302 0.9017 0.9305 0.4885 0.6072 0.67130.6 6.1680 6.6752 9.2186 0.8144 0.8925 0.9240 0.4731 0.6006 0.66910.7 5.6773 6.1941 8.5191 0.7964 0.8820 0.9166 0.4565 0.5933 0.66640.8 5.1464 5.6727 7.7653 0.7742 0.8688 0.9072 0.4371 0.5844 0.66270.9 4.4804 5.0163 6.8253 0.7411 0.8491 0.8932 0.4102 0.5711 0.6564







ð1Þð3Þ

0.1 16.4552 17.0094 62.2930 0.9202 0.9539 0.9674 0.2281 0.2513 0.26420.2 14.9442 15.5355 61.2919 0.9064 0.9461 0.9619 0.2042 0.2298 0.24380.3 13.9436 14.5629 60.7030 0.8953 0.9397 0.9575 0.1876 0.2148 0.22970.4 13.1431 13.7867 60.2835 0.8849 0.9338 0.9533 0.1738 0.2024 0.21800.5 12.4369 13.1032 59.9573 0.8745 0.9278 0.9492 0.1613 0.1911 0.20740.6 11.7686 12.4572 59.6901 0.8634 0.9215 0.9447 0.1492 0.1802 0.19720.7 11.0924 11.8042 59.4637 0.8510 0.9144 0.9397 0.1368 0.1689 0.18650.8 10.3481 11.0859 59.2674 0.8356 0.9055 0.9335 0.1230 0.1563 0.17460.9 9.39247 10.1633 59.0939 0.8128 0.8923 0.9242 0.1053 0.1398 0.1589


References

[1] L. Rade, Reliability equivalence, studies in statistical quality control and reliability, mathematical statistics, Chalmers University ofTechnology, 1989.

[2] L. Rade, Reliability systems of 3-state components, studies in statistical quality control and reliability, mathematical statistics,Chalmers University of Technology, 1990.

[3] L. Rade, Performance measures for reliability systems with a cold standby with a random switch, studies in statistical quality controland reliability, mathematical statistics, Chalmers University of Technology, 1991.

[4] L. Rade, Reliability equivalence, Microelectron. Reliab. 33 (1993) 323–325.[5] L. Rade, Reliability survival equivalence, Microelectron. Reliab. 33 (1993) 881–894.[6] A.M. Sarhan, Reliability equivalence of independent and non-identical components series systems, Reliab. Eng. Saf. 67 (2000)

293–300.[7] A.M. Sarhan, Reliability equivalence with a basic series/parallel system, Appl. Math. Comput. 132 (2002) 115–133.[8] A.M. Sarhan, Reliability equivalence of a series–parallel system, Appl. Math. Comput. 154 (2004) 257–277.[9] A.M. Sarhan, Reliability equivalence factors of a parallel system, Reliab. Eng. Saf. 87 (2005) 405–411.

[10] Cao Jinhua, Chen Kan, Introduction to Reliability Mathematics, Science Press, Beijing, 1986.

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Reliability equivalence factors in Gamma distribution