- Email: [email protected]
On reliability analysis of a two-dependent-unit series system with a standby unit
Applied Mathematics and Computation 218 (2012) 7792–7797
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On reliability analysis of a two-dependent-unit series system with a standby unit Serkan Eryilmaz a,⇑, Fatih Tank b a b
Atilim University, Department of Industrial Engineering, 06836 Incek, Ankara, Turkey Ankara University, Faculty of Science, Department of Statistics, 06100 Tandogan, Ankara, Turkey
a r t i c l e
i n f o
Keywords: Copula Exchangeability Mean time to failure Positive dependence Reliability
a b s t r a c t In this paper we study a series system with two active components and a single cold standby unit. The two simultaneously working components are assumed to be dependent and this dependence is modeled by a copula function. In particular, we obtain an explicit expression for the mean time to failure of the system in terms of the copula function and marginal lifetime distributions. We also provide illustrative numerical results for different copula functions and marginal lifetime distributions. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Standby redundancy is an effective way to improve the reliability of an engineering system. In the study of system reliability under standby redundancy, the components are usually assumed to be independent. This assumption may not be valid in practice. Two components working in the common random environment may share the same load or may be subject to the same stress. In such a case, the lifetimes of the components are dependent. Most of the studies on systems with dependent components focus on the case when there is no standby component [7,8,1,9]. The notion of copulas has been found to be useful for modeling dependence in the context of system reliability [6,4,9,2]. In system reliability, copulas are used to create a multivariate lifetime distribution for modeling dependence among the components. In this paper, we study two-unit series system with a single cold standby component whenever the two simultaneously working components are dependent through a given copula. In a series system with a single standby unit, the system consists of two serially connected working units and a single cold standby unit. The unit is said to be in the case of cold standby if it does not fail while in standby. When one of the working unit fails, then a cold standby unit is immediately put into operation. Some recent discussions on systems with standby component are in Jia and Wu [3], Wu and Wu [12], Wang and Zhang [11]. The paper is organized as follows. In Section 2, we define the lifetime of the system and compute its expected value, i.e. mean time to failure (MTTF). This section also includes bounds for the MTTF when the components are positively (negatively) quadrant dependent. Section 3 contains extensive numerical calculations for different copula functions and marginal lifetime distributions.
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (S. Eryilmaz). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.01.046
S. Eryilmaz, F. Tank / Applied Mathematics and Computation 218 (2012) 7792–7797
7793
2. Lifetime and MTTF of the system Consider a two-unit series system with one cold standby component. Let X i and Z represent respectively the lifetime of the ith active component (i ¼ 1; 2) and the lifetime of the standby component. Then the lifetime of the system is represented as
T ¼ minðX 1 ; X 2 Þ þ minðX 1 ; ZÞ; where X 1 represents the residual lifetime of surviving active component after the first failure in the system, i.e. st
X 1 ¼ ðX 1 minðX 1 ; X 2 ÞjX 1 > minðX 1 ; X 2 ÞÞ: The dependence between the components working in the common random environment is inevitable, and if the components are identical the common random environment makes them exchangeable dependent. In the present paper we model this dependence by the symmetric copula function C; i.e. Cðu; v Þ ¼ Cðv ; uÞ for all u; v 2 ½0; 1. Let X 1 and X 2 be exchangeable dependent with an absolutely continuous joint distribution and this dependence is modeled by the copula function Cðu; v Þ: That is, the joint cumulative distribution function (cdf) of X 1 and X 2 is given by
Fðt 1 ; t 2 Þ ¼ PfX 1 6 t 1 ; X 2 6 t2 g ¼ CðFðt1 Þ; Fðt2 ÞÞ; where FðtÞ ¼ PfX i 6 tg; i ¼ 1; 2: In such a case Fðt 1 ; t 2 Þ ¼ Fðt2 ; t1 Þ; i.e. X 1 and X 2 are exchangeable. Under these assumptions, the cdf of the random variable X 1 is found to be
F ðtÞ ¼ P X 1 6 t ¼ 2PfX 1 6 X 2 þ tg 1
ð1Þ
for t P 0 (see Appendix). The latter cdf can be computed from the following equation using the joint distribution of X 1 and X 2 (see Appendix).
F ðtÞ ¼ 2
Z
1
0
Z
yþt
cðFðxÞ; FðyÞÞdFðxÞdFðyÞ 1;
ð2Þ
0
where
cðu; v Þ ¼
@2 Cðu; v Þ: @u@ v
Assume that the cold standby component has the same marginal distribution with X i s, i.e. PfZ 6 tg ¼ FðtÞ: After the first failure in the system, the standby component is put into operation and it works together with the remaining component whose cdf is given by (1). Thus the joint distribution of X 1 and Z are no longer exchangeable. They are again assumed to be dependent with the joint cdf
P X 1 6 t1 ; Z 6 t2 ¼ CðF ðt 1 Þ; Fðt2 ÞÞ for t 1 ; t 2 P 0: The joint survival functions of ðX 1 ; X 2 Þ and ðX 1 ; ZÞ are given respectively by
PfX 1 > t 1 ; X 2 > t2 g ¼ 1 Fðt1 Þ Fðt2 Þ þ CðFðt 1 Þ; Fðt 2 ÞÞ
ð3Þ
and
P X 1 > t 1 ; Z > t2 ¼ 1 F ðt1 Þ Fðt 2 Þ þ CðF ðt 1 Þ; Fðt 2 ÞÞ: Thus using (3) and (4), the survival functions of minðX 1 ; X 2 Þ and
ð4Þ minðX 1 ; ZÞ
are found to be
PfminðX 1 ; X 2 Þ > t g ¼ P fX 1 > t; X 2 > t g ¼ 1 2FðtÞ þ CðFðtÞ; FðtÞÞ and
P minðX 1 ; ZÞ > t ¼ 1 F ðtÞ FðtÞ þ CðF ðtÞ; FðtÞÞ for t P 0: Using these survival functions, the MTTF of the series system with a single cold standby unit is computed from
EðTÞ ¼
Z
1
½1 2FðtÞ þ CðFðtÞ; FðtÞÞdt þ
0
Z
1
½1 F ðtÞ FðtÞ þ CðF ðtÞ; FðtÞÞdt:
ð5Þ
0
There are various notions of dependence. A pair of random variables ðX; YÞ is said to be positively quadrant dependent (PQD) if
PfX 6 t 1 ; Y 6 t2 g P PfX 6 t1 gP fY 6 t 2 g
ð6Þ
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S. Eryilmaz, F. Tank / Applied Mathematics and Computation 218 (2012) 7792–7797
for all t 1 and t 2 ; or equivalently
PfX > t1 ; Y > t2 g P PfX > t 1 gPfY > t2 g: If the inequality in (6) is reversed, then ðX; YÞ is negatively quadrant dependent (NQD). In terms of a copula function, the inequality (6) is equivalent to
Cðu; v Þ P uv for all u; v 2 ½0; 1. If ðX 1 ; X 2 Þ and ðX 1 ; ZÞ are PQD, then (see Appendix)
EðTÞ P
Z
1
ð1 FðtÞÞ2 dt þ
Z
0
1
ð1 F ðtÞÞð1 FðtÞÞdt:
ð7Þ
0
The right hand-side of inequality (7) is the MTTF of the series system with one cold standby unit when all components are independent. Similarly, if ðX 1 ; X 2 Þ and ðX 1 ; ZÞ are NQD, then
EðTÞ 6
Z
1
ð1 FðtÞÞ2 dt þ
0
Z
1
ð1 F ðtÞÞð1 FðtÞÞdt:
0
It should be noted that similar bounds have been obtained in Kotz et al. [5] for the two unit parallel system without a standby. 3. Illustrations Assume that the dependence between the components is modeled by FGM copula which is given by
C 1 ðu; v Þ ¼ uv f1 þ að1 uÞð1 v Þg for 1 6 a 6 1 and u; v 2 ½0; 1: This model includes independence for a ¼ 0 and has PQD (NQD) property when a P(6)0: It is easy to see that,
c1 ðu; v Þ ¼
@2 C 1 ðu; v Þ ¼ 1 þ að1 2uÞð1 2v Þ: @u@ v
Using (2),
F ðtÞ ¼ 2
Z
¼2
1
Z0
Z
yþt
½1 þ að1 2FðxÞÞð1 2FðyÞÞdFðxÞdFðyÞ 1
0 1
Fðy þ tÞ½1 þ að1 2FðyÞÞð1 Fðy þ tÞÞdFðyÞ 1
0
for t P 0: Let the marginal distribution F be exponential with cdf FðtÞ ¼ 1 ekt ; t P 0: Then
a
F ðtÞ ¼ 1 ekt þ ekt ð1 ekt Þ 3 for t P 0: It should be noted that when a ¼ 0; i.e. X 1 and X 2 are independent we have F ðtÞ ¼ 1 ekt . That is, the residual lifetime distribution of surviving component does not change and it has again exponential distribution. This is a direct consequence of memoryless property of exponential distribution. For the exponential marginal distribution, the survival function of minðX 1 ; X 2 Þ is found to be
PfminðX 1 ; X 2 Þ > t g ¼ 1 2FðtÞ þ C 1 ðFðtÞ; FðtÞÞ ¼ ð1 þ aÞe2kt 2ae3kt þ ae4kt
ð8Þ
for t P 0; which is a generalized mixture of three exponential survival functions with the weights 1 þ a; 2a and a. On the other hand, the survival function of minðX 1 ; ZÞ is
P minðX 1 ; ZÞ > t ¼ 1 F ðtÞ FðtÞ þ C 1 ðF ðtÞ; FðtÞÞ h i h i a a ¼ 1 1 ekt þ ekt ð1 ekt Þ ð1 ekt Þ þ 1 ekt þ ekt ð1 ekt Þ ð1 ekt Þ 3 3 h i n o a kt kt kt kt 1 þ a e e ð1 e Þ e 3 for t P 0: Using (5) with (8) and (9), the MTTF of the system is found to be
EðTÞ ¼ 1 6 a 6 1:
1 a a2 a3 ; 1þ k 9 180 540
ð9Þ
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S. Eryilmaz, F. Tank / Applied Mathematics and Computation 218 (2012) 7792–7797 Table 1 MTTF values with and without standby component for FGM and Ali–Mikhail–Haq copulas (Exponential marginals).
a
MTTF 1
MTTF 2
MTTF 01
MTTF 02
0 0.3 0.5 0.7 0.9
1.0000 1.0328 1.0539 1.0744 1.0941
1.0000 1.0344 1.0585 1.0836 1.1094
0.5000 0.5250 0.5417 0.5583 0.5750
0.5000 0.5267 0.5468 0.5697 0.5969
Table 2 MTTF values with and without standby component for FGM and Ali–Mikhail–Haq copulas (Weibull marginals).
a
MTTF 1
MTTF 2
MTTF 01
MTTF 02
0 0.3 0.5 0.7 0.9
1.0444 1.0513 1.0550 1.0581 1.0603
1.0444 1.0513 1.0547 1.0564 1.0541
0.6267 0.6406 0.6499 0.6592 0.6685
0.6267 0.6419 0.6538 0.6682 0.6868
Next assume that the dependence between the components is modeled by Ali–Mikhail–Haq copula given by
C 2 ðu; v Þ ¼
uv 1 að1 uÞð1 v Þ
for 1 6 a < 1 and u; v 2 ½0; 1. For the case a ¼ 0 we obtain independence copula. The joint density associated with C 2 ðu; v Þ is found to be
c2 ðu; v Þ ¼
ð1 aÞ2 þ a2 ðuv u v Þ þ aðu þ v þ uv Þ ½1 að1 uÞð1 v Þ3
:
Since c2 ðu; v Þ has a complicated form, it is not easy to get an exact analytical expression for EðTÞ. However, it can be computed by means of numerical integration. The bivariate models considered in reliability theory are mostly PQD. In Table 1, we compute the MTTF of the system for FGM (MTTF1 ) and Ali–Mikhail–Haq (MTTF2 ) copulas when the marginal distributions of the components are exponential with unit mean and when the dependence parameter a takes nonnegative values. This Table also includes the MTTF values without a standby component, i.e. EðminðX 1 ; X 2 ÞÞ for FGM (MTTF01 ) and Ali–Mikhail–Haq (MTTF02 ) copulas. The MTTF values for a > 0 are greater than the MTTF value when a ¼ 0 (independence case). This is consistent with the inequality (7). On the other hand, we observe that the MTTF value under Ali–Mikhail–Haq copula is greater than the MTTF under FGM copula when there is no standby component. This can be explained as follows. For two copulas C 1 and C 2 , C 1 is said to be less positive dependent than a copula C 2 (written as C 1 6PQD C 2 ) if C 1 ðu; v Þ 6 C 2 ðu; v Þ for all u; v 2 ½0; 1. We have the following relation between Ali–Mikhail–Haq and FGM copulas (see, Nelsen [10]).
C 2 ðu; v Þ ¼ uv f1 þ að1 uÞð1 v Þg þ uv
1 X
½að1 uÞð1 v Þi ¼ C 1 ðu; v Þ þ uv
i¼2
1 X ½að1 uÞð1 v Þi P C 1 ðu; v Þ; i¼2
which implies that FGM copula is less positive dependent than Ali–Mikhail–Haq copula. That is, if C 1 6PQD C 2 then
MTTF 01 ¼ EðminðX 1 ; X 2 ÞjC 1 Þ ¼ ¼ EðminðX 1 ; X 2 ÞjC 2 Þ ¼
Z
1
½1 2FðtÞ þ C 1 ðFðtÞ; FðtÞÞdt 6
0
Z
1
½1 2FðtÞ þ C 2 ðFðtÞ; FðtÞÞ dt
0
MTTF 02 :
Thus the MTTF (without standby component) under FGM model is always a lower bound for the MTTF under Ali–Mikhail– Haq copula model as soon as these two models have the common dependence parameter. However this is not generally true for a system with a standby component since in this case the cdf F in the second part of (5) also depends on the copula function. In Table 2, we present the corresponding MTTF values for FGM and Ali–Mikhail–Haq copulas when the marginal 2 distributions of the components are Weibull with cdf FðtÞ ¼ 1 et ; t P 0: From Table 2, for a ¼ 0:9; we observe that the MTTF (with standby component) for FGM model is larger than the case of Ali–Mikhail–Haq model. It is obvious that both copula models are increasing in a, i.e. C i ðaÞ6PQD C i ða0 Þ for a 6 a0 ; i ¼ 1; 2: As it can be seen from the Tables 1 and 2, the more dependence between the components the larger MTTF. 4. Concluding Remarks In this paper, we have studied a series system with a single standby unit when the components are dependent. The dependence between the components has been modeled by copulas. We have presented explicit expression for the MTTF
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S. Eryilmaz, F. Tank / Applied Mathematics and Computation 218 (2012) 7792–7797
of the system consisting of two components. The findings of the paper are important and novel from various perspectives. This is the first study which considers the effect of components’ dependence through a copula function in the context of systems with a standby unit. In most real life situations, the components working in the common environment are dependent and this dependence should be considered in system design and management from a practical point of view. An engineer may want know how the system’s lifetime is effected by the dependence among the components. The results of the paper, with some modifications can be extended to a series system with n components. In this case, the lifetime of the system can be represented as
T ¼ minðX 1 ; . . . ; X n Þ þ minðX 1 ; . . . ; X n1 ; ZÞ; where
X i stðX i minðX 1 ; . . . ; X n ÞjX i > minðX 1 ; . . . ; X n ÞÞ: ¼
The cdf of X i is found to be
P X i 6 t ¼
1 ½nP fX 1 6 X 2 þ t; . . . ; X 1 6 X n þ tg 1: n1
Similar results can be obtained by choosing an n-variate copula function Cðu1 ; . . . ; un Þ:. 5. Appendix Proof of (1): For t P 0,
F ðtÞ ¼ P X 1 6 t ¼ PfX 1 minðX 1 ; X 2 Þ 6 tjX 1 > minðX 1 ; X 2 Þg ¼
PfminðX 1 ; X 2 Þ < X 1 6 t þ minðX 1 ; X 2 Þg P fX 1 > minðX 1 ; X 2 Þg
¼
PfX 1 6 t þ minðX 1 ; X 2 Þg PfX 1 6 minðX 1 ; X 2 Þg PfX 1 > minðX 1 ; X 2 Þg
¼
PfX 1 6 X 2 þ tg PfX 1 6 X 2 g : PfX 1 > X 2 g
Since X 1 and X 2 have absolutely continuous exchangeable joint distribution,
PfX 1 6 X 2 g ¼ PfX 1 > X 2 g ¼ 1=2: Thus the required result is obtained. h Proof of (2): It is obvious that
F ðtÞ ¼ 2
Z
1 0
Z
yþt
f ðx; yÞdxdy 1;
0
where f ðt 1 ; t 2 Þ is the joint density of X 1 and X 2 : The proof follows noting that
f ðt 1 ; t 2 Þ ¼
@2 @2 Fðt 1 ; t2 Þ ¼ CðFðt1 Þ; Fðt 2 ÞÞ ¼ cðFðt 1 Þ; Fðt 2 ÞÞf ðt1 Þf ðt 2 Þ; @t 1 @t2 @t1 @t2 2
@ where f is the pdf associated with F and cðu; v Þ ¼ @u@ v Cðu; v Þ: Proof of (7): If ðX 1 ; X 2 Þ is PQD, then
CðFðtÞ; FðtÞÞ P F 2 ðtÞ and hence we have
EðminðX 1 ; X 2 ÞÞ ¼
Z
1
ð1 2FðtÞÞdt þ 0
Z
1
CðFðtÞ; FðtÞÞdt P
0
Similarly, if ðX 1 ; ZÞ is PQD, then
EðminðX 1 ; ZÞÞ ¼ ¼
Z
1
½1 F ðtÞ FðtÞdt þ
Z 0
1
1
ð1 2FðtÞÞdt þ
0
0
Z
Z
1
CðF ðtÞ; FðtÞÞdt P
Z
1
F 2 ðtÞdt ¼
Z
0
Z
1
½1 F ðtÞ FðtÞdt þ
0
ð1 F ðtÞÞð1 FðtÞÞdt:
0
Thus the inequality (7) is proved since EðTÞ ¼ EðminðX 1 ; X 2 ÞÞ þ EðminðX 1 ; ZÞÞ. h
1
ð1 FðtÞÞ2 dt:
0
Z 0
1
F ðtÞFðtÞdt
S. Eryilmaz, F. Tank / Applied Mathematics and Computation 218 (2012) 7792–7797
7797
Acknowledgement The authors thank the referees for their useful comments and suggestions. References [1] S. Eryilmaz, Reliability properties of consecutive k-out-of-n systems of arbitrarily dependent components, Reliability Engineering and System Safety 94 (2009) 350–356. [2] S. Eryilmaz, Estimation in coherent reliability systems through copulas, Reliability Engineering and System Safety 96 (2011) 564–568. [3] J. Jia, S. Wu, Optimizing replacement policy for a cold-standby system with waiting repair times, Applied Mathematics and Computation 214 (2009) 133–141. [4] X. Jia, L. Cui, J. Yan, A study on the reliability of consecutive k-out-of-n: G systems based on copula, Communications in Statistics – Theory and Methods 39 (2010) 2455–2472. [5] S. Kotz, C.D. Lai, M. Xie, On the effect of redundancy for systems with dependent components, IIE Transactions 35 (2003) 1103–1110. [6] C. Lai, M. Xie, Concepts of stochastic dependence in reliability analysis, in: H. Pham (Ed.), Handbook of Reliability Engineering, Springer, 2003, pp. 141– 156. [7] J. Navarro, J.M. Ruiz, C.J. Sandoval, A note on comparisons among coherent systems with dependent components using signatures, Statistics and Probability Letters 72 (2005) 179–185. [8] J. Navarro, J.M. Ruiz, C.J. Sandoval, Properties of coherent systems with dependent components, Communications in Statistics – Theory and Methods 36 (2007) 175–191. [9] J. Navarro, F. Spizzichino, Comparisons of series and parallel systems with components sharing the same copula, Applied Stochastic Models in Business and Industry 26 (2010) 775–791. [10] R.B. Nelsen, An Introduction to Copulas, second ed., Springer-Verlag, New York, 2006. [11] G.J. Wang, Y.L. Zhang, A bivariate optimal replacement policy for a cold standby repairable system with preventive repair, Applied Mathematics and Computation 218 (2011) 3158–3165. [12] Q. Wu, S. Wu, Reliability analysis of two-unit cold standby repairable systems under Poisson shocks, Applied Mathematics and Computation 218 (2011) 171–182.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On reliability analysis of a two-dependent-unit series system with a standby unit Serkan Eryilmaz a,⇑, Fatih Tank b a b
Atilim University, Department of Industrial Engineering, 06836 Incek, Ankara, Turkey Ankara University, Faculty of Science, Department of Statistics, 06100 Tandogan, Ankara, Turkey
a r t i c l e
i n f o
Keywords: Copula Exchangeability Mean time to failure Positive dependence Reliability
a b s t r a c t In this paper we study a series system with two active components and a single cold standby unit. The two simultaneously working components are assumed to be dependent and this dependence is modeled by a copula function. In particular, we obtain an explicit expression for the mean time to failure of the system in terms of the copula function and marginal lifetime distributions. We also provide illustrative numerical results for different copula functions and marginal lifetime distributions. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Standby redundancy is an effective way to improve the reliability of an engineering system. In the study of system reliability under standby redundancy, the components are usually assumed to be independent. This assumption may not be valid in practice. Two components working in the common random environment may share the same load or may be subject to the same stress. In such a case, the lifetimes of the components are dependent. Most of the studies on systems with dependent components focus on the case when there is no standby component [7,8,1,9]. The notion of copulas has been found to be useful for modeling dependence in the context of system reliability [6,4,9,2]. In system reliability, copulas are used to create a multivariate lifetime distribution for modeling dependence among the components. In this paper, we study two-unit series system with a single cold standby component whenever the two simultaneously working components are dependent through a given copula. In a series system with a single standby unit, the system consists of two serially connected working units and a single cold standby unit. The unit is said to be in the case of cold standby if it does not fail while in standby. When one of the working unit fails, then a cold standby unit is immediately put into operation. Some recent discussions on systems with standby component are in Jia and Wu [3], Wu and Wu [12], Wang and Zhang [11]. The paper is organized as follows. In Section 2, we define the lifetime of the system and compute its expected value, i.e. mean time to failure (MTTF). This section also includes bounds for the MTTF when the components are positively (negatively) quadrant dependent. Section 3 contains extensive numerical calculations for different copula functions and marginal lifetime distributions.
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (S. Eryilmaz). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.01.046
S. Eryilmaz, F. Tank / Applied Mathematics and Computation 218 (2012) 7792–7797
7793
2. Lifetime and MTTF of the system Consider a two-unit series system with one cold standby component. Let X i and Z represent respectively the lifetime of the ith active component (i ¼ 1; 2) and the lifetime of the standby component. Then the lifetime of the system is represented as
T ¼ minðX 1 ; X 2 Þ þ minðX 1 ; ZÞ; where X 1 represents the residual lifetime of surviving active component after the first failure in the system, i.e. st
X 1 ¼ ðX 1 minðX 1 ; X 2 ÞjX 1 > minðX 1 ; X 2 ÞÞ: The dependence between the components working in the common random environment is inevitable, and if the components are identical the common random environment makes them exchangeable dependent. In the present paper we model this dependence by the symmetric copula function C; i.e. Cðu; v Þ ¼ Cðv ; uÞ for all u; v 2 ½0; 1. Let X 1 and X 2 be exchangeable dependent with an absolutely continuous joint distribution and this dependence is modeled by the copula function Cðu; v Þ: That is, the joint cumulative distribution function (cdf) of X 1 and X 2 is given by
Fðt 1 ; t 2 Þ ¼ PfX 1 6 t 1 ; X 2 6 t2 g ¼ CðFðt1 Þ; Fðt2 ÞÞ; where FðtÞ ¼ PfX i 6 tg; i ¼ 1; 2: In such a case Fðt 1 ; t 2 Þ ¼ Fðt2 ; t1 Þ; i.e. X 1 and X 2 are exchangeable. Under these assumptions, the cdf of the random variable X 1 is found to be
F ðtÞ ¼ P X 1 6 t ¼ 2PfX 1 6 X 2 þ tg 1
ð1Þ
for t P 0 (see Appendix). The latter cdf can be computed from the following equation using the joint distribution of X 1 and X 2 (see Appendix).
F ðtÞ ¼ 2
Z
1
0
Z
yþt
cðFðxÞ; FðyÞÞdFðxÞdFðyÞ 1;
ð2Þ
0
where
cðu; v Þ ¼
@2 Cðu; v Þ: @u@ v
Assume that the cold standby component has the same marginal distribution with X i s, i.e. PfZ 6 tg ¼ FðtÞ: After the first failure in the system, the standby component is put into operation and it works together with the remaining component whose cdf is given by (1). Thus the joint distribution of X 1 and Z are no longer exchangeable. They are again assumed to be dependent with the joint cdf
P X 1 6 t1 ; Z 6 t2 ¼ CðF ðt 1 Þ; Fðt2 ÞÞ for t 1 ; t 2 P 0: The joint survival functions of ðX 1 ; X 2 Þ and ðX 1 ; ZÞ are given respectively by
PfX 1 > t 1 ; X 2 > t2 g ¼ 1 Fðt1 Þ Fðt2 Þ þ CðFðt 1 Þ; Fðt 2 ÞÞ
ð3Þ
and
P X 1 > t 1 ; Z > t2 ¼ 1 F ðt1 Þ Fðt 2 Þ þ CðF ðt 1 Þ; Fðt 2 ÞÞ: Thus using (3) and (4), the survival functions of minðX 1 ; X 2 Þ and
ð4Þ minðX 1 ; ZÞ
are found to be
PfminðX 1 ; X 2 Þ > t g ¼ P fX 1 > t; X 2 > t g ¼ 1 2FðtÞ þ CðFðtÞ; FðtÞÞ and
P minðX 1 ; ZÞ > t ¼ 1 F ðtÞ FðtÞ þ CðF ðtÞ; FðtÞÞ for t P 0: Using these survival functions, the MTTF of the series system with a single cold standby unit is computed from
EðTÞ ¼
Z
1
½1 2FðtÞ þ CðFðtÞ; FðtÞÞdt þ
0
Z
1
½1 F ðtÞ FðtÞ þ CðF ðtÞ; FðtÞÞdt:
ð5Þ
0
There are various notions of dependence. A pair of random variables ðX; YÞ is said to be positively quadrant dependent (PQD) if
PfX 6 t 1 ; Y 6 t2 g P PfX 6 t1 gP fY 6 t 2 g
ð6Þ
7794
S. Eryilmaz, F. Tank / Applied Mathematics and Computation 218 (2012) 7792–7797
for all t 1 and t 2 ; or equivalently
PfX > t1 ; Y > t2 g P PfX > t 1 gPfY > t2 g: If the inequality in (6) is reversed, then ðX; YÞ is negatively quadrant dependent (NQD). In terms of a copula function, the inequality (6) is equivalent to
Cðu; v Þ P uv for all u; v 2 ½0; 1. If ðX 1 ; X 2 Þ and ðX 1 ; ZÞ are PQD, then (see Appendix)
EðTÞ P
Z
1
ð1 FðtÞÞ2 dt þ
Z
0
1
ð1 F ðtÞÞð1 FðtÞÞdt:
ð7Þ
0
The right hand-side of inequality (7) is the MTTF of the series system with one cold standby unit when all components are independent. Similarly, if ðX 1 ; X 2 Þ and ðX 1 ; ZÞ are NQD, then
EðTÞ 6
Z
1
ð1 FðtÞÞ2 dt þ
0
Z
1
ð1 F ðtÞÞð1 FðtÞÞdt:
0
It should be noted that similar bounds have been obtained in Kotz et al. [5] for the two unit parallel system without a standby. 3. Illustrations Assume that the dependence between the components is modeled by FGM copula which is given by
C 1 ðu; v Þ ¼ uv f1 þ að1 uÞð1 v Þg for 1 6 a 6 1 and u; v 2 ½0; 1: This model includes independence for a ¼ 0 and has PQD (NQD) property when a P(6)0: It is easy to see that,
c1 ðu; v Þ ¼
@2 C 1 ðu; v Þ ¼ 1 þ að1 2uÞð1 2v Þ: @u@ v
Using (2),
F ðtÞ ¼ 2
Z
¼2
1
Z0
Z
yþt
½1 þ að1 2FðxÞÞð1 2FðyÞÞdFðxÞdFðyÞ 1
0 1
Fðy þ tÞ½1 þ að1 2FðyÞÞð1 Fðy þ tÞÞdFðyÞ 1
0
for t P 0: Let the marginal distribution F be exponential with cdf FðtÞ ¼ 1 ekt ; t P 0: Then
a
F ðtÞ ¼ 1 ekt þ ekt ð1 ekt Þ 3 for t P 0: It should be noted that when a ¼ 0; i.e. X 1 and X 2 are independent we have F ðtÞ ¼ 1 ekt . That is, the residual lifetime distribution of surviving component does not change and it has again exponential distribution. This is a direct consequence of memoryless property of exponential distribution. For the exponential marginal distribution, the survival function of minðX 1 ; X 2 Þ is found to be
PfminðX 1 ; X 2 Þ > t g ¼ 1 2FðtÞ þ C 1 ðFðtÞ; FðtÞÞ ¼ ð1 þ aÞe2kt 2ae3kt þ ae4kt
ð8Þ
for t P 0; which is a generalized mixture of three exponential survival functions with the weights 1 þ a; 2a and a. On the other hand, the survival function of minðX 1 ; ZÞ is
P minðX 1 ; ZÞ > t ¼ 1 F ðtÞ FðtÞ þ C 1 ðF ðtÞ; FðtÞÞ h i h i a a ¼ 1 1 ekt þ ekt ð1 ekt Þ ð1 ekt Þ þ 1 ekt þ ekt ð1 ekt Þ ð1 ekt Þ 3 3 h i n o a kt kt kt kt 1 þ a e e ð1 e Þ e 3 for t P 0: Using (5) with (8) and (9), the MTTF of the system is found to be
EðTÞ ¼ 1 6 a 6 1:
1 a a2 a3 ; 1þ k 9 180 540
ð9Þ
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S. Eryilmaz, F. Tank / Applied Mathematics and Computation 218 (2012) 7792–7797 Table 1 MTTF values with and without standby component for FGM and Ali–Mikhail–Haq copulas (Exponential marginals).
a
MTTF 1
MTTF 2
MTTF 01
MTTF 02
0 0.3 0.5 0.7 0.9
1.0000 1.0328 1.0539 1.0744 1.0941
1.0000 1.0344 1.0585 1.0836 1.1094
0.5000 0.5250 0.5417 0.5583 0.5750
0.5000 0.5267 0.5468 0.5697 0.5969
Table 2 MTTF values with and without standby component for FGM and Ali–Mikhail–Haq copulas (Weibull marginals).
a
MTTF 1
MTTF 2
MTTF 01
MTTF 02
0 0.3 0.5 0.7 0.9
1.0444 1.0513 1.0550 1.0581 1.0603
1.0444 1.0513 1.0547 1.0564 1.0541
0.6267 0.6406 0.6499 0.6592 0.6685
0.6267 0.6419 0.6538 0.6682 0.6868
Next assume that the dependence between the components is modeled by Ali–Mikhail–Haq copula given by
C 2 ðu; v Þ ¼
uv 1 að1 uÞð1 v Þ
for 1 6 a < 1 and u; v 2 ½0; 1. For the case a ¼ 0 we obtain independence copula. The joint density associated with C 2 ðu; v Þ is found to be
c2 ðu; v Þ ¼
ð1 aÞ2 þ a2 ðuv u v Þ þ aðu þ v þ uv Þ ½1 að1 uÞð1 v Þ3
:
Since c2 ðu; v Þ has a complicated form, it is not easy to get an exact analytical expression for EðTÞ. However, it can be computed by means of numerical integration. The bivariate models considered in reliability theory are mostly PQD. In Table 1, we compute the MTTF of the system for FGM (MTTF1 ) and Ali–Mikhail–Haq (MTTF2 ) copulas when the marginal distributions of the components are exponential with unit mean and when the dependence parameter a takes nonnegative values. This Table also includes the MTTF values without a standby component, i.e. EðminðX 1 ; X 2 ÞÞ for FGM (MTTF01 ) and Ali–Mikhail–Haq (MTTF02 ) copulas. The MTTF values for a > 0 are greater than the MTTF value when a ¼ 0 (independence case). This is consistent with the inequality (7). On the other hand, we observe that the MTTF value under Ali–Mikhail–Haq copula is greater than the MTTF under FGM copula when there is no standby component. This can be explained as follows. For two copulas C 1 and C 2 , C 1 is said to be less positive dependent than a copula C 2 (written as C 1 6PQD C 2 ) if C 1 ðu; v Þ 6 C 2 ðu; v Þ for all u; v 2 ½0; 1. We have the following relation between Ali–Mikhail–Haq and FGM copulas (see, Nelsen [10]).
C 2 ðu; v Þ ¼ uv f1 þ að1 uÞð1 v Þg þ uv
1 X
½að1 uÞð1 v Þi ¼ C 1 ðu; v Þ þ uv
i¼2
1 X ½að1 uÞð1 v Þi P C 1 ðu; v Þ; i¼2
which implies that FGM copula is less positive dependent than Ali–Mikhail–Haq copula. That is, if C 1 6PQD C 2 then
MTTF 01 ¼ EðminðX 1 ; X 2 ÞjC 1 Þ ¼ ¼ EðminðX 1 ; X 2 ÞjC 2 Þ ¼
Z
1
½1 2FðtÞ þ C 1 ðFðtÞ; FðtÞÞdt 6
0
Z
1
½1 2FðtÞ þ C 2 ðFðtÞ; FðtÞÞ dt
0
MTTF 02 :
Thus the MTTF (without standby component) under FGM model is always a lower bound for the MTTF under Ali–Mikhail– Haq copula model as soon as these two models have the common dependence parameter. However this is not generally true for a system with a standby component since in this case the cdf F in the second part of (5) also depends on the copula function. In Table 2, we present the corresponding MTTF values for FGM and Ali–Mikhail–Haq copulas when the marginal 2 distributions of the components are Weibull with cdf FðtÞ ¼ 1 et ; t P 0: From Table 2, for a ¼ 0:9; we observe that the MTTF (with standby component) for FGM model is larger than the case of Ali–Mikhail–Haq model. It is obvious that both copula models are increasing in a, i.e. C i ðaÞ6PQD C i ða0 Þ for a 6 a0 ; i ¼ 1; 2: As it can be seen from the Tables 1 and 2, the more dependence between the components the larger MTTF. 4. Concluding Remarks In this paper, we have studied a series system with a single standby unit when the components are dependent. The dependence between the components has been modeled by copulas. We have presented explicit expression for the MTTF
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S. Eryilmaz, F. Tank / Applied Mathematics and Computation 218 (2012) 7792–7797
of the system consisting of two components. The findings of the paper are important and novel from various perspectives. This is the first study which considers the effect of components’ dependence through a copula function in the context of systems with a standby unit. In most real life situations, the components working in the common environment are dependent and this dependence should be considered in system design and management from a practical point of view. An engineer may want know how the system’s lifetime is effected by the dependence among the components. The results of the paper, with some modifications can be extended to a series system with n components. In this case, the lifetime of the system can be represented as
T ¼ minðX 1 ; . . . ; X n Þ þ minðX 1 ; . . . ; X n1 ; ZÞ; where
X i stðX i minðX 1 ; . . . ; X n ÞjX i > minðX 1 ; . . . ; X n ÞÞ: ¼
The cdf of X i is found to be
P X i 6 t ¼
1 ½nP fX 1 6 X 2 þ t; . . . ; X 1 6 X n þ tg 1: n1
Similar results can be obtained by choosing an n-variate copula function Cðu1 ; . . . ; un Þ:. 5. Appendix Proof of (1): For t P 0,
F ðtÞ ¼ P X 1 6 t ¼ PfX 1 minðX 1 ; X 2 Þ 6 tjX 1 > minðX 1 ; X 2 Þg ¼
PfminðX 1 ; X 2 Þ < X 1 6 t þ minðX 1 ; X 2 Þg P fX 1 > minðX 1 ; X 2 Þg
¼
PfX 1 6 t þ minðX 1 ; X 2 Þg PfX 1 6 minðX 1 ; X 2 Þg PfX 1 > minðX 1 ; X 2 Þg
¼
PfX 1 6 X 2 þ tg PfX 1 6 X 2 g : PfX 1 > X 2 g
Since X 1 and X 2 have absolutely continuous exchangeable joint distribution,
PfX 1 6 X 2 g ¼ PfX 1 > X 2 g ¼ 1=2: Thus the required result is obtained. h Proof of (2): It is obvious that
F ðtÞ ¼ 2
Z
1 0
Z
yþt
f ðx; yÞdxdy 1;
0
where f ðt 1 ; t 2 Þ is the joint density of X 1 and X 2 : The proof follows noting that
f ðt 1 ; t 2 Þ ¼
@2 @2 Fðt 1 ; t2 Þ ¼ CðFðt1 Þ; Fðt 2 ÞÞ ¼ cðFðt 1 Þ; Fðt 2 ÞÞf ðt1 Þf ðt 2 Þ; @t 1 @t2 @t1 @t2 2
@ where f is the pdf associated with F and cðu; v Þ ¼ @u@ v Cðu; v Þ: Proof of (7): If ðX 1 ; X 2 Þ is PQD, then
CðFðtÞ; FðtÞÞ P F 2 ðtÞ and hence we have
EðminðX 1 ; X 2 ÞÞ ¼
Z
1
ð1 2FðtÞÞdt þ 0
Z
1
CðFðtÞ; FðtÞÞdt P
0
Similarly, if ðX 1 ; ZÞ is PQD, then
EðminðX 1 ; ZÞÞ ¼ ¼
Z
1
½1 F ðtÞ FðtÞdt þ
Z 0
1
1
ð1 2FðtÞÞdt þ
0
0
Z
Z
1
CðF ðtÞ; FðtÞÞdt P
Z
1
F 2 ðtÞdt ¼
Z
0
Z
1
½1 F ðtÞ FðtÞdt þ
0
ð1 F ðtÞÞð1 FðtÞÞdt:
0
Thus the inequality (7) is proved since EðTÞ ¼ EðminðX 1 ; X 2 ÞÞ þ EðminðX 1 ; ZÞÞ. h
1
ð1 FðtÞÞ2 dt:
0
Z 0
1
F ðtÞFðtÞdt
S. Eryilmaz, F. Tank / Applied Mathematics and Computation 218 (2012) 7792–7797
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Acknowledgement The authors thank the referees for their useful comments and suggestions. References [1] S. Eryilmaz, Reliability properties of consecutive k-out-of-n systems of arbitrarily dependent components, Reliability Engineering and System Safety 94 (2009) 350–356. [2] S. Eryilmaz, Estimation in coherent reliability systems through copulas, Reliability Engineering and System Safety 96 (2011) 564–568. [3] J. Jia, S. Wu, Optimizing replacement policy for a cold-standby system with waiting repair times, Applied Mathematics and Computation 214 (2009) 133–141. [4] X. Jia, L. Cui, J. Yan, A study on the reliability of consecutive k-out-of-n: G systems based on copula, Communications in Statistics – Theory and Methods 39 (2010) 2455–2472. [5] S. Kotz, C.D. Lai, M. Xie, On the effect of redundancy for systems with dependent components, IIE Transactions 35 (2003) 1103–1110. [6] C. Lai, M. Xie, Concepts of stochastic dependence in reliability analysis, in: H. Pham (Ed.), Handbook of Reliability Engineering, Springer, 2003, pp. 141– 156. [7] J. Navarro, J.M. Ruiz, C.J. Sandoval, A note on comparisons among coherent systems with dependent components using signatures, Statistics and Probability Letters 72 (2005) 179–185. [8] J. Navarro, J.M. Ruiz, C.J. Sandoval, Properties of coherent systems with dependent components, Communications in Statistics – Theory and Methods 36 (2007) 175–191. [9] J. Navarro, F. Spizzichino, Comparisons of series and parallel systems with components sharing the same copula, Applied Stochastic Models in Business and Industry 26 (2010) 775–791. [10] R.B. Nelsen, An Introduction to Copulas, second ed., Springer-Verlag, New York, 2006. [11] G.J. Wang, Y.L. Zhang, A bivariate optimal replacement policy for a cold standby repairable system with preventive repair, Applied Mathematics and Computation 218 (2011) 3158–3165. [12] Q. Wu, S. Wu, Reliability analysis of two-unit cold standby repairable systems under Poisson shocks, Applied Mathematics and Computation 218 (2011) 171–182.