American Journal of Systems Science 2015, 4(1): 1-10 DOI: 10.5923/j.ajss.20150401.01
Some Additive Failure Rate Models Related with MOEU Distribution Salah H. Abid*, Heba A. Hassan Mathematics department, Education College, Al-Mustansiriya University, Baghdad, Iraq
Abstract In reliability theory, a combination of two distributions failure rate model for reliability studies is paid much
attention. In this paper, we will derive the failure rate model of (Marshall-Olkin Extended Uniform distribution) MOEU (𝛼𝛼, 𝜃𝜃) and every one of MOEU (𝑎𝑎, 𝑏𝑏) , MOEU (𝑎𝑎, 𝜃𝜃) , uniform( 𝜃𝜃 ), truncated exponential (𝜆𝜆, 𝜃𝜃) , truncated Weibull (𝜆𝜆, 𝑘𝑘, 𝜃𝜃) , truncated Frechet (𝑎𝑎, 𝑏𝑏, 𝜃𝜃) , truncated Rayleigh (𝜎𝜎 2 , 𝜃𝜃) , doubly truncated Cauchy (𝑎𝑎, 𝑏𝑏, 𝜃𝜃) and doublytruncated Gumbel (𝑎𝑎, 𝑏𝑏, 𝜃𝜃) distributions.
Keywords
Reliability, MOEU, Additive rate model, Truncated distribution
1. Introduction In reliability studies, combinations of components forming series, parallel and k out of n systems are quite popular. The reliability probabilities of such systems are evaluated either by the system as a whole or through the reliability probabilities of the components that define the system. It is well known that in a series system of a finite number of components with independent life time random, the system reliability is equal to the product of the component reliabilities. If 𝑓𝑓(𝑥𝑥), 𝐹𝐹(𝑥𝑥), ℎ(𝑥𝑥) respectively indicate the failure density, failure probability and failure rate of a component with life time random variable x, then we know that the reliability is given by, Where
𝑅𝑅(𝑥𝑥) = 1 − 𝐹𝐹(𝑥𝑥) =
𝑥𝑥 𝐸𝐸𝐸𝐸𝐸𝐸�− ∫0 𝑥𝑥
ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 �
𝐹𝐹(𝑥𝑥) = 𝑝𝑝(𝑋𝑋 ≤ 𝑥𝑥) = ∫0 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑑𝑑
(1) (2)
If a series system has two component with independent but non-identical life patterns explained by two distinct random variables say 𝑥𝑥1 and 𝑥𝑥2 with respective failure densities, failure probabilities and failure rates as 𝑓𝑓1 (𝑥𝑥), 𝑓𝑓2 (𝑥𝑥) ; 𝐹𝐹1 (𝑥𝑥), 𝐹𝐹2 (𝑥𝑥) ; ℎ1 (𝑥𝑥), ℎ2 (𝑥𝑥) , then the system reliability is given by, 𝑅𝑅(𝑥𝑥) =
𝑥𝑥 𝐸𝐸𝐸𝐸𝐸𝐸�− ∫0 [ℎ1 (𝑥𝑥)
+ ℎ2 (𝑥𝑥)]𝑑𝑑𝑑𝑑 �
(3)
From the above expression we can get the failure density and the failure rate of the series system whose reliability is given by (3), such models are already studied in the past * Corresponding author:
[email protected] (Salah H. Abid) Published online at http://journal.sapub.org/ajss Copyright © 2015 Scientific & Academic Publishing. All Rights Reserved
with different choices of ℎ1 (𝑥𝑥) and ℎ2 (𝑥𝑥) by Rao, Nagendram and Rosaiah (2013), Rao, Kantam, Rosaiah and Baba (2013) [4] and Rosaiah, Nagarjuna, Kumar and Rao (2014) [3]. In this paper a combination of MOEU(α, θ) and some other distributions will studied.
2. MOEU Distribution and Its Properties Marshall and Olkin (1997) [2] introduced a new family of distributions in an attempt to add a parameter to a family of � (x) = P(X > 𝑥𝑥) be the reliability distributions. Let G function of a random variable X and α > 0 be a parameter. Then F� (x, α) =
� (x) αG � (x) 1−(1−α)G
, −∞ < 𝑥𝑥 < ∞, 𝛼𝛼 > 0, (4)
is a proper reliability function. F� (x, α) is called Marshall-Olkin family of distributions. The probability density function (p.d.f) corresponding to (4) is given by α g(x)
f(x, α) = [1−(1−α)G�
(x)]2
, −∞ < 𝑥𝑥 < ∞ , 𝛼𝛼 > 0 , (5)
�(x). The hazard where g(x) is the p.d.f. corresponding to G (failure) rate function is given by �(x)] , h(x, α) = r(x)⁄[1 − (1 − α)G
�(x). where r(x) = g(x)⁄G Now, Let X follows U(0,θ) distribution, where θ > 0. �(x) = 1 − (x⁄θ). Substituting in (1) we get a new Then G distribution denoted by MOEU ( α, θ ) with reliability function [1]. F� (x, α, θ) = α(θ − x)⁄(αθ + (1 − α)x) , 0 < 𝑥𝑥 < 𝜃𝜃, 𝛼𝛼 > 0.
(6)
The corresponding pdf is obtained as
f(x, α, θ) = αθ⁄(αθ + (1 − α)x)2 ,0 < 𝑥𝑥 < 𝜃𝜃, 𝛼𝛼 > 0. (7)
Salah H. Abid et al.:
2
Some Additive Failure Rate Models Related with MOEU Distribution
and the corresponding cumulative distribution function is, F(x, α, θ) = 1 − F� (x, α, θ) = x⁄(αθ + (1 − α)x) ,0 < 𝑥𝑥 < 𝜃𝜃, 𝛼𝛼 > 0.
(8)
Note that α is the shape parameter and θ is the scale parameter of the distribution. The hazard rate function of a random variable X with MOEU(α, θ) distribution is h(x, α, θ) = θ⁄[αθ + (1 − α)x](θ − x)
The higher-order moments is [1], E(𝑋𝑋
r)
θ
= � xr ∙ 0
αθ dx (αθ + (1 − α)x)2 r
θ
s=0
0
αθ r! (−αθ)s (αθ + (1 − α)x)r−s−1 = �� � (r − s)! s! (r − s − 1) (1 − α)r+1
=
r!(−αθ )s ∑r (1−α)r +1 s=0 (r−s)! s! (r−s−1) αθ
∙ [(θ)r−s−1 − (αθ)r−s−1 ]
(9)
(10)
Specially, the mean and the variance of a random variable X with MOEU(α, θ) distribution are, respectively [1], αθ 2
αθ
μ́ 1 = (1−α)2 (α − logα − 1), μ2 = (1−α)4 [(1 − α)2 − α(logα)2 ].
So, the coefficient of variation is, Cv = �(1 − α)2 − α(logα)2 ��√α (α − logα − 1)� , α > 0 . The qth quantile of a random variable X with MOEU(α, θ) distribution is given by [1], xq = F −1 (q) = qαθ⁄(1 − q(1 − α)) , 0 ≤ q ≤ 1, Where F −1 (∙) is the inverse distribution function. The median is Me = αθ⁄(α + 1), and the mode is Mo = x =
Sk =
Kr =
μx −mo σx
μ4 −3 μ2 2
=
√α[−ln α+2(α−1)] �(1−α)2 −α(ln α)2
and the kurtosis is
αθ
. The skewness is
(1−α)
αθ4 16 1 α2 θ4 3 1 { �4α3 lnα + α3 �α + 3 � + 6α2 − 2α + 3� − 4 �−3α2 lnα + α2 �α + 2� − 3α + 2� [−lnα + α − 1] 5 (1 − α) (1 − α)5 α3 θ4 α4 θ16 [2αlnα − α2 + 1][−lnα + α − 1] 2 − 3 +6 [−lnα + α − 1] 4 } (1 − α)7 (1 − α)20 = −3 αθ2 [(1 − α)2 − α(lnα)2 ])2 ( 4 (1 − α)
In the following sections we will derive some additive failure rate models related with MOEU distribution.
3. MOEU-MOEU Additive Failure Rate Model Here we choice 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) for ℎ1 (𝑥𝑥) and 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝑎𝑎, 𝑏𝑏) for ℎ2 (𝑥𝑥), then 𝑡𝑡
𝑡𝑡
∫0 [ℎ1 (𝑥𝑥) + ℎ2 (𝑥𝑥)] 𝑑𝑑𝑑𝑑 = ∫0 𝑡𝑡
𝜃𝜃
(𝜃𝜃 −𝑥𝑥)(𝜃𝜃𝜃𝜃 +(1−𝛼𝛼)𝑥𝑥) 𝑡𝑡
𝑡𝑡
𝑑𝑑𝑑𝑑 + ∫0
𝑏𝑏
(𝑏𝑏−𝑥𝑥)(𝑏𝑏𝑏𝑏 +(1−𝑎𝑎)𝑥𝑥)
𝑑𝑑𝑑𝑑 .
𝜃𝜃 𝜃𝜃 − (𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑥𝑥) + (𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑥𝑥) 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑑𝑑𝑑𝑑 = � 𝑑𝑑𝑑𝑑 (𝜃𝜃 − 𝑥𝑥)(𝜃𝜃𝜃𝜃 + (1 − 𝛼𝛼)𝑥𝑥) (𝜃𝜃 − 𝑥𝑥)(𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑥𝑥) 0
0 𝑡𝑡
=� 0
𝑡𝑡
(𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑥𝑥) 𝜃𝜃(1 − 𝛼𝛼) − (1 − 𝛼𝛼)𝑥𝑥 + 𝑑𝑑𝑑𝑑 (𝜃𝜃 − 𝑥𝑥)(𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑥𝑥) (𝜃𝜃 − 𝑥𝑥)(𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑥𝑥) 𝑡𝑡
1 (1 − 𝛼𝛼) 𝑑𝑑𝑑𝑑 + � 𝑑𝑑𝑑𝑑 =� (𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑥𝑥) (𝜃𝜃 − 𝑥𝑥) 0
so one can write (11)as ln �
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼 )𝑡𝑡) 𝛼𝛼(𝜃𝜃−𝑡𝑡)
� + ln �
(𝑏𝑏𝑏𝑏 +(1−𝑎𝑎)𝑡𝑡) 𝑎𝑎(𝑏𝑏−𝑡𝑡)
0
= [ln|(𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑥𝑥)|]𝑡𝑡0 + [− ln|(𝜃𝜃 − 𝑥𝑥)|]𝑡𝑡0 = ln|(𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑡𝑡)| − ln|𝜃𝜃𝜃𝜃| − ln|(𝜃𝜃 − 𝑡𝑡)| + ln|𝜃𝜃| (𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑡𝑡) � = ln � 𝛼𝛼(𝜃𝜃 − 𝑡𝑡) �, and then by (3) can get,
(11)
American Journal of Systems Science 2015, 4(1): 1-10
𝑡𝑡
𝑅𝑅(𝑥𝑥) = 𝑒𝑒 − ∫0 ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 = 𝑒𝑒 =�
3
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡) (𝑏𝑏𝑏𝑏 +(1−𝑎𝑎)𝑡𝑡) �+ln � �� 𝛼𝛼(𝜃𝜃−𝑡𝑡) 𝑎𝑎(𝑏𝑏−𝑡𝑡)
−�ln �
𝛼𝛼(𝜃𝜃−𝑡𝑡)
(𝜃𝜃𝜃𝜃 +(1−𝛼𝛼)𝑡𝑡)
∙
𝑎𝑎(𝑏𝑏−𝑡𝑡)
(𝑏𝑏𝑏𝑏 +(1−𝑎𝑎)𝑡𝑡)
�
(12)
Which is mean, 𝑅𝑅 = 𝑅𝑅1 ∙ 𝑅𝑅2 . So for two additive failure rates,ℎ1 (𝑥𝑥) of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃)and ℎ2 (𝑥𝑥) of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝑎𝑎, 𝑏𝑏), one can get the distribution of the system as. 𝑓𝑓(𝑡𝑡) = −𝜕𝜕𝜕𝜕/𝜕𝜕𝜕𝜕 = 𝑅𝑅1 ∙ 𝑓𝑓2 + 𝑅𝑅2 ∙ 𝑓𝑓1 , where, 𝜕𝜕𝑅𝑅 𝜕𝜕𝑡𝑡
= 𝑅𝑅1 ∙
𝜕𝜕𝑅𝑅2 𝜕𝜕𝑡𝑡
+ 𝑅𝑅2 ∙
𝜕𝜕𝑅𝑅1 𝜕𝜕𝑡𝑡
𝑓𝑓(𝑡𝑡) =
= −𝑅𝑅1 ∙ 𝑓𝑓2 − 𝑅𝑅2 ∙ 𝑓𝑓1 , so,
𝑎𝑎𝑎𝑎 𝑎𝑎(𝑏𝑏 − 𝑡𝑡) 𝛼𝛼𝛼𝛼 𝛼𝛼(𝜃𝜃 − 𝑡𝑡) ∙ + ∙ (𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑡𝑡) (𝑎𝑎𝑎𝑎 + (1 − 𝑎𝑎)𝑡𝑡)2 (𝑎𝑎𝑎𝑎 + (1 − 𝑎𝑎)𝑡𝑡) (𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑡𝑡)2
𝑓𝑓(𝑡𝑡) =
𝑎𝑎𝑎𝑎
𝑏𝑏(𝜃𝜃−𝑡𝑡)
�
+
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)(𝑎𝑎𝑎𝑎 +(1−𝑎𝑎)𝑡𝑡) (𝑎𝑎𝑎𝑎 +(1−𝑎𝑎)𝑡𝑡)
𝜃𝜃 (𝑏𝑏−𝑡𝑡)
�
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)
(13)
According to the same argument, if we have for two additive failure rates ℎ1 (𝑥𝑥) of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) and ℎ3 (𝑥𝑥) of (𝑎𝑎, 𝜃𝜃), then, one can get the distribution of the system as. 𝑎𝑎𝑎𝑎 𝑎𝑎(𝜃𝜃 − 𝑡𝑡) 𝛼𝛼𝛼𝛼 𝛼𝛼(𝜃𝜃 − 𝑡𝑡) ∙ + ∙ 𝑓𝑓(𝑡𝑡) = (𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑡𝑡) (𝑎𝑎𝑎𝑎 + (1 − 𝑎𝑎)𝑡𝑡)2 (𝑎𝑎𝑎𝑎 + (1 − 𝑎𝑎)𝑡𝑡) (𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑡𝑡)2
=
𝛼𝛼𝛼𝛼𝛼𝛼 (𝜃𝜃−𝑡𝑡)
1
�
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)(𝑎𝑎𝑎𝑎 +(1−𝑎𝑎)𝑡𝑡) (𝑎𝑎𝑎𝑎 +(1−𝑎𝑎)𝑡𝑡)
+
1
�
(14)
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)
4. MOEU-Uniform Additive Failure Rate Model
Here we choice 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) for ℎ1 (𝑥𝑥) and 𝑢𝑢(0, 𝜃𝜃) for ℎ2 (𝑥𝑥) . So, Since 𝑓𝑓2 (𝑥𝑥) = 1⁄𝜃𝜃 , 0 < 𝑥𝑥 < 𝜃𝜃 , then 𝐹𝐹2 (𝑥𝑥) = 𝑥𝑥 ⁄𝜃𝜃 , 𝑅𝑅2 (𝑥𝑥) = (𝜃𝜃 − 𝑥𝑥)⁄𝜃𝜃 , and ℎ2 (𝑥𝑥) = 1⁄(𝜃𝜃 − 𝑥𝑥). We have, 𝑡𝑡
𝑡𝑡
𝑡𝑡
𝜃𝜃 1 � [ℎ1 (𝑥𝑥) + ℎ2 (𝑥𝑥)]𝑑𝑑𝑑𝑑 = � 𝑑𝑑𝑑𝑑 + � 𝑑𝑑𝑑𝑑 (𝜃𝜃 − 𝑥𝑥)(𝜃𝜃𝜃𝜃 + (1 − 𝛼𝛼)𝑥𝑥) (𝜃𝜃 − 𝑥𝑥) 0 𝑡𝑡
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 0
0
1 𝑑𝑑𝑑𝑑 = [− ln|(𝜃𝜃 − 𝑥𝑥)|]𝑡𝑡0 (𝜃𝜃 − 𝑥𝑥)
= − ln|(𝜃𝜃 − 𝑡𝑡)| + ln|𝜃𝜃|
= ln �
then we get,
0
𝜃𝜃
(𝜃𝜃−𝑡𝑡)
𝑡𝑡
∫0 [ℎ1 (𝑥𝑥) + ℎ2 (𝑥𝑥)]𝑑𝑑𝑑𝑑 = ln �
So, by (3), one can get the system Reliability as, 𝑡𝑡
𝑅𝑅 = 𝑒𝑒 − ∫0 ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 ⟹ 𝑅𝑅 = 𝑒𝑒
�,
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡) 𝛼𝛼(𝜃𝜃 −𝑡𝑡)
− ln �
� +ln �
𝜃𝜃 (𝜃𝜃𝜃𝜃 +(1−𝛼𝛼 )𝑡𝑡) � 𝛼𝛼 (𝜃𝜃 −𝑡𝑡)2
𝜃𝜃
� = ln �
(𝜃𝜃−𝑡𝑡)
=�
𝛼𝛼(𝜃𝜃−𝑡𝑡)2
𝜃𝜃(𝜃𝜃𝜃𝜃 +(1−𝛼𝛼)𝑡𝑡) 𝛼𝛼(𝜃𝜃−𝑡𝑡)2
�
�
𝜃𝜃(𝜃𝜃𝜃𝜃 +(1−𝛼𝛼)𝑡𝑡)
(15)
For two additive failure rates, ℎ1 (𝑥𝑥) of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) and ℎ2 (𝑥𝑥) of (0, 𝜃𝜃), then, one can get the distribution of the system as. 𝑓𝑓(𝑡𝑡) = −
=
𝜃𝜃(𝜃𝜃𝜃𝜃 + (1 − 𝛼𝛼)𝑡𝑡)(−2𝛼𝛼(𝜃𝜃 − 𝑡𝑡)) − 𝛼𝛼(𝜃𝜃 − 𝑡𝑡)2 𝜃𝜃(1 − 𝛼𝛼) 𝜕𝜕𝜕𝜕 =− 𝜃𝜃 2 (𝜃𝜃𝜃𝜃 + (1 − 𝛼𝛼)𝑡𝑡)2 𝜕𝜕𝜕𝜕
𝛼𝛼(𝑡𝑡−𝜃𝜃){𝛼𝛼(𝑡𝑡−𝜃𝜃)−(𝑡𝑡+𝜃𝜃 )} 𝜃𝜃(𝜃𝜃𝜃𝜃 +(1−𝛼𝛼)𝑡𝑡)2
,
0 < 𝑡𝑡 < 𝜃𝜃 ,
5. MOEU-Truncated Exponential Additive Failure Rate Model The probability density function of truncated exponential distribution from the right can be derived as,
(16)
Salah H. Abid et al.:
4
so the cumulative distribution is
𝐹𝐹
∗ (𝑥𝑥)
𝑥𝑥
=� 0
Some Additive Failure Rate Models Related with MOEU Distribution
𝑓𝑓(𝑥𝑥)
𝑓𝑓 ∗ (𝑥𝑥) =
𝐹𝐹(𝑏𝑏)−𝐹𝐹(𝑎𝑎)
=
𝜆𝜆𝑒𝑒 −𝜆𝜆𝜆𝜆
𝑤𝑤ℎ𝑒𝑒𝑒𝑒 𝑎𝑎 = 0 𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏 = 𝜃𝜃 ,
1−𝑒𝑒 −𝜆𝜆𝜆𝜆
−1 −𝑒𝑒 −𝜆𝜆𝜆𝜆 + 𝑒𝑒 −𝜆𝜆(0) 1 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 𝜆𝜆𝑒𝑒 −𝜆𝜆𝜆𝜆 −𝜆𝜆𝜆𝜆 𝑥𝑥 ∗ (𝑥𝑥) 𝑑𝑑𝑑𝑑 = �𝑒𝑒 � = ⟹ 𝐹𝐹 . = 0 1 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 1 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 1 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 1 − 𝑒𝑒 −𝜆𝜆𝜆𝜆
and then the reliability function is
1 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 , 0 < 𝑥𝑥 < 𝜃𝜃 . 1 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 1 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 − 1 + 𝑒𝑒 −𝜆𝜆𝜆𝜆 𝑒𝑒 −𝜆𝜆𝜆𝜆 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 ∗ (𝑥𝑥) = ⟹ 𝑅𝑅 , 𝑠𝑠𝑠𝑠 𝑡𝑡ℎ𝑒𝑒 ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑏𝑏𝑏𝑏, = 1 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 1 − 𝑒𝑒 −𝜆𝜆𝜆𝜆
𝑅𝑅∗ (𝑥𝑥) = 1 − 𝐹𝐹 ∗ (𝑥𝑥) = 1 −
ℎ∗ (𝑥𝑥) =
𝑓𝑓 ∗ (𝑥𝑥)
𝑅𝑅 ∗ (𝑥𝑥)
=
𝜆𝜆𝑒𝑒 −𝜆𝜆𝜆𝜆 �(1−𝑒𝑒 −𝜆𝜆𝜆𝜆 )
=
(𝑒𝑒 −𝜆𝜆𝜆𝜆 −𝑒𝑒 −𝜆𝜆𝜆𝜆 )⁄(1−𝑒𝑒 −𝜆𝜆𝜆𝜆 )
𝜆𝜆
1−𝑒𝑒 −𝜆𝜆 (𝜃𝜃 −𝑥𝑥 )
.
(17)
Here we choice 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) for ℎ1 (𝑥𝑥) and truncated exponential from the right for ℎ2 (𝑥𝑥). Now, since 𝑡𝑡
� 0
1 − 𝑒𝑒
𝑡𝑡
𝜆𝜆
𝑑𝑑𝑑𝑑 = � −𝜆𝜆(𝜃𝜃−𝑥𝑥) 0
𝜆𝜆𝑒𝑒 −𝜆𝜆𝜆𝜆 𝑑𝑑𝑑𝑑 (𝑒𝑒 −𝜆𝜆𝜆𝜆 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 )
𝑡𝑡
= �− ln�(𝑒𝑒 −𝜆𝜆𝜆𝜆 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 )��0
= − ln�(𝑒𝑒 −𝜆𝜆𝜆𝜆 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 )� + ln�(1 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 )�
1 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 = ln � −𝜆𝜆𝜆𝜆 � (𝑒𝑒 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 )
So, we can write 𝑡𝑡
𝑡𝑡
𝑡𝑡
𝑡𝑡
𝑡𝑡
𝜃𝜃 𝜆𝜆 � ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 = � [ℎ1 (𝑥𝑥) + ℎ2 (𝑥𝑥)]𝑑𝑑𝑑𝑑 = � [ℎ1 (𝑥𝑥) + ℎ2 (𝑥𝑥)]𝑑𝑑𝑑𝑑 = � 𝑑𝑑𝑑𝑑 + � 𝑑𝑑𝑑𝑑 −𝜆𝜆(𝜃𝜃−𝑥𝑥) (𝜃𝜃 − 𝑥𝑥)(𝜃𝜃𝜃𝜃 + (1 − 𝛼𝛼)𝑥𝑥) 1 − 𝑒𝑒 0 0 0
We get ln �
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡) 𝛼𝛼(𝜃𝜃−𝑡𝑡)
� +ln �
1−𝑒𝑒 −𝜆𝜆𝜆𝜆
(𝑒𝑒 −𝜆𝜆𝜆𝜆 −𝑒𝑒 −𝜆𝜆𝜆𝜆 )
� = ln �
0
(1−𝑒𝑒 −𝜆𝜆𝜆𝜆 )(𝜃𝜃𝜃𝜃 +(1−𝛼𝛼)𝑡𝑡) 𝛼𝛼(𝑒𝑒 −𝜆𝜆𝜆𝜆 −𝑒𝑒 −𝜆𝜆𝜆𝜆 )(𝜃𝜃 −𝑡𝑡)
And then the reliability function of the system can be written by (3) as,
𝑅𝑅 = 𝑒𝑒 =�
=
𝑡𝑡
− ∫0 ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑
= 𝑒𝑒
�
(1−𝑒𝑒 −𝜆𝜆𝜆𝜆 )(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡) � 𝛼𝛼(𝜃𝜃−𝑡𝑡)(𝑒𝑒 −𝜆𝜆𝜆𝜆 −𝑒𝑒 −𝜆𝜆𝜆𝜆 )
− 𝑒𝑒 −𝜆𝜆𝜆𝜆 ) 𝛼𝛼(𝜃𝜃 − 𝑡𝑡)(𝑒𝑒 � (1 − 𝑒𝑒 −𝜆𝜆𝜆𝜆 )(𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑡𝑡) 𝛼𝛼
1−𝑒𝑒 −𝜆𝜆𝜆𝜆
�
−𝜆𝜆𝜆𝜆
− ln �
0
(𝜃𝜃−𝑡𝑡)(𝑒𝑒 −𝜆𝜆𝜆𝜆 −𝑒𝑒 −𝜆𝜆𝜆𝜆 ) (𝜃𝜃𝜃𝜃 +(1−𝛼𝛼)𝑡𝑡)
�
(18)
It follows that, for two additive failure rates ℎ1 (𝑥𝑥) of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) and ℎ2 (𝑥𝑥) of truncated exponential(𝜆𝜆) at, then one can get the distribution of the system as follows,
𝑓𝑓(𝑡𝑡) = −R̀ =
Where, R̀ =
𝛼𝛼
1−𝑒𝑒 −𝜆𝜆𝜆𝜆
�
𝛼𝛼
1−𝑒𝑒 −𝜆𝜆𝜆𝜆
�
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)�𝜆𝜆𝑒𝑒 −𝜆𝜆𝜆𝜆 (𝜃𝜃−𝑡𝑡)+�𝑒𝑒 −𝜆𝜆𝜆𝜆 −𝑒𝑒 −𝜆𝜆𝜆𝜆 ��+(𝜃𝜃−𝑡𝑡)(1−𝛼𝛼)�𝑒𝑒 −𝜆𝜆𝜆𝜆 −𝑒𝑒 −𝜆𝜆𝜆𝜆 � (𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)2
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)�−𝜆𝜆(𝜃𝜃 −𝑡𝑡)𝑒𝑒 −𝜆𝜆𝜆𝜆 −�𝑒𝑒 −𝜆𝜆𝜆𝜆 −𝑒𝑒 −𝜆𝜆𝜆𝜆 ��−(𝜃𝜃−𝑡𝑡)(1−𝛼𝛼)(𝑒𝑒 −𝜆𝜆𝜆𝜆 −𝑒𝑒 −𝜆𝜆𝜆𝜆 ) (𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)2
6. MOEU-Truncated Weibull Additive Failure Rate Model The pdf of the truncated Weibull from the right at 𝜃𝜃 can be derived as
�
� , 0 < 𝑡𝑡 < 𝜃𝜃 . (19)
American Journal of Systems Science 2015, 4(1): 1-10
𝑘𝑘
𝑘𝑘 𝑥𝑥 𝑘𝑘−1 −�𝑥𝑥𝜆𝜆 � ( ) 𝑒𝑒 𝑓𝑓(𝑥𝑥) 𝑓𝑓 ∗ (𝑥𝑥) = = 𝜆𝜆 𝜆𝜆 𝜃𝜃 𝑘𝑘 𝐹𝐹(𝜃𝜃) − 𝐹𝐹(0) −� � 1 − 𝑒𝑒 𝜆𝜆 so, the distribution function can be defined as
𝐹𝐹
∗ (𝑥𝑥)
𝑡𝑡 𝑘𝑘 −� �
𝑘𝑘 𝑡𝑡
𝑘𝑘−1 𝑒𝑒 𝜆𝜆 𝑥𝑥 ( ) ∫0 𝜆𝜆 𝜆𝜆 𝑘𝑘 𝜃𝜃 −� � 1−𝑒𝑒 𝜆𝜆
=
and then the reliability function will be,
𝑑𝑑𝑑𝑑 =
−1
𝑘𝑘 𝜃𝜃 −� � 1−𝑒𝑒 𝜆𝜆
𝑅𝑅∗ (𝑥𝑥) = 1 − 𝐹𝐹 ∗ (𝑥𝑥) = 1 − =
=
so the failure rate function will be,
ℎ∗ (𝑥𝑥) =
𝑓𝑓 ∗ (𝑥𝑥)
𝑅𝑅 ∗ (𝑥𝑥)
=
1−
𝜃𝜃 𝑘𝑘 −� � 𝑒𝑒 𝜆𝜆
1−
, 0 < 𝑥𝑥 < 𝜃𝜃 . 𝑡𝑡 𝑘𝑘 𝜆𝜆
−� �
𝑥𝑥
� = 0
1−𝑒𝑒
𝑥𝑥 𝑘𝑘 −� � 𝜆𝜆
𝜃𝜃 −� � 1−𝑒𝑒 𝜆𝜆
𝑘𝑘
,
𝑥𝑥 𝑘𝑘
1 − 𝑒𝑒 −�𝜆𝜆 �
1 − 𝑒𝑒
𝜃𝜃 𝑘𝑘 −� � 𝜆𝜆
𝑥𝑥 𝑘𝑘
− 1 + 𝑒𝑒 −�𝜆𝜆 �
𝜃𝜃 𝑘𝑘 −� � 𝑒𝑒 𝜆𝜆
𝑘𝑘 𝜃𝜃 𝑥𝑥 𝑘𝑘 −� � −� � 𝜆𝜆 𝜆𝜆 𝑒𝑒 −𝑒𝑒 𝜃𝜃 −� � 1−𝑒𝑒 𝜆𝜆
�𝑒𝑒
5
𝑘𝑘
,
𝑘𝑘 𝜃𝜃 𝑥𝑥 𝑘𝑘 −� � 𝑘𝑘 𝑥𝑥 𝑘𝑘−1 −� � ( ) 𝑒𝑒 𝜆𝜆 �(1−𝑒𝑒 𝜆𝜆 ) 𝜆𝜆 𝜆𝜆 𝑘𝑘 𝑘𝑘 𝜃𝜃 𝜃𝜃 𝑥𝑥 𝑘𝑘 −� � −� � −� � 𝜆𝜆 𝜆𝜆 𝜆𝜆 𝑒𝑒 −𝑒𝑒 �(1−𝑒𝑒 )
=
𝑘𝑘 𝑥𝑥 𝑘𝑘−1 ( ) 𝜆𝜆 𝜆𝜆 𝜃𝜃 −𝑥𝑥 𝑘𝑘 −( ) 𝜆𝜆 1−𝑒𝑒
(20)
So if we choice 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) for ℎ1 (𝑥𝑥) and truncated Weibull(𝜆𝜆, 𝜅𝜅) from the right at 𝜃𝜃 for ℎ2 (𝑥𝑥) 𝑡𝑡 𝑡𝑡 𝑘𝑘 𝑥𝑥 𝑘𝑘−1 𝑡𝑡 𝑡𝑡 ( ) 𝜃𝜃 𝑑𝑑𝑑𝑑 + � 𝜆𝜆 𝜆𝜆 𝜃𝜃−𝑥𝑥 𝑑𝑑𝑑𝑑 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 � ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 = � [ℎ1 (𝑥𝑥) + ℎ2 (𝑥𝑥)]𝑑𝑑𝑑𝑑 = � (𝜃𝜃 − 𝑥𝑥)(𝜃𝜃𝜃𝜃 + (1 − 𝛼𝛼)𝑥𝑥) −( )𝑘𝑘 0 0 𝜆𝜆 0 0 1 − 𝑒𝑒 𝑡𝑡 𝑡𝑡 𝑘𝑘 𝑥𝑥 𝑘𝑘−1 −(𝑥𝑥 )𝑘𝑘 𝑘𝑘 𝑥𝑥 𝑘𝑘−1 � � ( ) 𝑒𝑒 𝜆𝜆 𝜆𝜆 𝜆𝜆 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑑𝑑𝑑𝑑 = � 𝜆𝜆 𝜆𝜆 𝑑𝑑𝑑𝑑 1 − 𝑒𝑒
0
−�
𝜃𝜃−𝑥𝑥 𝑘𝑘 � 𝜆𝜆
0
=
𝑥𝑥 𝑘𝑘 𝑒𝑒 −�𝜆𝜆 �
−
𝜃𝜃 𝑘𝑘 −� � 𝑒𝑒 𝜆𝜆
𝑥𝑥 𝑘𝑘 �− ln �𝑒𝑒 −�𝜆𝜆 � 𝑡𝑡 𝑘𝑘
−
𝜃𝜃 𝑘𝑘 −� � 𝑒𝑒 𝜆𝜆 ��
= − ln �𝑒𝑒 −�𝜆𝜆 � − 𝑒𝑒 = ln �
𝑡𝑡
Then ∫0 ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 = ln �
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)
= ln �
𝛼𝛼(𝜃𝜃−𝑡𝑡)
� + ln �
𝜃𝜃 −( )𝑘𝑘 1−𝑒𝑒 𝜆𝜆
𝜃𝜃 𝑡𝑡 𝑘𝑘 −� � −� � 𝑒𝑒 𝜆𝜆 −𝑒𝑒 𝜆𝜆
(𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑡𝑡)(1 − 𝑡𝑡 𝑘𝑘 𝑡𝑡)(𝑒𝑒 −�𝜆𝜆 �
𝛼𝛼(𝜃𝜃 − − so by (3) the reliability function of the system is,
𝜃𝜃 𝑘𝑘 −� � 𝑒𝑒 𝜆𝜆 )
𝜃𝜃 𝑘𝑘 −� � 𝑒𝑒 𝜆𝜆 )
�,
𝑘𝑘
𝜃𝜃 𝑘𝑘
1 − 𝑒𝑒 −(𝜆𝜆 )
𝑡𝑡 𝑘𝑘 𝑒𝑒 −�𝜆𝜆 �
�
𝜃𝜃 −� � 𝜆𝜆
−
𝜃𝜃 𝑘𝑘 −� � 𝑒𝑒 𝜆𝜆
�
𝑘𝑘
𝑡𝑡
0
𝜃𝜃 𝑘𝑘
� + ln �1 − 𝑒𝑒 −(𝜆𝜆 ) �
Salah H. Abid et al.:
6
Some Additive Failure Rate Models Related with MOEU Distribution
𝑅𝑅 = 𝑒𝑒
𝑡𝑡 − ∫0 ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑
=�
𝛼𝛼(𝜃𝜃 −𝑡𝑡)(𝑒𝑒
𝑘𝑘 𝜃𝜃 𝑡𝑡 𝑘𝑘 −� � −� � 𝜆𝜆 −𝑒𝑒 𝜆𝜆 )
𝜃𝜃 −� � (𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)(1−𝑒𝑒 𝜆𝜆
𝑘𝑘
�
(21)
)
for two additive failure rates, ℎ1 (𝑥𝑥) of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) and ℎ2 (𝑥𝑥) of truncated weibull(𝜆𝜆, 𝜅𝜅) from the right at 𝜃𝜃, then, one can get the distribution of the system as,
𝑓𝑓(𝑡𝑡) = −
𝜕𝜕𝜕𝜕
=
𝜕𝜕𝜕𝜕
𝑡𝑡 𝑘𝑘
𝛼𝛼
𝑘𝑘 𝜃𝜃 −� � 𝜆𝜆 (1−𝑒𝑒 )
⎧(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)�(𝜃𝜃 −𝑡𝑡)𝑘𝑘 � 𝑡𝑡 �𝑘𝑘−1 𝑒𝑒 −�𝜆𝜆 � 𝜆𝜆 𝜆𝜆 ⎪ ⎨ ⎪ ⎩
𝜃𝜃 𝑡𝑡 𝑘𝑘 −� � −� � +�𝑒𝑒 𝜆𝜆 −𝑒𝑒 𝜆𝜆
𝑘𝑘
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)2
𝑘𝑘 𝜃𝜃 𝑡𝑡 𝑘𝑘 −� � −� � 𝜆𝜆 𝜆𝜆 ��+(𝜃𝜃−𝑡𝑡)(1−𝛼𝛼)�𝑒𝑒 −𝑒𝑒 �
7. MOEU-Truncated Frechet Additive Failure Rate Model The pdf of truncated Frechet from the right at 𝜃𝜃 can be derived as, 𝑓𝑓 ∗ (𝑥𝑥) =
so the distribution function can be derived as,
𝑥𝑥 𝑎𝑎𝑎𝑎 𝑡𝑡 −(𝑏𝑏+1) 𝑒𝑒 −𝑎𝑎 𝑡𝑡
𝐹𝐹 ∗ (𝑥𝑥) = ∫0
𝑒𝑒 −𝑎𝑎 𝜃𝜃
−𝑏𝑏
−𝑏𝑏
and then the reliability function as, 𝑅𝑅∗ (𝑥𝑥) = 1 − 𝐹𝐹 ∗ (𝑥𝑥) = So the hazard function will be
ℎ∗ (𝑥𝑥) =
𝑓𝑓 ∗ (𝑥𝑥) 𝑅𝑅 ∗ (𝑥𝑥)
=
𝑑𝑑𝑑𝑑 = 𝑒𝑒 −𝑎𝑎 𝜃𝜃
(𝑎𝑎𝑎𝑎 𝑥𝑥 −(𝑏𝑏 +1) 𝑒𝑒 −𝑎𝑎 𝑥𝑥
(𝑒𝑒 −𝑎𝑎 𝜃𝜃
−𝑏𝑏
−𝑒𝑒 −𝑎𝑎 𝑥𝑥
−𝑏𝑏
−𝑏𝑏
−𝑏𝑏
1
−𝑏𝑏
𝑓𝑓(𝑥𝑥) 𝐹𝐹(𝜃𝜃 )−𝐹𝐹(0)
�𝑒𝑒 −𝑎𝑎𝑡𝑡 � −𝑎𝑎 𝜃𝜃 −𝑏𝑏
𝑒𝑒
−𝑒𝑒 −𝑎𝑎 𝑥𝑥
−𝑏𝑏
)�𝑒𝑒 −𝑎𝑎 𝜃𝜃
−𝑏𝑏
−𝑏𝑏 𝑒𝑒 −𝑎𝑎 𝜃𝜃
+1)⁄𝑒𝑒 −𝑎𝑎 𝜃𝜃
−𝑏𝑏
=
𝑒𝑒 −𝑎𝑎 𝑥𝑥
−𝑏𝑏
𝑒𝑒 −𝑎𝑎 𝜃𝜃
−1
𝑒𝑒 −𝑎𝑎 𝜃𝜃
−𝑏𝑏
(𝑎𝑎𝑎𝑎 𝑥𝑥 −(𝑏𝑏 +1) 𝑒𝑒 −𝑎𝑎 𝑥𝑥
−𝑏𝑏
+1
=
𝑥𝑥
𝑎𝑎𝑎𝑎 𝑥𝑥 −(𝑏𝑏 +1) 𝑒𝑒 −𝑎𝑎 𝑥𝑥
=
0
(𝑒𝑒 −𝑎𝑎 𝜃𝜃
−𝑏𝑏
−𝑒𝑒 −𝑎𝑎 𝑥𝑥
−𝑏𝑏
−𝑏𝑏
𝑡𝑡
𝑡𝑡
⎬ ⎪ ⎭
(22)
, 0 < 𝑥𝑥 < 𝜃𝜃 .
,
)
(23)
+1)
Now, if we choice ℎ1 (𝑥𝑥) of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) and ℎ2 (𝑥𝑥) of truncated Frechet(𝑎𝑎, 𝑏𝑏)from the right at, 𝑡𝑡
−𝑏𝑏
⎫ ⎪
𝑡𝑡
−𝑏𝑏
𝜃𝜃 (𝑎𝑎𝑎𝑎𝑥𝑥 −(𝑏𝑏+1) 𝑒𝑒 −𝑎𝑎𝑥𝑥 ) 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 � ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 = � [ℎ1 (𝑥𝑥) + ℎ2 (𝑥𝑥)]𝑑𝑑𝑑𝑑 = � 𝑑𝑑𝑑𝑑 + � 𝑑𝑑𝑑𝑑 −𝑏𝑏 −𝑏𝑏 (𝜃𝜃 − 𝑥𝑥)(𝜃𝜃𝜃𝜃 + (1 − 𝛼𝛼)𝑥𝑥) (𝑒𝑒 −𝑎𝑎𝜃𝜃 − 𝑒𝑒 −𝑎𝑎𝑥𝑥 + 1) 0 0 𝑡𝑡
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 0
𝑒𝑒
𝑎𝑎𝑎𝑎𝑥𝑥
−(𝑏𝑏+1) −𝑎𝑎𝑥𝑥 −𝑏𝑏
−𝑎𝑎𝜃𝜃 −𝑏𝑏
𝑡𝑡
Then ∫0 ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 = ln �
− 𝑒𝑒
𝑒𝑒
−𝑎𝑎𝑥𝑥 −𝑏𝑏
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡) 𝛼𝛼(𝜃𝜃−𝑡𝑡)
+1
0
𝑑𝑑𝑑𝑑 = �− ln �𝑒𝑒 −𝑎𝑎𝜃𝜃 = − ln �𝑒𝑒 −𝑎𝑎𝜃𝜃
� + ln �
𝑒𝑒
−𝑏𝑏
−𝑏𝑏
− 𝑒𝑒 −𝑎𝑎𝑥𝑥
− 𝑒𝑒 −𝑎𝑎𝑡𝑡 −𝑏𝑏
−𝑏𝑏
−𝑏𝑏
+ 1��
0
𝑡𝑡
0
+ 1� + ln �𝑒𝑒 −𝑎𝑎𝜃𝜃
𝑒𝑒 −𝑎𝑎𝜃𝜃 = ln � � −𝑏𝑏 −𝑏𝑏 𝑒𝑒 −𝑎𝑎𝜃𝜃 − 𝑒𝑒 −𝑎𝑎𝑡𝑡 + 1 𝑒𝑒 −𝑎𝑎 𝜃𝜃
−𝑎𝑎 𝜃𝜃 −𝑏𝑏
(𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑡𝑡)𝑒𝑒 −𝑎𝑎𝜃𝜃
−𝑏𝑏
−𝑒𝑒 −𝑎𝑎 𝑡𝑡
−𝑏𝑏
−𝑏𝑏
+1
−𝑏𝑏
− 1 + 1�
�
� , so the reliability function of the system is, = ln � −𝑏𝑏 −𝑏𝑏 𝛼𝛼(𝜃𝜃 − 𝑡𝑡)(𝑒𝑒 −𝑎𝑎𝜃𝜃 − 𝑒𝑒 −𝑎𝑎𝑡𝑡 + 1)
𝑅𝑅 = 𝑒𝑒
𝑡𝑡 − ∫0 ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑
=
−𝑏𝑏
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)𝑒𝑒 −𝑎𝑎 𝜃𝜃 − ln � � −𝑎𝑎 𝜃𝜃 −𝑏𝑏 −𝑒𝑒 −𝑎𝑎 𝑡𝑡 −𝑏𝑏 +1) 𝛼𝛼(𝜃𝜃−𝑡𝑡)(𝑒𝑒 𝑒𝑒
=�
𝛼𝛼(𝜃𝜃 −𝑡𝑡)(𝑒𝑒 −𝑎𝑎 𝜃𝜃
−𝑏𝑏
−𝑒𝑒 −𝑎𝑎 𝑡𝑡
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)𝑒𝑒 −𝑎𝑎 𝜃𝜃
−𝑏𝑏
−𝑏𝑏
+1)
�
(24)
For two additive failure rates ℎ1 (𝑥𝑥) of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) and ℎ2 (𝑥𝑥) of truncated Frechet(𝑎𝑎, 𝑏𝑏), then, one can get the probability distribution of the system as,
𝑓𝑓(𝑡𝑡) =
−𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
=
𝛼𝛼
−𝑏𝑏 𝑒𝑒 −𝑎𝑎 𝜃𝜃
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)�(𝜃𝜃−𝑡𝑡)𝑎𝑎𝑎𝑎 𝑡𝑡 −(𝑏𝑏 +1) 𝑒𝑒 −𝑎𝑎 𝑡𝑡
�
−𝑏𝑏
+(𝑒𝑒 −𝑎𝑎 𝜃𝜃
−𝑏𝑏
−𝑒𝑒 −𝑎𝑎 𝑡𝑡
−𝑏𝑏
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)2
+1)�+(𝜃𝜃−𝑡𝑡)(1−𝛼𝛼)(𝑒𝑒 −𝑎𝑎 𝜃𝜃
−𝑏𝑏
−𝑒𝑒 −𝑎𝑎 𝑡𝑡
−𝑏𝑏
+1)
� (25)
American Journal of Systems Science 2015, 4(1): 1-10
7
8. MOEU-Truncated Rayleigh Additive Failure Rate Model The pdf of truncated Rayleigh from the right at 𝜃𝜃 can be derived as, 2
𝑥𝑥 𝑥𝑥 −2𝜎𝜎 𝑒𝑒 2 𝑓𝑓(𝑥𝑥) 2 𝑓𝑓 ∗ (𝑥𝑥) = = 𝜎𝜎 , 0 < 𝑥𝑥 < 𝜃𝜃 . 𝜃𝜃 2 𝐹𝐹(𝜃𝜃) − 𝐹𝐹(0) − 1 − 𝑒𝑒 2𝜎𝜎 2
so the distribution function is,
and the reliability function is,
𝐹𝐹 ∗ (𝑥𝑥) =
−
𝑡𝑡
𝑡𝑡 2 2
𝑥𝑥 2 𝑒𝑒 2𝜎𝜎 ∫0 𝜎𝜎 𝜃𝜃 2 − 1− 𝑒𝑒 2𝜎𝜎 2
𝑑𝑑𝑑𝑑 =
𝑅𝑅∗ (𝑥𝑥) = 1 − 𝐹𝐹 ∗ (𝑥𝑥) = 1 −
So, the hazard function will be
ℎ∗ (𝑥𝑥) =
2
∗ (𝑥𝑥)
𝑓𝑓 = 𝑅𝑅∗ (𝑥𝑥)
𝑒𝑒
1
−
�−𝑒𝑒
𝜃𝜃 2
1− 𝑒𝑒 2𝜎𝜎 2
1−
1−
𝑥𝑥 2 − 2 2𝜎𝜎 𝑒𝑒 𝜃𝜃 2 − 2 2𝜎𝜎 𝑒𝑒
𝑡𝑡 2 2𝜎𝜎 2
−
− 𝑒𝑒
=
− 𝑒𝑒
0
1−
2
𝜃𝜃 2 − 2 2𝜎𝜎 �1
� =
𝑥𝑥 2 − 2 2𝜎𝜎 𝑒𝑒
𝑥𝑥 𝜃𝜃 𝑥𝑥 −2𝜎𝜎 − 2 �1 − 𝑒𝑒 2𝜎𝜎 2 𝑒𝑒 2 𝜎𝜎
𝑥𝑥 2 − 2 2𝜎𝜎
𝑥𝑥
𝑥𝑥 2
−
𝜃𝜃 2
,
1− 𝑒𝑒 2𝜎𝜎 2
−
𝜃𝜃 2 − 2 2𝜎𝜎 𝑒𝑒
𝜃𝜃 2 − 2 2𝜎𝜎 𝑒𝑒
𝑥𝑥 ⁄𝜎𝜎 2
=
𝜃𝜃 2 − 2 2𝜎𝜎
−
1− 𝑒𝑒 2𝜎𝜎 2
1 − 𝑒𝑒
(𝜃𝜃 2 −𝑥𝑥 2) − 2𝜎𝜎 2
Now, if we choice 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) for ℎ1 (𝑥𝑥) and truncated Rayleigh(𝜎𝜎) from the right at 𝜃𝜃 for ℎ2 (𝑥𝑥) 𝑡𝑡
𝑡𝑡
𝑡𝑡
then, ∫0 ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 = ∫0 [ℎ1 (𝑥𝑥) + ℎ2 (𝑥𝑥)]𝑑𝑑𝑑𝑑 = ∫0 𝑡𝑡
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 0
𝑥𝑥 ⁄𝜎𝜎 2
1 − 𝑒𝑒
𝜃𝜃 2 −𝑥𝑥 2 − 2𝜎𝜎 2
𝑡𝑡
𝑑𝑑𝑑𝑑 = � 0
𝑒𝑒
𝑥𝑥 𝜎𝜎 2
= ln � 𝑡𝑡
𝑒𝑒
−
𝑥𝑥 2 − 𝑒𝑒 2𝜎𝜎 2
− 𝑒𝑒
𝑥𝑥 2 − 2 2𝜎𝜎
1− 𝑡𝑡 2 2𝜎𝜎 2
𝜃𝜃 2 − 2 2𝜎𝜎
− 𝑒𝑒
𝜃𝜃 2 − 𝑒𝑒 2𝜎𝜎 2
− 𝑒𝑒
−
𝑑𝑑𝑑𝑑
𝑡𝑡 𝜃𝜃 2 − 2 2𝜎𝜎 ��
𝜃𝜃 2 2𝜎𝜎 2
= − ln �𝑒𝑒
0
�,
(𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑡𝑡) 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 � ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 = ln � � + ln � 𝛼𝛼(𝜃𝜃 − 𝑡𝑡) 0
𝑒𝑒
𝜃𝜃 2 − (𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)(1− 𝑒𝑒 2𝜎𝜎 2 )
= ln �
so the reliability function of the system is ,
𝑅𝑅 = 𝑒𝑒
𝑡𝑡 − ∫0 ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑
= 𝑒𝑒
−
𝑡𝑡 2
−
𝜃𝜃 2
𝛼𝛼 (𝜃𝜃−𝑡𝑡)(𝑒𝑒 2𝜎𝜎 2 − 𝑒𝑒 2𝜎𝜎 2 )
𝜃𝜃 2 − (𝛼𝛼𝛼𝛼 +(1−𝛼𝛼 )𝑡𝑡)(1− 𝑒𝑒 2𝜎𝜎 2 ) − ln � � 𝜃𝜃 2 𝑡𝑡 2 − − 𝛼𝛼 (𝜃𝜃 −𝑡𝑡)(𝑒𝑒 2𝜎𝜎 2 − 𝑒𝑒 2𝜎𝜎 2 )
𝑥𝑥 ⁄𝜎𝜎 2
𝑡𝑡
𝑑𝑑𝑑𝑑 + ∫0
(𝜃𝜃−𝑥𝑥)(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑥𝑥)
𝑥𝑥 2 − 2 2𝜎𝜎
= �− ln �𝑒𝑒
𝜃𝜃
−
�,
1 − 𝑒𝑒 𝑡𝑡 2 2𝜎𝜎 2
−
1− 𝑒𝑒
𝑡𝑡 2 − 2 2𝜎𝜎
𝜃𝜃 2 2𝜎𝜎 2
− 𝑒𝑒
=�
−
−
𝜃𝜃 2 2𝜎𝜎 2
(𝜃𝜃 2 −𝑥𝑥 2 ) 2𝜎𝜎 2
− 𝑒𝑒
𝑑𝑑𝑑𝑑
𝜃𝜃 2 − 2 2𝜎𝜎 �
+ ln �1 − 𝑒𝑒
𝜃𝜃 2 − 2 2𝜎𝜎 �
�
−
𝑡𝑡 2
−
𝜃𝜃 2
𝛼𝛼(𝜃𝜃−𝑡𝑡)(𝑒𝑒 2𝜎𝜎 2 − 𝑒𝑒 2𝜎𝜎 2 ) −
𝜃𝜃 2
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)(1− 𝑒𝑒 2𝜎𝜎 2 )
� ,
(26)
For two additive failure rates, ℎ1 (𝑥𝑥) of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) and ℎ2 (𝑥𝑥) of truncated Rayleigh(𝜎𝜎) from the right at 𝜃𝜃 then, one can get the distribution of the system as,
Salah H. Abid et al.:
8
f(t) =
− ∂R ∂t
=
𝛼𝛼
(1−𝑒𝑒
−
𝜃𝜃 2 2𝜎𝜎 2 )
Some Additive Failure Rate Models Related with MOEU Distribution
⎧(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)�(𝜃𝜃 −𝑡𝑡)𝜎𝜎𝑡𝑡2 𝑒𝑒 ⎨ ⎩
−
𝑡𝑡 2 𝑡𝑡 2 𝜃𝜃 2 𝑡𝑡 2 𝜃𝜃 2 − − − − 2𝜎𝜎 2 +(𝑒𝑒 2𝜎𝜎 2 −𝑒𝑒 2𝜎𝜎 2 )�+(𝜃𝜃 −𝑡𝑡)(1−𝛼𝛼)(𝑒𝑒 2𝜎𝜎 2 −𝑒𝑒 2𝜎𝜎 2 )
⎫
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)2
9. MOEU- Doubly Truncated Cauchy Additive Failure Rate Model
⎬ ⎭
(27)
The pdf of doublytruncated Cauchy from the right at 𝜃𝜃 and from the left at zero can be derived as, 𝑏𝑏⁄{𝜋𝜋(𝑏𝑏 2 + (𝑥𝑥 − 𝑎𝑎)2 } 𝑓𝑓(𝑥𝑥) = , 𝐹𝐹(𝜃𝜃) − 𝐹𝐹(0) 1 + 1 tan−1 (𝜃𝜃 − 𝑎𝑎) − 1 − 1 tan−1 (−𝑎𝑎) 2 𝜋𝜋 2 𝜋𝜋 𝑏𝑏 𝑏𝑏 𝑏𝑏⁄{𝜋𝜋(𝑏𝑏 2 + (𝑥𝑥 − 𝑎𝑎)2 } = , 0 < 𝑥𝑥 < 𝜃𝜃 . 1 𝜃𝜃 − 𝑎𝑎 1 −𝑎𝑎 tan−1 ( ) − tan−1 ( ) 𝜋𝜋 𝜋𝜋 𝑏𝑏 𝑏𝑏
𝑓𝑓 ∗ (𝑥𝑥) =
So the distribution function is
𝐹𝐹
∗ (𝑥𝑥)
And the reliability function is 𝑅𝑅
∗ (𝑥𝑥)
and then the hazard function will be,
𝑥𝑥
𝑏𝑏⁄{𝜋𝜋(𝑏𝑏 2 + (𝑡𝑡 − 𝑎𝑎)2 } 𝑑𝑑𝑑𝑑 1 −1 (𝜃𝜃 − 𝑎𝑎 ) − 1 tan−1 (−𝑎𝑎 ) tan 0 𝜋𝜋 𝜋𝜋 𝑏𝑏 𝑏𝑏 𝑥𝑥 𝑡𝑡 − 𝑎𝑎 𝑥𝑥 − 𝑎𝑎 −𝑎𝑎 �tan−1 ( )� ) − tan−1 ( ) tan−1 ( 𝑏𝑏 0 𝑏𝑏 𝑏𝑏 , = = 𝜃𝜃 − 𝑎𝑎 1 −𝑎𝑎 𝜃𝜃 − 𝑎𝑎 −𝑎𝑎 1 −1 −1 −1 −1 tan ( ) − tan ( ) tan ( ) − tan ( ) 𝜋𝜋 𝜋𝜋 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏
=�
= 1 − 𝐹𝐹
∗ (𝑥𝑥)
𝜃𝜃 − 𝑎𝑎 𝑥𝑥 − 𝑎𝑎 � − tan−1 � � 𝑏𝑏 𝑏𝑏 , = 𝜃𝜃 − 𝑎𝑎 −𝑎𝑎 � − tan−1 � � tan−1 � 𝑏𝑏 𝑏𝑏
𝑓𝑓 ∗ (𝑥𝑥) 𝑅𝑅 ∗ (𝑥𝑥)
ℎ∗ (𝑥𝑥) =
tan−1 �
=
𝑏𝑏 ⁄�𝑏𝑏 2 +(𝑥𝑥−𝑎𝑎)2 �
𝜃𝜃 −𝑎𝑎 𝑥𝑥−𝑎𝑎 )−tan −1 ( ) 𝑏𝑏 𝑏𝑏
tan −1 (
,
(28)
Now, if we choice 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) for ℎ1 (𝑥𝑥) and doublytruncated Cauchy(𝑎𝑎, 𝑏𝑏) from the right at 𝜃𝜃 and from the left at zero for ℎ2 (𝑥𝑥) then, 𝑡𝑡
𝑡𝑡
𝑡𝑡
𝑡𝑡
𝜃𝜃 𝑏𝑏⁄{𝑏𝑏2 + (𝑥𝑥 − 𝑎𝑎)2 } � ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 = � [ℎ1 (𝑥𝑥) + ℎ2 (𝑥𝑥)]𝑑𝑑𝑑𝑑 = � 𝑑𝑑𝑑𝑑 + � 𝑑𝑑𝑑𝑑 (𝜃𝜃 − 𝑥𝑥)(𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑥𝑥) −1 (𝜃𝜃 − 𝑎𝑎 ) − tan−1 (𝑥𝑥 − 𝑎𝑎 ) 0 0 tan 0 0 𝑏𝑏 𝑏𝑏 𝑡𝑡 𝑡𝑡 2 2 𝑏𝑏⁄{𝑏𝑏 + (𝑥𝑥 − 𝑎𝑎) } 𝜃𝜃 − 𝑎𝑎 𝑥𝑥 − 𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑑𝑑𝑑𝑑 = �− ln �tan−1 � � − tan−1 � ��� 𝜃𝜃 − 𝑎𝑎 𝑥𝑥 − 𝑎𝑎 𝑏𝑏 𝑏𝑏 −1 � 0 � − tan−1 � � 0 tan 𝑏𝑏 𝑏𝑏 𝜃𝜃 − 𝑎𝑎 𝑡𝑡 − 𝑎𝑎 𝜃𝜃 − 𝑎𝑎 −𝑎𝑎 = − ln �tan−1 � � − tan−1 � �� + ln �tan−1 ( ) − tan−1 ( )� 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝜃𝜃 − 𝑎𝑎 −𝑎𝑎 ) − tan−1 ( ) tan−1 ( 𝑏𝑏 𝑏𝑏 � = ln � 𝜃𝜃 − 𝑎𝑎 𝑡𝑡 − 𝑎𝑎 −1 −1 ) − tan ( ) tan ( 𝑏𝑏 𝑏𝑏 Then, 𝑡𝑡 ∫0
ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 = ln �
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡) 𝛼𝛼(𝜃𝜃−𝑡𝑡)
� + ln �
𝜃𝜃 −𝑎𝑎 −𝑎𝑎 �−tan −1 � �) 𝑏𝑏 𝑏𝑏 𝜃𝜃 −𝑎𝑎 𝑡𝑡−𝑎𝑎 𝛼𝛼(𝜃𝜃 −𝑡𝑡)(tan −1 � �−tan −1 � �) 𝑏𝑏 𝑏𝑏
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)(tan −1 �
= ln �
𝜃𝜃 −𝑎𝑎 −𝑎𝑎 )−tan −1 ( ) 𝑏𝑏 𝑏𝑏 𝜃𝜃 −𝑎𝑎 𝑡𝑡−𝑎𝑎 tan −1 ( )−tan −1 ( ) 𝑏𝑏 𝑏𝑏
tan −1 (
So, the reliability of the system can be written as
�
�,
(29)
American Journal of Systems Science 2015, 4(1): 1-10
𝑅𝑅 = 𝑒𝑒
−𝑎𝑎 𝜃𝜃 −𝑎𝑎 (𝛼𝛼𝛼𝛼 +(1−𝛼𝛼 )𝑡𝑡)(tan −1 � �−tan −1 � �) 𝑏𝑏 𝑏𝑏 � 𝜃𝜃 −𝑎𝑎 𝑡𝑡−𝑎𝑎 �−tan −1 � �) 𝛼𝛼 (𝜃𝜃 −𝑡𝑡)(tan −1 � 𝑏𝑏 𝑏𝑏
− ln �
9
𝜃𝜃 −𝑎𝑎 𝑡𝑡−𝑎𝑎 �−tan −1 � �) 𝑏𝑏 𝑏𝑏 𝜃𝜃 −𝑎𝑎 −𝑎𝑎 −1 −1 (𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)(tan � �−tan � �) 𝑏𝑏 𝑏𝑏
𝛼𝛼(𝜃𝜃 −𝑡𝑡)(tan −1 �
=
(30)
So, for two additive failure rates, ℎ1 (𝑥𝑥) of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) and ℎ2 (𝑥𝑥) of doublytruncated Cauchy(𝑎𝑎, 𝑏𝑏) from the right at 𝜃𝜃 and from the left at zero, one can get the distribution of the system as −𝜕𝜕𝜕𝜕 𝑓𝑓(𝑡𝑡) = 𝜕𝜕𝜕𝜕
=
𝑏𝑏 (𝜃𝜃 −𝑡𝑡)
𝜃𝜃 −𝑎𝑎
𝑡𝑡−𝑎𝑎
𝜃𝜃 −𝑎𝑎
𝑡𝑡−𝑎𝑎
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)� 2 +(tan −1 � �−tan −1 � �)�+(𝜃𝜃−𝑡𝑡)(1−𝛼𝛼)(tan −1 � �−tan −1 � �) 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏 +(𝑡𝑡−𝑎𝑎 )2 � 𝜃𝜃 −𝑎𝑎 −𝑎𝑎 2 (𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡) (tan −1 � �−tan −1 � �) 𝑏𝑏
𝛼𝛼
𝑏𝑏
�
(31)
10. MOEU-Doublytruncated Gumbel Additive Failure Rate Model The pdf of doublytruncated Gumbel from the right, at 𝜃𝜃 and from the left at zero, can be derived as, 𝑥𝑥−𝑎𝑎
𝑥𝑥−𝑎𝑎
−�
�
1 −�� 𝑏𝑏 �+𝑒𝑒 𝑏𝑏 � 𝑒𝑒 𝑓𝑓(𝑥𝑥) 𝑓𝑓 ∗ (𝑥𝑥) = = 𝑏𝑏 𝜃𝜃 −𝑎𝑎 , −𝑎𝑎 −� � 𝐹𝐹(𝜃𝜃) − 𝐹𝐹(0) −� � 𝑏𝑏 𝑏𝑏 −𝑒𝑒 −𝑒𝑒 𝑒𝑒 − 𝑒𝑒 So the distribution function is
0 < 𝑥𝑥 < 𝜃𝜃 .
𝑡𝑡−𝑎𝑎 𝑡𝑡−𝑎𝑎 � −� 𝑏𝑏 � −�� �+𝑒𝑒 𝑏𝑏
𝑥𝑥 −𝑎𝑎 −𝑎𝑎 1 � −� −� � 𝑏𝑏 −𝑒𝑒 −𝑒𝑒 𝑏𝑏 𝑡𝑡−𝑎𝑎 𝑥𝑥 𝑒𝑒 � −� 1 𝑒𝑒 − 𝑒𝑒 𝐹𝐹 ∗ (𝑥𝑥) = � 𝑏𝑏 𝜃𝜃 −𝑎𝑎 𝑑𝑑𝑑𝑑 = �𝑒𝑒 −𝑒𝑒 𝑏𝑏 � = 𝜃𝜃 −𝑎𝑎 𝜃𝜃 −𝑎𝑎 −𝑎𝑎 −𝑎𝑎 −𝑎𝑎 � � � −� −� −� −� � −� � −� � 0 𝑏𝑏 𝑏𝑏 0 𝑒𝑒 −𝑒𝑒 𝑒𝑒 −𝑒𝑒 − 𝑒𝑒 −𝑒𝑒 𝑏𝑏 − 𝑒𝑒 −𝑒𝑒 𝑏𝑏 𝑒𝑒 −𝑒𝑒 𝑏𝑏 − 𝑒𝑒 −𝑒𝑒 𝑏𝑏
𝑥𝑥
And the reliability function is
and then the hazard function will be
𝑅𝑅
∗ (𝑥𝑥)
= 1 − 𝐹𝐹
ℎ∗ (𝑥𝑥) =
𝑓𝑓 ∗ (𝑥𝑥) 𝑅𝑅 ∗ (𝑥𝑥)
∗ (𝑥𝑥)
=
1 𝑒𝑒 𝑏𝑏
= −��
𝑒𝑒 −𝑒𝑒
𝜃𝜃 −𝑎𝑎 � −� 𝑏𝑏
𝑒𝑒 −𝑒𝑒
𝑒𝑒 −𝑒𝑒
−𝑒𝑒 −𝑒𝑒
𝜃𝜃 −𝑎𝑎 � −� 𝑏𝑏
−�
−𝑒𝑒 −𝑒𝑒
𝑥𝑥 −𝑎𝑎 � 𝑏𝑏
−�
−𝑎𝑎 � 𝑏𝑏
,
𝑥𝑥−𝑎𝑎 � −� 𝑥𝑥−𝑎𝑎 𝑏𝑏 � �+𝑒𝑒 𝑏𝑏
𝜃𝜃 −𝑎𝑎 � −� 𝑏𝑏
−𝑒𝑒 −𝑒𝑒
−�
(32)
𝑥𝑥−𝑎𝑎 � 𝑏𝑏
Now, if we choice 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) for ℎ1 (𝑥𝑥) and doublytruncated Gumbel(a,b) from the right at 𝜃𝜃 and from the left at zero for ℎ2 (𝑥𝑥) then, 𝑥𝑥−𝑎𝑎
𝑥𝑥−𝑎𝑎
−�
�
1 −�� 𝑏𝑏 �+𝑒𝑒 𝑏𝑏 � 𝑡𝑡 𝑡𝑡 𝑒𝑒 𝜃𝜃 � ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 = � [ℎ1 (𝑥𝑥) + ℎ2 (𝑥𝑥)]𝑑𝑑𝑑𝑑 = � 𝑑𝑑𝑑𝑑 + � 𝑏𝑏 𝜃𝜃 −𝑎𝑎 𝑑𝑑𝑑𝑑 , 𝑥𝑥−𝑎𝑎 −� � (𝜃𝜃 − 𝑥𝑥)(𝛼𝛼𝛼𝛼 + (1 − 𝛼𝛼)𝑥𝑥) −� � 0 0 𝑏𝑏 𝑏𝑏 −𝑒𝑒 0 0 𝑒𝑒 −𝑒𝑒 − 𝑒𝑒 𝑡𝑡
𝑡𝑡
𝑥𝑥 −𝑎𝑎 𝑥𝑥−𝑎𝑎 � −� 𝑏𝑏 � −�� �+𝑒𝑒 𝑏𝑏
1 𝑡𝑡 𝜃𝜃 −𝑎𝑎 𝑥𝑥 −𝑎𝑎 𝑒𝑒 � −� � −� 𝑏𝑏 𝑏𝑏 𝑏𝑏 −𝑒𝑒 −𝑒𝑒 �� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � 𝑑𝑑𝑑𝑑 = �− ln �𝑒𝑒 − 𝑒𝑒 𝜃𝜃 −𝑎𝑎 𝑥𝑥−𝑎𝑎 � −� � −� 𝑏𝑏 𝑏𝑏 0 −𝑒𝑒 −𝑒𝑒 0 𝑒𝑒 − 𝑒𝑒 𝑡𝑡
= − ln
�𝑒𝑒 −𝑒𝑒
𝑒𝑒 −𝑒𝑒
𝜃𝜃 −𝑎𝑎 � −� 𝑏𝑏
𝜃𝜃 −𝑎𝑎 � −� 𝑏𝑏
− 𝑒𝑒
−𝑒𝑒
− 𝑒𝑒 −𝑒𝑒
−�
−�
𝑡𝑡−𝑎𝑎 � 𝑏𝑏
−𝑎𝑎 � 𝑏𝑏
� = ln � 𝜃𝜃 −𝑎𝑎 𝑡𝑡−𝑎𝑎 � −� � −� 𝑏𝑏 𝑏𝑏 𝑒𝑒 −𝑒𝑒 − 𝑒𝑒 −𝑒𝑒
� + ln
�𝑒𝑒 −𝑒𝑒
−�
𝜃𝜃 −𝑎𝑎 � 𝑏𝑏
− 𝑒𝑒 −𝑒𝑒
−�
−𝑎𝑎 � 𝑏𝑏
�
Salah H. Abid et al.:
10
Some Additive Failure Rate Models Related with MOEU Distribution
𝜃𝜃 −𝑎𝑎 −� � 𝑏𝑏
𝑡𝑡
−𝑎𝑎
−�
�
− 𝑒𝑒 −𝑒𝑒 𝑏𝑏 (𝜃𝜃𝜃𝜃 + (1 − 𝛼𝛼)𝑡𝑡) 𝑒𝑒 −𝑒𝑒 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 � ℎ(𝑥𝑥)𝑑𝑑𝑑𝑑 = ln � � + ln � � 𝜃𝜃 −𝑎𝑎 𝑡𝑡−𝑎𝑎 −� � 𝛼𝛼(𝜃𝜃 − 𝑡𝑡) −� � 0 𝑏𝑏 𝑏𝑏 −𝑒𝑒 −𝑒𝑒 𝑒𝑒 − 𝑒𝑒
= ln �
(𝜃𝜃𝜃𝜃 +(1−𝛼𝛼)𝑡𝑡)(𝑒𝑒 −𝑒𝑒 𝛼𝛼(𝜃𝜃 −𝑡𝑡)(𝑒𝑒 −𝑒𝑒
So the reliability function of the system can be written as
𝑅𝑅 = 𝑒𝑒
−�
𝜃𝜃 −𝑎𝑎 −𝑎𝑎 � −� −� � 𝑏𝑏 𝑏𝑏 ) (𝜃𝜃𝜃𝜃 +(1−𝛼𝛼 )𝑡𝑡)(𝑒𝑒 −𝑒𝑒 −𝑒𝑒 −𝑒𝑒 − ln � � 𝜃𝜃 −𝑎𝑎 𝑡𝑡−𝑎𝑎 � −� � −� 𝑏𝑏 𝑏𝑏 ) 𝛼𝛼 (𝜃𝜃 −𝑡𝑡)(𝑒𝑒 −𝑒𝑒 −𝑒𝑒 −𝑒𝑒
𝜃𝜃 −𝑎𝑎 � −� 𝑏𝑏
𝜃𝜃 −𝑎𝑎 � 𝑏𝑏
, =
−𝑒𝑒
−𝑒𝑒 −𝑒𝑒
−�
−𝑎𝑎 � 𝑏𝑏
𝑡𝑡−𝑎𝑎 � −� 𝑏𝑏 ) −𝑒𝑒
)
𝛼𝛼(𝜃𝜃−𝑡𝑡)(𝑒𝑒 −𝑒𝑒
�
𝜃𝜃 −𝑎𝑎 � −� 𝑏𝑏
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)(𝑒𝑒
−𝑒𝑒 −𝑒𝑒
𝜃𝜃 −𝑎𝑎 � −� 𝑏𝑏 −𝑒𝑒
(33)
−�
−𝑒𝑒
𝑡𝑡−𝑎𝑎 � 𝑏𝑏
)
−𝑎𝑎 −� � 𝑏𝑏 −𝑒𝑒
,
(34)
)
For two additive failure rates, ℎ1 (𝑥𝑥) of 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀(𝛼𝛼, 𝜃𝜃) and ℎ2 (𝑥𝑥) of doublytruncated Gmbel(a,b) from the right at 𝜃𝜃 and from the left at zero, one can get the distribution of the system as. −𝜕𝜕𝜕𝜕 𝑓𝑓(𝑡𝑡) = 𝜕𝜕𝜕𝜕
=
�𝑒𝑒 −𝑒𝑒
𝛼𝛼
𝜃𝜃 −𝑎𝑎 −� � 𝑏𝑏
−𝑒𝑒 −𝑒𝑒
−𝑎𝑎 � 𝑏𝑏
−�
𝑡𝑡−𝑎𝑎 𝜃𝜃 −𝑎𝑎 𝜃𝜃 −𝑎𝑎 𝑡𝑡−𝑎𝑎 𝑡𝑡−𝑎𝑎 −� � 𝑡𝑡−𝑎𝑎 −� −� � � −� � −� � 𝑏𝑏 𝑒𝑒 𝑏𝑏 𝑏𝑏 � 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑒𝑒 − +(𝑒𝑒 −𝑒𝑒 −𝑒𝑒 −𝑒𝑒 )�+(𝜃𝜃−𝑡𝑡)(1−𝛼𝛼)(𝑒𝑒 −𝑒𝑒 −𝑒𝑒 −𝑒𝑒 )
⎧(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)�−(𝜃𝜃 −𝑡𝑡)1 𝑒𝑒 −� 𝑏𝑏 ⎪ ⎨ ⎩
�⎪
(𝛼𝛼𝛼𝛼 +(1−𝛼𝛼)𝑡𝑡)2
⎫ ⎪ ⎬ ⎪ ⎭
(35)
11. Summary and Conclusions
REFERENCES
In spite of the great importance of the uniform distribution uses, but unfortunately the form of the distribution and its properties reduced the distribution applications, especially in real life. This issue has made us think to construct other distributions based on the uniform distribution, So that the new distributions have flexible forms and properties to represent a lot of other applications. A combination of (Marshall-Olkin Extended Uniform distribution) MOEU(𝛼𝛼, 𝜃𝜃) model and every one of some probability models are developed on lines of the well known linear failure rate model .We derive here the additive failure rate model of MOEU(𝛼𝛼, 𝜃𝜃) and every one of MOEU(𝑎𝑎, 𝑏𝑏), MOEU(𝑎𝑎, 𝜃𝜃), uniform(𝜃𝜃 ), truncated exponential (𝜆𝜆, 𝜃𝜃), truncated Weibull (𝜆𝜆, 𝑘𝑘, 𝜃𝜃) , truncated Frechet (𝑎𝑎, 𝑏𝑏, 𝜃𝜃) , truncated Rayleigh(𝜎𝜎 2 , 𝜃𝜃), truncated Cauchy (𝑎𝑎, 𝑏𝑏, 𝜃𝜃) and truncated Gumbel(𝑎𝑎, 𝑏𝑏, 𝜃𝜃) distributions.
[1]
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Srinivasa Rao, B., Nagendram, S. and Rosaiah, K. (2013) "Exponential–Half logistic additive failure rate model", International Journal of Scientific and Research, 3(5), p.1-10.
