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Reliability Prediction ……
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Reliability Prediction ……
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Reliability Prediction ……
Reliability and Prediction for Traffic Accident in Sulaymaniyh Governorate Dr.Nawzad Muhamad Ahmed 𝟏 𝟏
&
Aras Jalal Mhamad 𝟐
Kurdistan Institution for strategic Studyies and Scientific Research
(KISSR), Sulaymaniyah City, Kurdistan Region – Iraq 𝟐
Statistic Department-School of Administration & Economics- Sulaymaniyah University, Sulaymaniyah City, Kurdistan Region – Iraq
Abstract Reliability analysis is accepted as the most commonly used statistical analysis technique
in the social and engineering field to determine a rate of failure and guarantee
according working in the system. Traffic problem is one of the most important system which must be study on it, because of traffic accidents are among the major and most dangerous problems all over the world, which cause loss of life, and thousands of people lose their lives on the roads every day. Many millions more are left with disabilities. Any contradiction or simple accident occur in the traffic system is define as an event (traffic accident). In this paper, a reliability technique is proposed to determine the probability of traffic accident occurrence per time and location and the same time determine the rate of failure for them in the Sulaymaniyh city, Iraq as the research goal. This analysis is applied specifically to a case study in Sulaymanyh in Kurdistan region of Iraq. The statistical analysis results shows that the time between (9:00 – 9:14) had a biggest rate of failure and the zone (Malik – Mahmood) street also had a biggest failure rate with a smallest guarantee for occurrence, that is means this zone with this time have a worse survival probability for traffic system.
Keywords: Reliability Analysis, Analysis of Traffic Accident 3
Reliability Prediction …… 1. Introduction The word reliability is associated with the civilization of mankind to compare one item/person with another. Trustworthy, dependable and consistent are the words, which can be used to give an indication of why the characteristic of reliability is valued. Reliability cannot be precisely measured with respect to human behavior but can give an indication that a particular person is more reliable than the other. The judgment can be easier when it is associated with human behavior or function [9]. The main target of work in reliability is to identify or to forecast the expected failure behavior of a product
[3]
, the characteristic of reliability is usually used to
describe some function or task or in widest sense, it may be said to be a measure of performance. A person who completes his work in time is said to be more reliable than the other who does not [9]
. Reliability is an abstract term meaning endurance, dependability, and good performance. For
any systems, however, it is more than an abstract term; it is something that can be computed, measured, evaluated, planned, and designed into a piece of system. Reliability means the ability of a system to perform the function it is designed for under the operating conditions encountered during its projected lifetime. Reliability is the probability of not failing in a specified time interval. Applied to a population of equipment, it is equal to the proportion of the original population that survives after that given time [2].
2. Materials and Methods 2.1 Reliability Function Reliability can also be defined as the probability of non-failure. If F(t) is the failure probability, then [1– F(t)] gives the non-failure probability. Thus, the reliability of device for time T = t (i.e., the device functions satisfactorily for T ≥ t) is [4] [9]. R (t) = 1 – F(t) 𝑡
= 1 − ∫ 𝑓(𝜏)𝑑(𝜏) 0 ∞
= ∫ 𝑓(𝜏)𝑑(𝜏) 0
= 𝐹(∞) = 1
4
… … … … … (2.1)
Reliability Prediction ……
Figure2.1 Above figure shows the shape of reliability function. Corresponding to reliability function R(t), F(t) is the unreliability function and represented by Q(t). The probability density F(t) was defined as the derivative of the failure distribution function F(t). Since F(t) = 1 – R(t) 𝑑𝐹(𝑡) 𝑑𝑅(𝑡) =− … … … (2.2) 𝑑𝑡 𝑑𝑡 The failure density is defined as the probability of failure occurring in a unit interval of time at 𝑓 (𝑡 ) =
time t [9].
2.2 Failure Rate Function It is typical to model component reliability parameters by a single scalar value. For example, a power transformer might be modeled with a failure rate of 0.03 per year. These scalar values, though useful, might not tell the entire story. Perhaps the most obvious example is the observation that the failure rates of certain components tend to vary with age. It might seem reasonable to conclude that new equipment fails less than old equipment. When dealing with complex components, this is usually not the case. In fact, newly installed electrical equipment has a relatively high failure rate due to the possibility that the equipment has manufacturing flaws, was damaged during shipping, was damaged during installation or was installed incorrectly. This period of high failure rate is referred to as the infant mortality period or the equipment break-in period [11]. If a piece of equipment survives its break-in period, it is likely that there are no manufacturing defects, that the equipment is properly installed, and that the equipment is being used within its design specifications. It now enters a period referred to as its useful life, 5
Reliability Prediction …… characterized by a nearly constant failure rate that can be accurately modeled by a single scalar number. As the useful life of a piece of equipment comes to an end, the previously constant failure rate will start to increase as the component starts to wear out. That is why this time is referred to as the wear out period of the equipment. During the wear out period, the failure rate of a component tends to increase exponentially until the component fails. Upon failure, the component should be replaced [1]. To describe the failure behavior with the failure rate λ(t), the failures at the point in time t or in a class i are not divided by the sum of total failures: [3]. 𝑎 𝜆 (𝑡 ) = … … … (2.3) 𝑏 Where: (a) is Failures (at the point in time t or in class i), and (b) is sum of unit still intact (at the point in time t or in class i). 2.3 Distribution Function (Failure Density) The density function f(t) describes the number of failures and the survival probability R(t) describes the number of units still intact. Therefore, the failure rate λ(t) can be calculated as the quotient of these two functions: [3] 𝜆 (𝑡 ) =
𝑓(𝑡) 𝑅(𝑡)
… … … (2.4)
Figure 2.2: Determination of the failure rate out of the density function and survival Probability
Figure 2.2 shows a graphical representation of equation (2.4) for the failure time tx. The failure rate at time t can be interpreted as a measurement for the risk that a part will fail, with the prerequisite that the component has already survived up to this point in time t. The failure rate at 6
Reliability Prediction …… a point in time specifies how many of the still intact parts will fail in the next unit of time. The failure rate λ(t) is used very often not only to describe wear out failures, and also early and random failures. The goal is to collect the complete failure behavior of a part or system. The result is always a similar and typical characteristic of the curve, see Figure 2.3. This curve is called the “bathtub curve”. The bathtub curve can be divided into three distinct sections: section 1 for early failures, section 2 for random failures, and section 3 for wear out failures [3] [5].
Figure 2.3: The “bathtub curve” Section 1 is characterized by a decreasing failure rate. The risk that a part will fail decreases with increasing time. Such early failures are mainly caused by failures in the assembly, production, material or by a definite design flaw. The failure rate is constant in section 2. Thus, the failure risk remains the same. Most of the time, this risk is also relatively low. Such failures are provoked for example by operating or maintenance failures or by dirt particles. Normally, such failures are difficult to pre-estimate. The failure rate increases rapidly in the section for wear out failures (section 3). The risk that a part will fail increases rapidly with time. Wear out failures are caused by fatigue failures, aging, pettings, etc [8]. Each of the three sections corresponds to different failure causes. Accordingly, different actions must be taken for an improvement in reliability in each respective section, see Figure 2.3. For 7
Reliability Prediction …… section 1 many trials and pilot-run series are recommended. The production and quality of the parts should also be controlled. In section 2, correct operation and maintenance should be considered and the established use and application of the product must be ensured. Section 3 requires either very exact calculations for components or corresponding practical trials. The actions taken in sections 1 and 2 must be ensured by appropriate steps taken early on in the design process. The improvements in section 3, however, take place in the stage of constructive dimensioning. Thus, the designer can have a strong influence on this section. In addition to representing the most decisive section for reliability, section 3 is the only section which can be calculated. Thus, a prognosis of the expected system reliability is often limited to just this section[3].
2.4 Generalized Gama Distribution Function While not as frequently used for modeling life data as the previous distributions, the generalized gamma distribution does have the ability to mimic the attributes of other distributions such as the Weibull or lognormal, based on the values of the distribution's parameters. While the generalized gamma distribution is not often used to model life data by itself (Mostly due to its mathematical complexity and its requirement of large sample sizes (>30) for convergence), its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data [6] [7]. The generalized gamma function is a 3-parameter distribution. One version of the generalized gamma distribution uses the parameters k, β, and θ. The pdf for this form of the generalized gamma distribution is given by: 𝛽 𝑡 𝑘𝛽−1 −( 𝑡 )𝛽 𝑓 (𝑡 ) = ( ) 𝑒 𝜃 Γ(𝑘 ) ∗ 𝜃 𝜃
… … … (2.5)
Where 𝜃 > 0 is a scale parameter, 𝛽 > 0 and 𝑘 > 0 are shape parameters and Γ(𝑥 ) is the gamma function of x, which is defined by: ∞
Γ(𝑘) = ∫ 𝑠𝑥−1 𝑒−𝑠 𝑑𝑠 0
8
… … … (2.6)
Reliability Prediction …… With this version of the distribution, however, convergence problems arise that severely limit its usefulness. Further adding to the confusion is the fact that distributions with widely different values of k, β, and θ may appear almost identical. In order to overcome these difficulties, also reparameterization the parameters k, β, and θ [7], where: 𝜇 = ln(𝜃) +
𝜎=
𝜆=
1 1 ∗ ln ( 2 ) 𝛽 𝜆
… … … (2.7)
1
… … … (2.8)
𝛽√𝑘 1
… … … (2.9)
√𝑘
Where −∞ < 𝜇 < ∞ , 𝜎 > 0 , 𝜆 > 0 While this makes the distribution converge much more easily in computations, it does not facilitate manual manipulation of the equation. By allowing
to become negative, the pdf of
the re-parameterized distribution is given by: [7] 𝜆∗
𝜆 1 ∗ ∗ 𝑒[ 𝜎 ∗ 𝑡 Γ(1) 𝜆2 1 {
𝑡 ∗ 𝜎√2𝜋
ln(𝑡)−𝜇 ln(𝑡)−𝜇 1 𝜆∗ 𝜎 +ln( 2 )−𝑒 𝜎 𝜆 2 𝜆
] 𝑖𝑓𝜆
≠0 … … … (2.10)
2
1 ln(𝑡)−𝜇 − ( ) 𝑒 2 𝜎
𝑖𝑓 𝜆 = 0
}
2.4.1 Survival Probability (Reliability) in Generalized Gama Distribution The sum of failures and the sum of the intact units in each class i or at any point in time t always add up to 100%. The survival probability R(t) is thus the complement to the failure probability F(t) [3] R(t) =1− F(t)
9
…………. (2.11)
Reliability Prediction …… The reliability function for the generalized gamma distribution is given by [7] [10]: 𝑒 1 − Γ𝐼 (
𝑅 (𝑡 ) =
𝑒
√
𝑧
∫ 𝑒 2𝜋 −∞
−
𝑥2 2
;
1 ) 𝑖𝑓 𝜆 > 0 𝜆2
ln(𝑡) − 𝜇 ) 𝜎
ln(𝑡)−𝜇 𝜆( ) 𝜎
𝜆2
{ 1
ln(𝑡)−𝜇 ) 𝜎
𝜆2
1 −Φ(
Γ𝐼 (
Where: Φ(𝑧) =
𝜆(
;
𝑖𝑓 𝜆 = 0
1 ) 𝜆2
… … … (2.12)
𝑖𝑓 𝜆 < 0 }
𝑑𝑥 and Γ𝐼 (𝑘; 𝑥) is the incomplete gamma function of k and x,
which is given by: Γ𝐼 (𝑘; 𝑥 ) =
1 Γ(k)
𝑥
∫ 𝑠 𝑘−1 𝑒 −𝑠 𝑑𝑠
… … … (2.13)
0
Where Γ(x) is the gamma function of x. 2.4.2 Confidence Bounds The method was used in this paper for confidence bounds for the generalized gamma distribution is the Fisher matrix. The lower and upper bounds on the parameter 𝜇 are estimated from: 𝜇𝑈 = 𝜇̂ + 𝑘𝛼 √𝑣𝑎𝑟(𝜇̂ ) (𝑢𝑝𝑝𝑒𝑟 𝑏𝑜𝑢𝑛𝑑)
… … … (2.14)
𝜇𝑈 = 𝜇̂ − 𝑘𝛼 √𝑣𝑎𝑟(𝜇̂ ) (𝑙𝑜𝑤𝑒𝑟 𝑏𝑜𝑢𝑛𝑑)
… … … (2.15)
For the parameter 𝜎̂, ln(𝜎̂) is treated as normally distributed, and the bounds are estimated from: 𝜎𝑈 = 𝜎̂ ∗ 𝑒 𝜎𝐿 =
𝑘𝛼 √𝑣𝑎𝑟(𝜎 ̂) 𝜎 ̂
𝜎̂ 𝑘𝛼 √𝑣𝑎𝑟(𝜎 ̂) 𝜎 ̂ 𝑒
(𝑢𝑝𝑝𝑒𝑟 𝑏𝑜𝑢𝑛𝑑) (𝑙𝑜𝑤𝑒𝑟 𝑏𝑜𝑢𝑛𝑑)
10
… … … (2.16) … … … (2.17)
Reliability Prediction …… For the parameter 𝜆 the bounds are estimated from: 𝜆𝑈 = 𝜆̂ + 𝑘𝛼 √𝑣𝑎𝑟(𝜆̂) (𝑢𝑝𝑝𝑒𝑟 𝑏𝑜𝑢𝑛𝑑)
… … … (2.18)
𝜆𝑈 = 𝜆̂ − 𝑘𝛼 √𝑣𝑎𝑟(𝜆̂) (𝑙𝑜𝑤𝑒𝑟 𝑏𝑜𝑢𝑛𝑑)
… … … (2.19)
Where 𝑘𝛼 is defined by: 𝛼=
1 √2𝜋
If 𝛿 is the confidence level, then 𝛼 =
1−𝛿 2
∞
∫𝑒
−
𝑡2 2 𝑑𝑡
= 1 − Φ( 𝑘𝛼 )
… … … (2.20)
𝑘𝛼
for the two-sided bounds, and 𝛼 = 1 − 𝛿 for the one-
sided bounds. The variances and covariance of 𝜇̂ and 𝜎̂ are estimated as follows:
̂ ( 𝜇̂ ) 𝑉𝑎𝑟 ̂ (𝜎̂, 𝜇̂ ) ( 𝐶𝑜𝑣
̂ ( 𝜇̂ , 𝜎̂) 𝐶𝑜𝑣 ̂ ( 𝜎̂) 𝑉𝑎𝑟
̂ (𝜆̂, 𝜇̂ ) 𝐶𝑜𝑣
̂ (𝜆̂, 𝜎̂) 𝐶𝑜𝑣
𝜕2Λ − 2 𝜕𝜇 ̂ ( 𝜇̂ , 𝜆̂ ) 𝐶𝑜𝑣 𝜕2Λ ̂ (𝜎̂, 𝜆̂) ) = − 𝐶𝑜𝑣 𝜕𝜇𝜕𝜎 ̂ ( 𝜆̂) 𝑉𝑎𝑟 𝜕2Λ − ( 𝜕𝜇𝜕𝜆
𝜕2Λ − 𝜕𝜇𝜕𝜎 𝜕2Λ − 2 𝜕𝜎 𝜕2Λ − 𝜕𝜆𝜕𝜎
𝜕2Λ − 𝜕𝜇𝜕𝜆 𝜕2Λ − 𝜕𝜆𝜕𝜎 𝜕2Λ − 2 𝜕𝜆 )
−1
… … … (2.21)
Where Λ is the log-likelihood function of the generalized gamma distribution. 2.5 MTTF, MTBF and some other parameters as a tool for prediction For reliability analysis it is important to know mean time to failure rather than the complete failure details. The parameter will be assumed to be the same for all the components which are identical in the design and operate under identical conditions [9]. There are various possibilities to specify the lifetime of a non-repairable system. The mean for
the time without failures for an observed period of time is the expected value for the lifetime t, normally called (Mean Time To Failure). The MTTF can be calculated with integration [3]. If we have life-tests information on a population of N items with failure times t1, t2,.., tn then the MTTF is defined as: 𝑛
1 𝑀𝑇𝑇𝐹 = ∑ 𝑡𝑖 𝑁 𝑖=1
11
… … … (2.22)
Reliability Prediction …… However, if a component is described by its reliability function and hazard model, then the MTTF is given by mathematical expression of the random variable T describing the time to failure of component. Therefore, ∞
𝑀𝑇𝑇𝐹 = 𝐸 (𝑇) = ∫ 𝑡 ∗ 𝑓 (𝑡)𝑑𝑡 0
𝑓 (𝑡 ) =
𝑑𝐹 (𝑡) 𝑑𝑅 (𝑡) =− 𝑑𝑡 𝑑𝑡 ∞
𝑀𝑇𝑇𝐹 = − ∫ 𝑡 ∗ 𝑑𝑅(𝑡)
Hence,
0 ∞
∞
= − 𝑡 ∗ 𝑅(𝑡) ∫ + ∫ 𝑅(𝑡)𝑑𝑡 0
0
∞
𝑀𝑇𝑇𝐹 = ∫ 𝑅(𝑡)𝑑𝑡
… … … (2.23)
0
The MTTF can also be computed by using Laplace-transform of R(t) i.e. ∞
𝑡
𝑀𝑇𝑇𝐹 = ∫ 𝑅(𝑡)𝑑𝑡 = lim ∫ 𝑅(𝑥 )𝑑𝑥 𝑡→∞
0
… … … (2.24)
0 𝑡
𝐻𝑜𝑤𝑒𝑣𝑒𝑟,
lim ∫ 𝑅 (𝑥 )𝑑𝑥 = lim 𝑅(𝑠)
𝑡→∞
𝑡→∞
0
Where, R(s) is the Laplace-transform of R(t) [9] Thus,
𝑀𝑇𝑇𝐹 = lim 𝑅(𝑠) 𝑡→∞
Further definition of the lifetime after the first failure of a component can be described by the MTBF(Mean Time Between Failure), which determines the mean lifetime of a component until its next failure and thus until repair maintenance. Under the assumption that the element is as good as new after maintenance, then the next mean time to failure (MTBF) is the same as the previous mean time to first failure MTTFF after the end of maintenance
[3]
. Failure rate is the reciprocal of the mean life. Failure rate is usually denoted
by the letter f or the Greek letter lambda (𝜆). So [2]
12
Reliability Prediction …… 𝜆=
1 𝑀𝑇𝐵𝐹
… … … (2.25)
1 𝜆
… … … (2.26)
So, 𝑀𝑇𝐵𝐹 =
3. Data Analysis and Results The data set used in this study consists of a two sample the first is consist of 96 observation about time of accident occurring and the second is consist of 75 observations which is named for each location that contain accident. The subjects were selected from traffic police records at Sulaymaniyh city. The variables which are determined in this study are; Time of accident occurring and location of accident occurring. The results divided in to two sections according to the variables are:
3.1 Section One for Accident Time Variable: Each variables are tested to choose suitable distribution, the first variable (accident time) was distributed as Gama distribution, then estimate it parameters, after that calculate the (Failure Rate, Failure Distribution Function and Survival Function) for accident time factor as follows: To test the null hypothesis for variable (accident time) which states that “the sample follows a Gama Distribution” can be use Chi-square test as: Table (3-1): Chi-square test for goodness of fit Chi-square (Observed value)
Chi-square (Critical value)
p-value
alpha
9.196
14.067
0.239
0.050
The test results are shown that the p-value (0.239) for the test is greater than the significant level (α = 0.05), that is mean we cannot reject null hypothesis, means that the accident time variable is distributed as gamma distribution, and the risk to reject the null hypothesis H0 while it is true is 23.89%. Then the parameters estimation for Gama distribution to variable (accident time) are: Mu=1.79029853541834
,
and
K=1.42650086108753
13
Reliability Prediction …… 0.027277 −0.027283 ] −0.027283 0.038920 The variance and covariance matrix for parameters in Gama distribution for variable time is Var − Cov = [
indicated that these parameters are really independent because of the weakness of these value for covariance that is an evidence that the distribution of these case are actually Gama distribution. Then the survival function, failure function and failure rate are shown: Table (3-2): Reliability, Failure Function, and Failure Rate for accident time Time
Reliability
Failure Function
Failure Rate
:111 – :1:1
0.250932
0.749068
0.143438/Hr
:1:1 – :1:1
0.854717
0.145283
0.109207/Hr
:1:1 – :111
0.854717
0.145283
0.109207/Hr
:111– :1:1
0.331026
0.668974
0.140402/Hr
:1:1– :111
0.854717
0.145283
0.109207/Hr
:111– :1:1
0.781729
0.218271
0.117022/Hr
1111– 11:1
0.854717
0.145283
0.109207/Hr
11:1– 11:1
0.854717
0.145283
0.109207/Hr
11:1– 1111
0.854717
0.145283
0.109207/Hr
1111– 11:1
0.704568
0.295432
0.122814/Hr
11:1– 1111
0.781729
0.218271
0.117022/Hr
1111– 1111
0.854717
0.145283
0.109207/Hr
4111– 41:1
0.854717
0.145283
0.109207/Hr
41:1– 41:1
0.629078
0.370922
0.127316/Hr
41:1– 4111
0.854717
0.145283
0.109207/Hr
4111– 4111
0.854717
0.145283
0.109207/Hr
4111– 41:1
0.492284
0.507716
0.133915/Hr
41:1– 41:1
0.629078
0.370922
0.127316/Hr
41:1– 4111
0.1889
0.8111
0.145835/Hr
4111– 4111
0.492284
0.507716
0.133915/Hr
4111– 41:1
0.250932
0.749068
0.143438/Hr
14
Reliability Prediction …… 41:1– 41:1
0.704568
0.295432
0.122814/Hr
41:1– 4111
0.090919
0.909081
0.150094/Hr
4111– 4111
0.331026
0.668974
0.140402/Hr
1111– 11:1
0.027117
0.972883
0.154231/Hr
11:1– 11:1
0.492284
0.507716
0.133915/Hr
11:1– 1111
0.163553
0.836447
0.146855/Hr
1111– 1111
0.492284
0.507716
0.133915/Hr
:111 – :1:1
0.067438
0.932562
0.151902/Hr
:11:1 – :11:1
0.629078
0.370922
0.127316/Hr
:11:1 – :1111
0.331026
0.668974
0.140402/Hr
:1111 – :1111
0.492284
0.507716
0.133915/Hr
::111 – ::1:1
0.090919
0.909081
0.150094/Hr
::1:1 – ::1:1
0.432624
0.567376
0.136419/Hr
::1:1 – ::111
0.1889
0.8111
0.145835/Hr
::111 – ::111
0.432624
0.567376
0.136419/Hr
::111 – ::1:1
0.163553
0.836447
0.146855/Hr
::1:1 – ::1:1
0.331026
0.668974
0.140402/Hr
::1:1 – ::111
0.331026
0.668974
0.140402/Hr
::111 – ::111
0.378948
0.621052
0.138555/Hr
::111 – ::1:1
0.331026
0.668974
0.140402/Hr
::1:1 – ::1:1
0.1889
0.8111
0.145835/Hr
::1:1 – ::111
0.042868
0.957132
0.153355/Hr
::111 – ::111
0.217883
0.782117
0.144703/Hr
:1111 – :11:1
0.036817
0.963183
0.153355/Hr
:11:1 – :11:1
0.432624
0.567376
0.136419/Hr
:11:1 – :1111
0.05802
0.94198
0.152421/Hr
:1111 – :1111
0.629078
0.370922
0.127316/Hr
:1111 – :11:1
0.042868
0.957132
0.153355/Hr
:11:1 – :11:1
0.331026
0.668974
0.140402/Hr
15
Reliability Prediction …… :11:1 – :1111
0.105444
0.894556
0.149388/Hr
:1111 – :1111
0.781729
0.218271
0.117022/Hr
:4111 – :41:1
0.141437
0.858563
0.147779/Hr
:41:1 – :41:1
0.331026
0.668974
0.140402/Hr
:41:1 – :4111
0.078332
0.921668
0.150744/Hr
:4111 – :4111
0.557915
0.442085
0.130934/Hr
:4111 – :41:1
0.122181
0.877819
0.148620/Hr
:41:1 – :41:1
0.781729
0.218271
0.117022/Hr
:41:1 – :4111
0.250932
0.749068
0.143438/Hr
:4111 – :4111
0.090919
0.909081
0.150094/Hr
:4111 – :41:1
0.250932
0.749068
0.143438/Hr
:41:1 – :41:1
0.629078
0.370922
0.127316/Hr
:41:1 – :4111
0.704568
0.295432
0.122814/Hr
:4111 – :4111
0.557915
0.442085
0.130934/Hr
:1111 – :11:1
0.163553
0.836447
0.146855/Hr
:11:1 – :11:1
0.704568
0.295432
0.122814/Hr
:11:1 – :1111
0.557915
0.442085
0.130934/Hr
:1111 – :1111
0.492284
0.507716
0.133915/Hr
:1111– :11:1
0.331026
0.668974
0.140402/Hr
:11:1– :11:1
0.557915
0.442085
0.130934/Hr
:11:1– :1111
0.122181
0.877819
0.148620/Hr
:1111– :1111
0.704568
0.295432
0.122814/Hr
::111 – ::1:1
0.067438
0.932562
0.151902/Hr
::1:1 – ::111
0.288497
0.711503
0.142015/Hr
::111 – ::111
0.704568
0.295432
0.122814/Hr
::111– ::1:1
0.078332
0.921668
0.150744/Hr
::1:1– ::1:1
0.854717
0.145283
0.109207/Hr
::1:1– ::111
0.492284
0.507716
0.133915/Hr
::111– ::111
0.557915
0.442085
0.130934/Hr
16
Reliability Prediction …… ::111– ::1:1
0.067438
0.932562
0.151902/Hr
::1:1– ::1:1
0.629078
0.370922
0.127316/Hr
::1:1– ::111
0.432624
0.567376
0.136419/Hr
::111– ::111
0.854717
0.145283
0.109207/Hr
11111– 111:1
0.557915
0.442085
0.130934/Hr
111:1– 111:1
0.704568
0.295432
0.122814/Hr
111:1– 11111
0.854717
0.145283
0.109207/Hr
Table (3-2) shows the reliability, probability of occurring accident and risk for each time, the first time is started from (1:00 am) to (1:14 am), the guarantee for non-happening accident (reliability) at this time is equal to (0.25), when the probability of occurring accident (failure distribution) for the same time is equal to (0.75), and the risk (failure rate) at the same time is equal to (0.14) and so on for the another time until the last time in the day which is started from (00:30) to (00:45), in this time the guarantee for non-happening accident (reliability) is equal to (0.85), while the probability of occurring accident (failure distribution) is equal to (0.15) in the same time, finally the risk at this time is equal to (0.11).
(a)
(b) 17
Reliability Prediction ……
(c)
(d)
Figure (3.1): shows, (a) Survival Function, (b) Failure Density Function, (c) Failure Distribution Function, and (d) Failure Rate From the figure (3.1), it is clear that the survival function is decreased according time, and part (b) illustrated the probability density function for gamma distribution also explained the fitted data with the gamma distribution, the part (c) clarified the probability of occurring accident per time which is decreased according time, the last part in the figure (3.1) shows the risk at each time. Then the mean time to failure is (MTTF = 7.736841), that is mean number of traffic occurring is equal to eight accident in the one day. 3.2 Section Two for Accident Location Variable: The second variable (Accident Location) was distributed as Generalized Gamma distribution, then estimate it parameters, after that calculate the (Failure Rate, Failure Distribution Function and Survival Function) for the variable accident location as follows: To test the null hypothesis for accident location variable which states that “the sample follows a Generalized Gama Distribution” can be use Chi-square test as: Table (3-3): Chi-square test for goodness of fit Chi-square (Observed value)
Chi-square (Critical value)
p-value
alpha
7.105
9.167
0.120
0.050
18
Reliability Prediction …… The test results are shown that the p-value is equal to (0.120) which greater than the significant level (α = 0.05), that is mean we cannot reject null hypothesis, means that the variable accident location is distributed as generalized gamma distribution. Then the parameters estimation for generalized gamma distribution to accident location are: Mu = 1.712176,
Sigma = 0.996967, and Lambda = -0.165685 0.037698 Var − Cov = [0.002690 0.048085
0.002690 0.007046 0.004251
0.048085 0.004251] 0.096200
The variance and covariance matrix for parameters in generalized gamma distribution for accident location variable is indicated that these parameters are really independent because of the weakness of these value for covariance that is an evidence that the distribution of these case are actually generalized gamma distribution. Then the survival function, failure function and failure rate are shown: Table (3-4): Reliability, Failure Function, and Failure Rate for accident location Location
Reliability
Failure Function
Failure Rate
Zanko
0.752019
0.24798
0.145470/Hr
Rania
0.267889
0.73211
0.107883/Hr
Pwenjwen
0.100991
0.89901
0.073986/Hr
Salm
0.023479
0.97652
0.043909/Hr
Darbandikhan - Sulaimani
0.138229
0.86177
0.083360/Hr
MalikMahmood
0.006047
0.99395
0.028169/Hr
KhasrawKhal
0.100991
0.89901
0.073986/Hr
Qalawa
0.299107
0.70089
0.112638/Hr
Sarchnar Intersection
0.076274
0.92373
0.066643/Hr
Halabja
0.490296
0.5097
0.129221/Hr
SaidSadq
0.56298
0.43702
0.135285/Hr
Bazian
0.649831
0.35017
0.141076/Hr
ParkAzady
0.179809
0.82019
0.092286/Hr
Shakraka
0.37861
0.62139
0.117787/Hr
Kurdsat
0.267889
0.73211
0.107883/Hr
19
Reliability Prediction …… Sulaimani-Kalar
0.865696
0.1343
0.132620/Hr
KawaAsngar
0.37861
0.62139
0.117787/Hr
Piramerd
0.490296
0.5097
0.129221/Hr
Dastaraka
0.752019
0.24798
0.145470/Hr
Sulaimani-Arbat
0.055697
0.9443
0.059424/Hr
Orzdy
0.865696
0.1343
0.132620/Hr
MajedBag
0.37861
0.62139
0.117787/Hr
KarezaWshk
0.217773
0.78223
0.099462/Hr
Darbandikhan-Sangaw
0.968325
0.03168
0.081177/Hr
Sulaimani-Tasluja
0.241027
0.75897
0.103501/Hr
Darbandikhan-Kalar
0.138229
0.86177
0.083360/Hr
Sabunkaran
0.865696
0.1343
0.132620/Hr
Hawarabarza
0.968325
0.03168
0.081177/Hr
Qaladzee
0.37861
0.62139
0.117787/Hr
Riaia (UN)
0.56298
0.43702
0.135285/Hr
Sarchnar
0.093883
0.90612
0.071988/Hr
KhalaHajy
0.490296
0.5097
0.129221/Hr
Qanat
0.241027
0.75897
0.103501/Hr
Zargata
0.37861
0.62139
0.117787/Hr
Darbanikhan
0.030945
0.96906
0.048277/Hr
Shekhan
0.865696
0.1343
0.132620/Hr
Kaneskan
0.968325
0.03168
0.081177/Hr
Dabashan
0.241027
0.75897
0.103501/Hr
Tweemalik
0.267889
0.73211
0.107883/Hr
MamaResha
0.865696
0.1343
0.132620/Hr
Azady
0.299107
0.70089
0.112638/Hr
Baxtyary
0.197528
0.80247
0.095734/Hr
ShexMhedin
0.752019
0.24798
0.145470/Hr
Alwaka
0.490296
0.5097
0.129221/Hr
20
Reliability Prediction …… Sirwan
0.490296
0.5097
0.129221/Hr
IbrahimAhmed
0.490296
0.5097
0.129221/Hr
Sulaimani-Chamchamal
0.865696
0.1343
0.132620/Hr
Ablakh
0.865696
0.1343
0.132620/Hr
Aqary
0.429559
0.57044
0.123332/Hr
Sulaimani-SaidSadq
0.56298
0.43702
0.135285/Hr
Sulaimani-Qaradakh
0.649831
0.35017
0.141076/Hr
Sitak
0.865696
0.1343
0.132620/Hr
Tasluja
0.37861
0.62139
0.117787/Hr
Khabat
0.37861
0.62139
0.117787/Hr
IbrahimPasha
0.56298
0.43702
0.135285/Hr
Kanakawa
0.752019
0.24798
0.145470/Hr
HawaryShar
0.649831
0.35017
0.141076/Hr
MamaYara
0.752019
0.24798
0.145470/Hr
Zarayan
0.968325
0.03168
0.081177/Hr
AshabaSpy
0.649831
0.35017
0.141076/Hr
Dukan-Slemani
0.968325
0.03168
0.081177/Hr
Hawkary
0.968325
0.03168
0.081177/Hr
Baranan
0.865696
0.1343
0.132620/Hr
Bakrajo
0.37861
0.62139
0.117787/Hr
KanyKurda
0.56298
0.43702
0.135285/Hr
Rapareen
0.081598
0.9184
0.068326/Hr
HawaryTaza
0.865696
0.1343
0.132620/Hr
Rzgary
0.490296
0.5097
0.129221/Hr
FulkayYakgrtn
0.217773
0.78223
0.099462/Hr
Arbat-warmawa
0.1176
0.8824
0.078370/Hr
Zerinok
0.865696
0.1343
0.132620/Hr
Qlyasan
0.490296
0.5097
0.129221/Hr
Chwarbakh
0.56298
0.43702
0.135285/Hr
21
Reliability Prediction …… Table (3-4) shows the reliability, probability of accident happening and risk for each location, the guarantee for non-accident occurring (reliability) for the first location (Zanko street) is equal to (0.75), when the probability of accident occurring (failure distribution) for the same street is equal to (0.25), and the risk for happening accident at the same street is equal to (0.15) and so on for the another streets (locations) which are explained in the table (), up to the last street in the table () which is named by (Chwarbakh) , at this street the guarantee for nonhappening accident (reliability) is equal to (0.56), while the probability of occurring accident (failure distribution) is equal to (0.44) at the same street, finally the risk at the Chwarbakh street is equal to (0.14).
(a)
(b)
(c)
(d)
Figure (3.2): shows, (a) Survival Function, (b) Failure Density Function, (c) Failure Distribution Function, and (d) Failure Rate
22
Reliability Prediction …… From the figure (3.2), it is clear that the survival function according location street, and part (b) illustrated the probability density function for generalized gamma distribution also explained the fitted data with the generalized gamma distribution, the part (c) clarified the probability of occurring accident per location, the last part in the figure (3.2) shows the risk at each location. Then the mean time to failure in this section is (MTTF = 10.275340), that is mean number of traffic occurring is equal to ten accident for each zone.
4. Conclusions (Discussion) In this study, the methodology provided a powerful technique to determine the worse and good location with time according to the survival function, probability of occurring accident and risk of each location. The main aim of the conducted study was to improve the traffic system. The result shows the guarantee of non-accident happening, probability of accident occurring, and failure rate of each location with time of accident as follows: the reliability of the first time which is started from (1:00 am) to (1:14 am) is equal to (0.25) that is mean the guarantee for nonhappening accident for this time is (0.25), when failure distribution function and failure rate for this time are equal to (0.75) and (0.14) respectively, upon for the another observation of time accident variable. Then the reliability of the first location which is named Zanko street is equal to (0.75) that is mean the guarantee for non-happening accident for this time is (0.75), when failure distribution function and failure rate for this time are equal to (0.25) and (0.15) respectively, upon for the another observation of location accident variable. The most important result of this paper is to determine the safety time and location with the worse of them, the safety time is time which has the lowest failure distribution function with the highest reliability that are equal to (0.109207), and (0.854717) respectively, while the worse time for occurring accident is the time between (9:00 – 9:14) which has the highest failure distribution function with the lowest reliability that are equal to (0.154231), and (0.027117) respectively. Also the safety zone is a location which has the lowest probability of accident occurring with the highest guarantee for non-accident occurring that are equal to (0.03168), and (0.968325) respectively, while the worse zone for occurring accident is the location which has the highest probability of accident occurring with the lowest guarantee for non-accident occurring that are equal to (0.99395), and (0.006047) respectively which named by (Malik-Mahmood) street.
23
Reliability Prediction …… 5. References
1- Brown E. Richard “Electric Power Distribution Reliability”,
Marchel Dekker, INC, New
York, Basel, (2002).
2- Benbow W. Donald & Broome W. Hugh, “The Certified Reliability Engineer Handbook”, American Society for Quality, Quality Press, Milwaukee 53203 by ASQ, (2009).
3- Bertsche B. “Reliability in Automotive and Mechanical Engineering”, VDI-Buch, doi: 10.1007/978-3-540-34282-3_1, Springer-Verlag Berlin Heidelberg (2008).
4- Birolini Alessandro “Reliability Engineering Theory and Practice”, Fifth edition, Springer Verlag Berlin Heidelberg (2007).
5- Chowdhury A. Ali & Koval O. Don “Power Distribution System Reliability Practical Methods and Applications”, Institute of Electrical and Electronics Engineers, Inc., Hoboken, New Jersey. Published simultaneously in Canada, (2009).
6- Jones L. Tyrone “Handbook of Reliability Engineering Prediction Procedures for Mechanical Equipment”, MechRel Program Manager, Naval Surface Warfare Center 9500 MacArthur Blvd, West Bethesds, (2010).
7- Lawless, J.F., “Statistical Models and Methods for Lifetime Data”, John Wiley & Sons, Inc., New York, (1982).
8- Mark. Levin, & Ted T., Kalal “Improving Product Reliability : Strategies and Implementation Wiley Series in Quality and Reliability Engineering”, John Wiley & Sons, Ltd. (UK), (2003).
9- Mishra C. R. & Sandilya Ankit, “Reliability and Quality Management”, New Age International (P) Ltd., Publishers Published by New Age International (P) Ltd., Publishers, (2009).
10- Pham Hoang “Handbook of Reliability Engineering”, Springer-Verlag London Berlin Heidelberg a member of BertelsmannSpringer Science+Business Media GmbH, (2003).
11- Yuan-Shun Dai Min Xie & Poh Leng Kim “Computing System Reliability Models and Analysis”, Kluwer Academic Publishers, Kluwer Academic/Plenum Publishers, New York, (2004).
24
Reliability Prediction ……
25

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