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Downtime Data Analysis Based on Maximum Likelihood (MLE) to Estimate Parameters of Reliability Distributions

Reliability analysis techniques are customary standard tools that are used for evaluating the performance of different equipment and devices in order to minimize their downtime. To predict the reliability, life data from a sample that is satisfactorily representative of the equipment should be fitted to the suitable statistical distribution. The parameterized distribution may be used to estimate essential characteristics such as failure rate; and probability at a precise time, as well as system reliability. In the current study, Weibull++/ALTA software package is used as a novel tool to fit the available data set to estimate the best fitted probability density function (PDF) using Maximum Likelihood (MLE) for parameter estimation. The determined distributions are then assessed using goodness-of-fit test to define how well it fits the available data set. There are multiple methods for determining goodness-of-fit. Weibull distributions and their special cases’ parameters have an effect on life times.


Introduction
"Reliability Life Data Analysis" means studying and modeling documented data of an equipment to estimate its significant characteristics such as device reliability, failure rate, or mean time to failure (MTTF). Several studies for reliability assessment were; and still are, conducted.

Reliability Approaches and Indices
A general approach to system reliability assessment is to determine a number (or more) of its reliability indices that measure some aspects of system reliability performance such as mean time between failure (MTBF), failure rate (ƛ) and mean time to repair (MTTR) [10]. Many studies considered analytic models that are based on Weibull, exponential, uniform, and other distributions. Reliability prediction approach was presented by Denson through analyzing the failure rate model of plastic encapsulated microcircuits. The approach depended upon the product development stages and its related reliability metric [11]. Various reliability prediction methods, their concepts of application, advantages, and disadvantages were investigated by Thakur & Sakravdia [12]. The classical approach fits equipment failure rates to statistical models [13]; however in the data-mining methodology, it is displayed utilizing a data-mining algorithmdecision tree induction, establishing logical, mathematical, and statistical relations between MTTF and its several factor of impact (equipment conditions, failure history, etc.). As product failure rates are timedependent, they can be considered as time series. Barabady and Kumar [14] applied several statistical distributions containing Weibull, exponential, normal, and log normal distribution to analyze the reliability of a crushing plant, in order to trace the components with low reliability for a specified performance.

Methodology
The data gathering plays important role in this study to be able to predict the appropriate distribution as the distributions will be used to analyze the failure rate, reliability, availability, and maintainability of the equipment. Hence, the accomplishment of such research depends on the availability of statistical data for the target equipment in addition to the knowledge of statistical modeling and reliability theories. The following subsections present different tools that are used to estimate the reliability of any mechanical or electric component/or system.

Basic Concept and Approach for Reliability Analysis
Accurate assessment of reliability is of vital importance in the study of the performance modern systems. There are several quantitative measures for the reliability of an item. This item can be anything from a small component to a large system. These measures are listed in the following sub-sections.

Mean time between failures (MTBF)
MTBF measures the average time equipment is expected to perform prior to the occurrence of an unplanned failure. Testing the system for a period of time T, registering the number of faults N that is repaired then system testing is resumed. The repair time is extracted from the total test time T. MTBF is then given by equation 1 [15], [16]: MTBF = = (hours), F = expected failure rate. (1) This index is liable to examining inaccuracies, since the system is viewed for just an example of its all-out life. Derivation from the outcome ought to consider these inaccuracies. Hence taking everything into description, the system with the best MTBF is considered the most reliable.

Mean time to repair (MTTR)
MTTR is a measure of time period that is required for the equipment to get back to normal service status when it fails, also known as "Down Time". MTTR is calculated as shown in the following equations [16], [17].

Availability (A)
Availability is defined as; "percentage of time that equipment is capable of producing its end product at some specified acceptable level". To calculate availability, the machine working time (Up time, UT) and faulty (or being repaired) time (Down time, DT) are first determined. The total period of observation is the sum of UT + DT and availability is ratio of UT/ total period as exhibited in equations 4&5 [15]]:

Reliability (R (t))
Reliability is defined as "the capability of apparatus to perform; satisfactorily, the essential function under specified situations through a specified period of time", Ireson et al. [17]. Reliability can also be defined as "the probability that the apparatus is operating without failure in the time period t" as shown in the following equation [17], [18].
Using equation (1) in equation (6), we have Where; t = specified failure-free operation period of time.

Data Fitting and Parameters' Estimation
To assess reliability realistic modeling of the system components is required. In practice, not all the assessments of the system under analysis could be complete captured in the model. This is due to the inherently random nature of a number of the phenomena occurring through the system life, and the insufficient information some of them. "Reliability Life Data Analysis" is generally denoted to as "Weibull Analysis". To predict life of all products in the population, data from a representative sample is fitted to the suitable statistical distribution, hence the parameters of the statistical distribution that is considered the best fit should be estimated. The "probability density function (PDF)" is a function that; mathematically, pronounces the distribution. The PDF of some statistical distributions are exhibited in the next subsections.

Lognormal distribution
The time to failure T of an item is said to be log normally distributed with parameters  and  2 , T  lognormal (,  2 ), if Y = In T is normally (Gaussian) distributed with mean  and variance 2 [i.e., Y -X(,  2 )]. The probability density function of T is: The lognormal probability density is sketched in Fig. 2.19 for selected values of  and. The mean time to failure is:
where, ( + ) is the gamma function assessed at the value of: ( + ) The gamma function is well-defined as: For the 2-parameter case, this can be reduced to: Note that specific practitioners incorrectly assume that is equal to the MTTF, ̅ . This is only accurate for the case of: = 1 or:

The Median
The median, ̌ of the Weibull distribution is given by:

The Mode
While the mode, ̃ is:

The Standard Deviation
The standard deviation is assumed by: Weibull Reliability Function 3-parameter Weibull cumulative density function, cdf, is given by equation 15: Some authors refer to the above equation as unreliability ( ).
As the reliability function of a distribution equals one minus the cdf, the 3-parameter Weibull distribution' reliability function is: The reliability function of 3-parameter Weibull conditional reliability is: Or: It is called conditional because reliability of a novel assignment is calculated built on the detail that the unit (or units) has already collected hours of operation successfully.

The Weibull Failure Rate Function
Equation (19) represents the "Weibull failure rate function", ( )

Gamma distribution
The gamma distribution may be considered as a generalized case of the exponential distribution with mean= 1/¸ > 0. The mean = 1/¸ signifies the time elapsed till the first event takes place, where  is the mean of the Poisson process that generate the events. While the waiting time until the a th event to occur is represented by gamma random variable X. Hence, Where Y1, …. ; Yn are "independent exponential random variables" with mean= 1/.
The probability density function of Gamma distribution is given by (21): Where is the "shape parameter", β is the "scale parameter", and Γ is the "gamma function".

G-Gamma distribution
Generalized gamma (GG) distribution that is considered a flexible distribution, and has subfamilies; "exponential, gamma, and Weibull", and a limiting distribution "lognormal distribution" [1].
X-Generalized gamma (α, β, γ) representation is applied to specify the generalized gamma distribution with real positive parameters; scale parameter α, while (β & γ) are the shape parameters. The probability density function of generalized gamma random variable X has as following in equation (22).
The median time to failure satisfying R(tm) = 0.51 is = and the mode of the distribution is = − 2

The Loglogistic Distribution
The log logistic distribution is a 2-parameter distribution with parameters and . Equation (22) gives the pdf for this distribution: where:

The Loglogistic Reliability Function
The reliability for an operation of time , starting at 0, following the log logistic distribution is determined by: where: The "log logistic Failure Rate Function" The log logistic failure rate is given by:

The 2-Parameter Exponential Distribution
The pdf of the "2-parameter exponential" is given by: Where: is the location parameter.
Specific of the characteristics of the 2-parameter exponential distribution are discussed in Kececioglu [21,22]: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 16 October 2020 doi:10.20944/preprints202010.0355.v1 ▪ Positive location parameter; , shifts the beginning of the distribution right of the origin by a distance . This means that failures possibility starts only after hours of operation. ▪ The "scale parameter" is: =́− = − ▪ The distribution starts at = at the level of ( = ) = , decreasing exponentially and monotonically thereafter as increase beyond .
The Mean or MTTF The mean ̅ or (MTTF) is given by: It could be noted that MTTF is the inverse of the constant failure rate of the exponential distribution.

The Median
The median ̌ is:

The Mode
The mode ̃ is: The Standard Deviation The standard deviation is: The Exponential Reliability Function The equation for the 2-parameter exponential cumulative density function, or cdf, is given by: As the reliability function of a distribution equals one minus the cdf, then reliability function of the 2parameter exponential distribution is as follows: The "1-parameter exponential reliability function" is given by:

The Exponential Conditional Reliability Function
The equation of "exponential conditional reliability" expresses the reliability of an operation with a duration that accumulated hours till a new mission starts. The "exponential conditional reliability function" is: This means that the reliability for a assignment of t period started after the constituent has already accrued T hours of process from age zero are only a function of the assignment period, and not the age function at the beginning of the assignment. This is known as the memory less property.

The Exponential Failure Rate Function
The exponential failure rate function is:

CASE STUDY
To estimate the reliability, failure, and probability density functions of machines in a production facility, data are obtained from a previous study in Coldair Engineering Company for refrigerators and air conditioners in Sudan [23]. The available data are monthly down-time summery reports for 6 months of 2015 downtime causes of machines producing required plastic parts. After fitting these data to the suitable distribution of each cause of downtime; estimating their parameters, both failure and reliability versus time relations are then estimated. A sample of this data is exhibited in Table 1, while the other set of data are shown in reference [23]. The tables show the cause of the machine stopping and its duration in hours.

Procedures of Weibull++ ALTA Software Application
To predict the reliability performance of an item or component, available life data; down time or failure time is analyzed and fitted to an adequate statistical distribution. Thus it would be possible to estimation of numerous life characteristics, such as probability of failure, reliability, mean life, or failure rate. The software comprises a number of statistical distributions and analysis approaches [31] such as: • 1, 2 and 3 parameter Weibull • Logistic and Log logistic • 1 and 2 parameter Exponential • Bayesian-Weibull • Normal and Lognormal • Gamma and Generalized Gamma • 2, 3 and 4 subpopulation Mixed Weibull (Situations when there are different trends in the data and no distinct failure mode could be identified).
All above distributions were fitted to down time data (DT) as shown the following section.

Goodness of Fit Tests
Goodness-of-fit test is then applied on the probability distribution model of the fitted data (in our case it is down time).There is several approaches for defining goodness-of-fit. Several of the utmost prevalent methods utilized contain the chi-square, the "Kolmogorov-Smirnov test", the "Anderson-Darling test", and the "Stephens, M. A. test" [24]. In the current study, DT data for different causes of machine stoppage for 5 months are used as input data, as shown in table 1.
To determine the best fit the software package performs three "goodness of fit tests" that rank the fitted distributions; these tests are: • The Kolmogorov-Smirnov test (K-S): tests for statistical difference between the expected and results obtained from the fitted distribution.
• The correlation coefficient test (CC): measures how well the plotted points fit a straight line.
• The Likelihood Value test (LHV): calculates the value of the log-likelihood function assumed the parameters of the distribution.

Parameter Estimation
To estimate the reliability of each machine, the reliability function of the cause-of-downtime fitted distribution is used once the distribution parameters are estimated. There are several methods for parameter estimation that can be used such as: the maximum likelihood estimation (MLE) method, rank regression analysis, median ranks (MED), and Fisher matrix confidence bounds (FM). Using rank regression analysis regression line is fitted to the data points to obtain the distribution parameters. The closer regression lines to the plotted points, the better the fit. While maximum likelihood estimation (MLE) method obtains the solution using the likelihood function. Reliability of the entire machine and subsystems is then estimated using best fitted distribution and estimated parameters, as shown in section 4 of results

Best Fit Distribution
Using "Weibull++/ ALTA", components' data; shown in table 1 were first fitted using both MLE and RRX and it gave the same results. The resulted distributions are tested using K-S goodness of fit test, Correlation Coefficient (CC) test and Likelihood Value (LHV) test. Based on the three rankings; resulted from the three above mentioned tests, and on weights assigned to the individual criteria, an aggregate criterion (DESV) is applied to choose the best-fitting probability distribution. The described DESV aggregate criterion is shown in Equation 33 [24]. In this study, the used weights are the default values of weights selected by the software "Weibull++/ ALTA" manufacturer. These values resulted from many analyses conducted in applications based on engineering practice. The method assumes that the lowest DESV value corresponds to the best-fitting theoretical distribution. Using down time data, a sample of the results of this ranking procedure of one of the components; air drop is shown in table 2.The results for the other components are illustrated in appendix B. To rank the distributions, the three tests (K-S, CC, and LHV) are used in conjunction with their weights in the following equation to calculate aggregate criterion (DESV) where the distributions are ranked according to its value (rank 1 is that of minimum DESV value) [24 and 32]:

DESV= (K-S Rank × K-S Weight) + (CC Rank × CC Weight) + (LKV Rank × LKV Weight)
A summary of the resulted ranks of one of the causes of machine breakdown in Coldair Company; air drop, is presented in Tables 2. The results for Group2&3 are exhibited appendix A. In the above table: RAVGOF = K-S, RAVPLOT = CC, RLKV = ranking of the LKV test From table 2, the presented analysis of the down time of the machines due to air drop shows that the bestfitting distribution, according to the aggregate criterion, is the 2P-Exponentialdistribution. Nevertheless, with successive failures, different distribution may be indicated as the best-fit.

Characteristics of Each Cause of Breakdown
Data analysis of the breakdown causes (reasons of failure; listed in the first column of table 2) were found to follow one of these statistical distributions: Lognormal, 3-P Weibull, Gamma, and Generalized Gamma distributions. The characteristics of each of these distributions are presented in the following subsections.

i) Lognormal distribution characteristics [25]
The "lognormal distribution" is appropriate to random variates constrained by zero and a few very large values. In this case the distribution is "asymmetrical" and "positively skewed" [25 and 33]. The pdf increases from zero, till it reaches the "mode" then it decreases afterwards. For given ́, the degree of skewness increases with the increase of , while For the same , the skewness increases with the increase of ( ) . ́ is the mean of the natural logarithms of the time to failure which is the scale parameter, and is the standard deviation which is the shape parameter. Another characteristic is that if is considerably > 1, pdf sharply raise at the beginning.

ii) Cause of breakdown following Lognormal distribution (NSE power off)
Using ReliaSoft's software, the parameters repaid for the lognormal distribution are continuously logarithmic. Hence, as mentioned above (i), ́ signifies the mean of the natural logarithms of the times-tofailure, while ́ signifies the standard deviation of these data point logarithms. This means the value of the times-to-failure are not utilize as a parameter.  It could be seen in figure 1 that ́= 0.995768 (slightly < 1), and the mode (log mean) = 2.581665, hence pdf begins decreasing directly after the mode.
In figure 3 it is seen that the reliability is 90% at the beginning of operation (t=0) decreasing rapidly to about 28% after 24 min of operation to reach zero after168 min.

i) Weibull distribution characteristics
"Weibull distribution" is utilized a model variety of life behaviours according to the values of the parameters. The values of the "shape parameter " and the "scale parameter ", affect distribution characteristics; the curve shape, the reliability and the failure rate. The most general form "3-parameter Weibull distribution" is considered. The other forms, such as the 2-parameter distribution ( = 0) or the 1-parameter form ( = = , can be easily determined [27 and 34].

The Effect of beta on the pdf
The value of Weibull "shape parameter"; equals the slope of the regressed line in probability plots. Diverse values of"; can change the distribution behaviour. Some values of would cause the distribution equations to convert to those of other distributions. If = 1, the pdf of the "3-parameter Weibull distribution" changes to that of the "2-parameter exponential distribution" or: Where: 1 = = failure rate. The parameter is a pure number. The Figure 4 shows the effect of different values of , on the shape of the pdf. It could be seen that pdf shape can take on a variety of forms based on the value of .  ( ) Increases as →̃ then it decreases. ▪ For < 2.6 the Weibull pdf has a right tail (positively skewed) while for 2.6 < < 3.7 there is no tail (coefficient of skewness is almost zero). In case of > 3.7 a left tail exists (the curve is negatively skewed). Hence, different values of affect both reliability and failure rate functions as for = 0.999, (0) = ∞, while for = 1.001, (0) = 0. This sudden shift is why MLE estimation is complicated when is close to 1, [28].  As indicated in figure 5, downtime data with < 1 exhibit a failure rate that decreases with time, while that with = 1 have a constant failure rate (consistent with the exponential distribution) and populations with > 1 have a failure rate that increases with time. This means that all three life stages of reliability can be modelled with the Weibull distribution. After that failure rate, ( ) monotonically decreases, approaching the value of zero as → ∞ (∞) = 0 [27 , 28 and 29]. . Consequently, the failure rate increases at a constant rate as increases. Furthermore, if = 1 the slope becomes equal to 2, and when = 0, ( ), becomes a straight line which passes through the origin with a slope of 2. Note that at = 2 , the Weibull distribution equations reduce to that of the Rayleigh distribution.

The Effect of the shape parameter on the Weibull Failure Rate
When > 2 the ( ) curve is convex, the curve slope increases as increases. Consequently, the failure rate increases at an increasing rate as increases, indicating wear out life.
▪ For > 1, ( ) decreases as increases. Figure 6 shows the effect of β on Weibull Reliability plot

Figure 6: Effect of on Weibull Reliability plot [27]
From figure 6 it is seen that for = 0.5 (< 1), reliability decrease fast. For = 1, the reliability declines from 100% at the beginning of operation to reach 3.5% after 700 hrs. In case of > 1, reliability decreases slowly from 100% to about 50% then faster till it reaches 0.

− The Effect of the scale parameter on Weibull pdf
A change in the scale parameter has the same effect on the distribution as of changing the x-axis scale. If is increased; while holding constant, this will cause the expansion of the pdf. Then the zone below a pdf curve is a constant assessment of one, the "peak" of the pdf curve will moreover decrease with the increase of , as indicated in figure 7.
▪ If is increased while and are similar, the distribution becomes stretched out to the right and its height decreases, while keeping its shape and location. ▪ If is decreased while and are reserved similar, the distribution becomes pressed in towards the left (i.e., towards its beginning or towards 0 or ), and its height increases. ▪ has the similar units as , such as hours, miles, cycles, actuations, etc.

− Effects of the location parameter
It is obvious that the parameter, , positions the distribution along the time axis. As the value of changes, the distribution; and its associated function, slide either to the right (if > 0) or to the left (if < 0 ). The parameter has the same units of the x-axis. Figure 8 illustrates the effect of the location parameter on Weibull distribution.

Figure 8: Effect of the location parameter on Weibull distribution [27]
From figure 8 it is noted that: ▪ When = 0 the distribution starts at = 0 or at the origin. ▪ If > 0 the distribution starts at the right of the origin (at distance = ) . ▪ If < 0 the distribution starts the left of the origin (at distance = ) . ▪ A negative probably signifies that failures had occurred prior to the test beginning. This may be during production, in storage, transportation, or even before the actual use.
As delivers an estimation of the earliest time-to-failure of such units, the operating period from 0 to + is a failure free of such units.

ii) Cause of breakdowns following 3-P Weibull distribution
Three types of break down cause are found to be following 3-P; maintenance, Stocking, and others. Figures 9, 10, and 11 represent probability density function (pdf), failure rate vs. time, and reliability vs. time, of stocking cause of failure.

Figure 9: PDF function of the three causes of downtime following (3-P Weibull) Maintenance, Stoking and Others
It could be seen in figure 11 that maintenance pdf (red line) which has the value of = 19.794366 exhibits a different behaviour (shape) than the stocking and other causes pdf that have < < . It is also noted that for the maintenance cause of failure; red curve, < while stocking and other causes > . The characteristics of these cases are mentioned in "The Effect of beta on the pdf" subsection.

Figure 10: Failure rate vs. Time of the three causes of downtime following (3-P Weibull) Maintenance, stoking and others
As indicated in figure 10, downtime data with < 1 (other causes and stoking) exhibit a failure rate that decreases with time In case of down time due to maintenance where > 1, failure rate increases with time. This behaviour is previously explained in subsection (ii).

Figure 11: Reliability vs. Time of the three causes of downtime following (3-P Weibull) Maintenance, Stoking and Others
It could be seen in figure 11 that the reliability of stocking and other causes decreases sharply as 0 < < 1 for both of them; 0.396443 and 0.935702, respectively. While the decrease of maintenance reliability curve is less sharp as = 1.27 (>1) according to the characteristics previously mentioned in section 4.2.2 (i).

Gamma Distribution
i) Gamma distribution characteristics Gamma distribution has 2 parameters; scale parameter (k) and shape parameter ( ). Figure 12 shows probability density function (pdf) of Gamma distribution for k>1.
In case of (0 < < 1): Figures (13, 14, and 15) exhibit the effect of K parameter on Gamma pdf . It could be seen in figures 13, 14, and 15 that as increases, the pdf becomes stretched out to the right and its height decreases, while preserving its shape. While as decreases, the pdf shifts towards the left with increased height, and the mode does not exist.

i) Characteristics of Generalized Gamma
As stated previously, the generalized gamma distribution contains other distributions as distinct cases built on the values of the parameters [36]. Figure16 exhibits the probability density function (pdf) of Generalized Gamma distribution.

Figure16: Probability density Function (pdf) of Generalized Gamma distribution
− Effects of the parameters ,  and From figure 19 the following could be extracted. The exponential distribution is a distinct case when =1 and =1. The lognormal distribution is a special case when = 0. The gamma distribution is a distinctive case when  =  and Weibull > , = , < , < , = , > , respectively).

ii) Causes of breakdowns following Generalized Gamma distribution
Three cause of downtime are found to follow Generalized Gamma distribution, these are; Electric, heating up, and tools change. Figure 17 exhibits the (pdf) of the three causes of downtime, while figure 18 represents failure rate vs. Time of electric downtime causes and figure 19 shows Reliability vs. Time of Electric downtime. ▪ The shape of the loglogistic distribution is actual like to that of the lognormal distribution and the Weibull distribution. ▪ The pdf starts at zero, increases to its mode, and decreases subsequently. ▪ As increases, while is reserved similar, the pdf becomes stretched out to the right and its height decreases, while preserving its shape.  From figure 21 it can be deducted that the failure rate increases rapidly to 0.046 (in about 12 hr), then it decreases gradually to 0.0312 after 40 hrs, and finally reaches 0.046 after 200 hours.

i) Characteristics of the2-Parameters exponential distribution
Exponential distribution is used for modelling the behaviour of items with a constant failure rate. It is easy to manipulate as its mathematical form is quite simple with one parameter and it has no shape parameter. This fact may lead to the use of this model in conditions where it is not appropriate [37].  It could be seen in figure 23 that the exponential pdf is always convex and is stretched to the right as decreases in value and the value of the pdf function is always equal to the value of at = 0 ( = ). → ∞, ( ) → 0. The scale parameter is 1 = ̅ − = − . Figure 24 illustrates the pdf of Air Drop cause of failure that follows 2P-Exponential distribution. As explained in the previous section; Effect of  and on the exponential pdf, the air drop pdf curve; figure 24, does not begin at time (0) but is shifted to the right at (t =1 = ).

− Effect of  and  on the exponential reliability function
▪ The 1-parameter exponential reliability function starts at the value of 100% at t= 0 , declining afterward monotonically. ▪ The 2-parameter exponential reliability function remnants at the value of 100% for = 0 up to = , and declining afterward.
The reliability for task with duration of one MTTF (= 1 ) always equals 36.79%. This means that only 36.8% of the task will be completed successfully. In other words, only 36.8% of the equipment undertaking this mission will survive their mission. Figure 25 shows the effect of  on the reliability of air drop cause of failure.   Figure 27 shows that failure rate of Air Drop cause of failure that follows 2-p exponential distribution is constant and identical to the characteristics in reference [27] shown in figure 26. Table 3 presents a summary of different fitted distributions and their parameters for the different causes  of defects for Coldair Engineering Company for refrigerators and air conditioners in Sudan, while table 4 summarizes the different parameters.

Conclusions
To predict the reliability performance of an item or component, available life data; down time or failure time is analyzed and fitted to a novel statistical distributions using Weibull++ ALTA Software. For the case under study downtime data were fitted to different distributions and the parameters are estimated by two methods using both MLE and RRX and it gave the same results. The resulted distributions are then tested using K-S goodness of fit test, Correlation Coefficient (CC) test and Likelihood Value (LHV) test. Based on the three rankings; resulted from the three mentioned tests, and on weights assigned to the individual criteria, an aggregate criterion (DESV) is applied to choose the best-fitting probability distribution. In this study, the used weights are the default values of weights selected by the software "Weibull++/ ALTA" manufacturer.

Specific conclusions:
1. In the case under study the causes of downtime are: • Maintenance, Stoking and Others which were found to follow 3-P Weibull.
• Air drop following 2-P exponential distribution • Electric, Heating up, and Tools change causes following Generalized Gamma • NSE power off found to follow the log normal distribution 2. The three cause following 3-P Weibull, maintenance pdf which has ( = 19.794366 and β < 1) exhibits a different behaviour (shape) than the stocking and other causes pdf that have (1 < < 2 and β > 1). This different behaviour also applies for failure rate and reliability as for < 1 (stoking and other causes) the failure rate decreases with time and decreases sharply. In case of > 1 (maintenance), failure rate increases with time while the decrease in reliability curve is less sharp.
3. Weibull distribution is a special case of Generalized Gamma when = 1 and = 1 = ( ) 4. Downtime data with < 1 exhibit a failure rate that decreases with time, while that with = 1 have a constant failure rate (consistent with the exponential distribution) and populations with > 1 have a failure rate that increases with time. This means that all three life stages of reliability can be modeled with the Weibull distribution.