Synthesis of data
4.3 AN EXAMPLE OF EMPIRICAL–THEORETICAL INFERENCE ABOUT THE DISTRIBUTION OF A RANDOM VARIABLE
Many systems in mining are of the series type in a reliability sense, especially the mechanised ones, e.g., BWE—belt conveyors—stacker in surface mining; shearer—AFC—belt conveyors in underground coal longwall mining; crusher (sizer, Australian)—belt conveyor—dumping machine; screen—receiving conveyor. These systems consist of pieces of equipment that can be repaired (are renewable). There are also series systems in mining that consist of non-renewable objects.
In reliability theory, a system has a series structure if, and only if, the failure of any element of the system means the failure of the system.
This type of structure is the worst of all of those that are possible, i.e. such a system has the most inconvenient reliability parameters and characteristics.
If, for instance, any piece of equipment that is engaged in the excavation and haulage of the coal won in an underground mine fails, then the whole system will not have any stream of coal quite soon. An exception to this rule is when such a system has a bin in its structure and a certain amount of the mineral can be stored in it.
The reliability parameters of renewable series systems are calculated by applying the Markov processes theory if the times of the states (work and repair) can be satisfactorily described by exponential distributions. If, however, the probability distributions of the times of the states are not exponential but are still stochastically independent, then the theory of semi-Markov processes should be applied.
If a system is of a series structure and its components work to the first failure occurrence, which means that the whole system works to the first failure occurrence, then order statistics should be applied. A system of this type can be found, for instance, in electronics.
Table 4.3. (Continued).
xj F(e)(xj) F(t)(xj) max|F(e)(xj) − F0(xj)| F(t)(xj)
j [min] gamma Weibull
29 130 0.829 0.842 0.822
30 140 0.857 0.872 0.857
31 155 0.886 0.908 0.898
32 170 0.914 0.935 0.929
33 170 0.943 0.935 0.929
34 180 0.971 0.948 0.945
35 230 1 0.984 0.986
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Before a definition of order statistics is given, let us consider a random variable that describes the time of work of such a system that consists of identical elements. It is defined by the equation:
X1, X X X
X
X n in( (1),X(2), , ( )n) (4.36) where: X(i); i = 1, 2, …, n denote the work times of the elements
X1,n denotes the first (shortest) time amongst n possible.
Denote by F(i)(x) the probability distribution of the random variables X(i).
Assuming that the work times of the elements are stochastically independent, the prob-ability distribution of the work time for a series system is determined by the formula:
n F
which means that the reliability (survival) function is given by the pattern:
Rn R If the probability distributions of the work time of the elements are exponential, i.e.
F( )i( )xx =1 exp(−e p(−λ( )( )iiii))xx)))) xx≥0 λ( )i >0 (4.39) where λ(i) is the intensity of the failures of the i-th element then the probability distribution of the work time of the system is also exponential and given by the formula:
Fn
The formula above contains information that the intensity of the failures of the system is the sum of all of the intensities of the failures of all of its elements.
The mean time of work of the system is determined by the equation:
E n i E
Particularly, if all of the elements are the same in a reliability sense then R1,n x n
R
R ( )x =[ (R )]
and for the exponential law
λ λλλ E
■ Example 4.4
Often, durability tests of chains, which are not used only in mining, rely on the following scheme. In testing a device, a three-link or five-link segment of the chain is fixed and an alternating load is applied periodically. Such tests are usually normalised. The test is finished the moment that any link breaks off. The number of load cycles until the moment of the occurrence of the break is noted6.
When the data collected from strength tests are gathered, it is usually presumed that the number of cycles noted contains information about the durability of the links in the chain and because this number is a random variable, the probability distribution is usually esti-mated and—as a rule—a log-normal distribution is applied.
The presented reasoning is flawed.
The random variable that is observed in testing—the number of load cycles until the moment of the failure of one out of three chain links or one out of five chain links—is the first order statistic X1,n, n = 3 or 5. From a reliability point of view, a chain segment is a system consisting of three or five identical elements, stochastically taking. All of the elements work until the moment of the first failure occurrence. Thus, these systems are non-repairable ones.
Recall now the definition of an order statistic.
Let X1, X2, …, Xn be a random vector of size n and let x1, x2, …, xn be the realisation of this vector. An order statistic Xk,n is a function of the random variables X1, X2, …, Xn taking k-th largest value in every sequence x1, x2, …, xn.
If f(x) denotes the probability density function of the random variable x and F(x) is the cumulative function, then the probability density function fk,n(x) is determined by the pattern:
f x n
k n k f x x x
ffk n, x) ! k x n k
(k )!( )! x [ ()F )] [ FFF( )]
= n−
−
F n (4.42)
As you can see the probability distribution functions of order statistics are different.
If the estimation of the probability distribution of the random variable—the number of load cycles until the moment of the failure occurrence—is done, this distribution is F1,n(x).
Therefore, if there is interest in the probability distribution of the durability of the links in a chain F(x) then, bearing in mind formula (4.37) and rearranging it appropriately, we have:
F( )x = 1 (−( FFF1,nn(x))n (4.43)
6 Some other investigations use a slightly different scheme. A given segment of a chain is tested for a given number of load cycles. If there is no failure during this test, the investigation is finished and it is assumed that the chain fulfils the stipulated requirements.
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As the investigations show, the data gathered during durability tests can be satisfactorily described by a log-normal distribution and this distribution is F1,n(x). Now, the problem is what can be stated about the probability distribution F(x).
Let us analyse this problem in more depth.
Presume that as a result of testing, a sequence of numbers was obtained that created a sample of size n; x1, x2, …, xn. Knowing a priori that the theoretical model is log-normal, it is necessary to convert the sample into a sequence of logarithms log x
1, log x
2, …, log x
n. The next step is the estimation of the structural parameters, i.e. the estimation of the average value and the standard deviation.
Presume that the estimation gives the following result:
− The estimate of the average value of the number of load cycles until the moment of the first failure occurrence out of n links tested, equals 3 × 104 (μ = log 3 × 104= 4.477)
− The estimate of the standard deviation equals 1.5 × 104 (σ = log 1.5 × 104= 4.176).
Having the estimations of these parameters of the probability function, a test of the goodness-of-fit should now be applied in order to verify the hypothesis which states that the empirical and the theoretical functions are identical statistically. It should be possible to describe the distribution of logarithms by the Gaussian distribution.
Presume that the test that was used gave no ground to reject the verified supposition.
If so, the probability density function is given by the pattern:
f x c
x
x x
n
ff1
2
2 2 2 0 c 04343
, x) exp (log )
= exp⎛− − .
⎝⎜
⎛⎛
⎝⎝
⎞
⎠⎟
⎞⎞
⎠⎠ > 0 c σ π2
μ σ and this is shown in Figure 4.5.
Let us now start a statistical inference about the probability density function f(x) that already has the probability density function f1,n(x) determined. Unfortunately, it is easy to come to the conclusion—looking at formula (4.43)—that there is no way to present the func-tion that is being searched for in an explicit form.
We can only:
− state that the distribution that is being searched for is not a log-normal one
− find a plot of the probability density function f(x) by applying an appropriate computer program.
0 2 . 104 4 . 104 6 . 104 8 . 104 1 . 105 0
5 . 10–6 1 . 10–5
f1,n(x)
x
Figure 4.5. The probability density function of a log-normal distribution of the first order statistic of a chain link.
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A graph of the probability density function of the number of load cycles of a chain link until the moment it fails for the data being considered and only the three chain links that created the segment (i.e. the diameter of the links of the chain is large) is presented in Figure 4.6.
Although the course of the function is similar to the function from Figure 4.5, it does not have a log-normal distribution. You can only try to find the theoretical distribution that will describe this function satisfactorily from a statistical point of view. Obviously the expected
values and the standard deviations should be the same. ◀
0.006
0.004
0.002
0 f(x)
x
0 10 20 30 40 50
Figure 4.6. The probability density function of chain link durability for n = 3.
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