On the methodology for collecting and processing statistical information on accelerator operation
Abstract Generated abstract
The paper outlines a methodology for systematically recording and statistically processing information on accelerator operation in order to evaluate reliability and maintainability. It proposes a failure-registration log designed to capture operating times, failure events, failed components, fault-location and restoration times, causes, corrective actions, and failure type, distinguishing catastrophic failures from noncatastrophic ones. The article then summarizes statistical procedures for estimating mean time between failures, fault-location and repair times, standard deviations, distribution laws, goodness-of-fit checks, availability, utilization, idle-time, operating-cost coefficients, and confidence intervals. This framework is presented as a basis for identifying weak accelerator components and supporting operational decisions and modernization.
Full Text
On the Methodology for Collecting and Processing Statistical Information on Accelerator Operation
N. I. Sablin
Systematic collection and processing of statistical information on accelerator operation is of the utmost importance for obtaining a posteriori data on their reliability, which are then used as initial data for many operational problems.
To collect statistical information, failure-registration logs have been introduced at the NIIYaF EA for all accelerators; the form of these logs and an example of how to fill them in are presented in Table 1.
Table 1
| Date | Turn-on time (hours) | Shutoff time (hours) | Failure time (hours) | Failed element and its place in the circuit | Fault-location time (hours) | Restoration time (hours) | Cause of failure | Corrective action | Type of failure |
|---|---|---|---|---|---|---|---|---|---|
| 29/III-66 | 9 15 |
14 15.5 |
— 15.5 |
— L., circuit 2 |
— 0.5 |
— 1 |
— Short circuit of electrodes |
— L., replacement with a new l. |
— Catastrophic |
The form of the log is arranged in such a way as to obtain the maximum information that would make it possible to determine all the principal quantitative reliability characteristics, to qualitatively analyze and systematize the processes occurring during failures, and also to systematize failures. Columns 2, 3, and 4 of Table 1 serve to determine reliability criteria; column 5, to differentiate the failed elements; columns 6 and 7, to determine maintainability criteria; columns 8 and 9, for qualitative analysis and systematization of the processes occurring during a failure; column 10 serves to systematize failures.
By the time for locating a fault one should understand the time interval from the moment the failure is recorded to the localization of the faulty element.
By the restoration time one should understand the time interval spent on replacing or repairing the failed element.
Two types of failures should be distinguished: catastrophic failure and noncatastrophic failure. A failure is called catastrophic if its occurrence leads to disruption of the experiment; otherwise it is noncatastrophic. Such a systematization of failures is necessary in order to identify the most “narrow” points of accelerators and subsequently to justify modernization of the nodes that are “weak” in reliability.
The accumulated information is processed by methods of mathematical statistics and is treated in detail in many works [1, 2, 3, 4, 5, 6].
Below we briefly present the procedure for calculating the principal criteria.
- The determination of the time between failures is carried out by the formula [1]
\[ T_0^*=\frac{1}{n}\sum_{i=1}^{n} t_i, \tag{1} \]
where \(T_0^*\) is the time between failures;
\(t_i\) is the operating time of the apparatus between restoration of the apparatus after the \((i-1)\)-th failure and the \(i\)-th failure;
\(n\) is the number of failures that occurred in the system during the specified calendar period.
Similarly, the average statistical time for finding a fault is determined:
\[ T_{\mathrm{n}}^*=\frac{1}{m}\sum_{j=1}^{m}\Theta_j, \tag{2} \]
where \(\Theta_j\) is the time spent searching for the \(j\)-th failed element;
\(m\) is the number of failed elements; and the average statistical restoration time is
\[ T_{\mathrm{в}}^*=\frac{1}{l}\sum_{\kappa=1}^{l}\tau_{\kappa}, \tag{3} \]
where \(\tau_{\kappa}\) is the restoration time of the \(\kappa\)-th failed element,
\(\kappa\) is the number of failed elements.
- The root-mean-square deviation for the time between failures is calculated by the formula [1]
\[ s_0=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}\left(t_i-T_0^*\right)^2}, \tag{4} \]
where \(s_0\) is the root-mean-square deviation for the time between failures.
Similarly, the root-mean-square deviations for \(T_{\mathrm{n}}^*\) and \(T_{\mathrm{в}}^*\) are calculated:
\[ s_{\mathrm{n}}=\sqrt{\frac{1}{m-1}\sum_{j=1}^{m}\left(\Theta_j-T_{\mathrm{n}}^*\right)^2}, \tag{5} \]
\[ s_b=\sqrt{\frac{1}{l-1}\sum_{\kappa=1}^{l}\left(\tau_{\kappa}-T_b^*\right)^2}. \tag{6} \]
- The experimental determination of the unknown distribution laws of the time between failures and the restoration time should be carried out in the following sequence [2]:
a) Pass to a new statistical population of realizations of the quantities
\[ x_{i0}=\frac{t_i}{T_0^*}\quad \text{or}\quad x_{ib}=\frac{\tau_{\kappa}}{T_b^*}; \tag{7} \]
b) determine the size of the grouping interval \(\Delta x\) and the total number of these intervals \(\kappa\) from the formulas
\[ \left. \begin{aligned} \Delta x &= \frac{x_{\max}-x_{\min}}{1+3.3\lg n},\\ \kappa &= \frac{x_{\max}-x_{\min}}{\Delta x} \end{aligned} \right\}; \tag{8} \]
c) construct a polygon of random numbers \(\Delta n_i^*\) (the number of failures falling in the \(i\)-th interval) and choose an approximating function for smoothing the statistical distribution;
d) check the agreement of the chosen approximating function with the experimental data; in doing so, the goodness-of-fit test should be carried out using 2–3 conformity criteria (for example, by means of the criteria \(\chi^2\) and \(\omega^2\)) [3, 4, 5, 6].
- The availability coefficient is calculated by the formula
\[ K_{\mathrm{g}}=\frac{T_0^*}{T_0^*+T_{\mathrm{p}}^*+T_{\mathrm{v}}^*}. \tag{9} \]
- The utilization coefficient is calculated by the formula
\[ K_{\mathrm{i}}= \frac{\displaystyle \sum_{i=1}^{n} t_i} {\displaystyle \sum_{j=1}^{m}\Theta_j+\sum_{\kappa=1}^{l}\tau_\kappa+\sum_{i=1}^{n}t_i}. \tag{10} \]
- The idle-time coefficient is calculated by the formula
\[ K_{\mathrm{p}}= \frac{\displaystyle \sum_{\kappa=1}^{l}\tau_\kappa+\sum_{j=1}^{m}\Theta_j} {\displaystyle \sum_{j=1}^{m}\Theta_j+\sum_{\kappa=1}^{l}\tau_\kappa+\sum_{i=1}^{n}t_i}. \tag{11} \]
The idle-time coefficient and the utilization coefficient are related to one another by the simple relation [1]:
\[ K_{\mathrm{p}}=1-K_{\mathrm{i}}. \tag{12} \]
- The economic evaluation is performed by means of the operating-cost coefficient [1]:
\[ K_x=\frac{C_1}{C_0}, \tag{13} \]
where \(C_1\) is the cost of operation for one year,
\(C_0\) is the cost of the apparatus.
The usual value of this coefficient is \(0.1 \div 0.2\).
- The calculation of confidence intervals and probabilities is carried out differently depending on the distribution law.
For example, for the normal distribution, calculation of the confidence probability reduces to calculation of the integral Student distribution [1]:
\[ P=\int_{-t_P}^{t_P}\varphi(t)\,dt, \tag{14} \]
where
\[ \varphi(t)= \frac{\Gamma\left(\dfrac{n}{2}\right)} {\sqrt{(n-1)\pi}\,\Gamma\left(\dfrac{n-1}{2}\right)} \left(1+\frac{t^2}{n-1}\right)^{-\frac{n}{2}} \]
\[
\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}\,dt
\]
is the gamma function,
\(t_P\) is the function of the selected probability.
The values of integral (14) are usually tabulated; the tables give the value \(t_P\) for a specified confidence probability \(P\) and for a specified number of trials \(n\). Using such tables, one can solve both direct and inverse problems, i.e., both the problem of calculating the confidence interval and the problem of determining the confidence probability.
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