EFFICIENCY CRITERION FOR A FAULT DETECTION SYSTEM IN LONG-TERM USE DEVICES

V. M. Razin

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

As is known from the literature [1], the mean economic losses \(\overline{W}_{\mathrm R}\) in systems for processing discrete information of central importance with a continuous process are determined by the relation

\[ \overline{W}_{\mathrm R} = \sum_{i=1}^{\alpha} \overline{\lambda}_i \overline{W}_i + \overline{W}_0 = \sum_{i=1}^{\alpha} \overline{\lambda}_i \left[ \int p_i(t_{\mathrm p})\, w(t_{\mathrm p})\, dt_{\mathrm p} + B_i \right] + W_0, \tag{1} \]

where \(\alpha\) is the number of possible types of failures;

\(\overline{W}_i\) is the mean loss per failure of the \(i\)-th type;

\(\overline{\lambda}_i\) is the mean intensity of occurrence of a failure of the \(i\)-th type, taking repair time into account;

\(p_i(t_{\mathrm p})\) is the density of the conditional probability of a downtime of duration \(t_{\mathrm p}\) in the case of the \(i\)-th failure;

\(w(t_{\mathrm p})\) is the loss function;

\(B_i\) are losses independent of information distortions, for example, losses for repair and for eliminating accidents occurring in the case of a failure of the \(i\)-th type;

\(W_0\) is the expenditure per unit time for routine technical maintenance.

The quantity \(\overline{\lambda}_i\) is determined by an expression of the form

\[ \overline{\lambda}_i = \frac{1}{T_{pi}+t_{ni}}, \tag{2} \]

where \(T_{pi}\) is the mean operating time of the device between failures of the \(i\)-th type;

\(t_{ni}\) is the mean repair time for the \(i\)-th failure. In many cases, for systems with a continuous process,

\[ w(t_{\mathrm p}) = A_0 t_{\mathrm p}, \tag{3} \]

where \(A_0\) is the specific loss (loss per unit time) during system downtime, equal to the mean revenue that the system would have yielded in the case of proper operation.

Taking (3) into account, we have

\[ \overline{W}_{\mathrm R} = \sum_{i=1}^{\alpha} \overline{\lambda}_i \left(A_0 t_{ni}+B_i\right) + W_0. \tag{4} \]

By analogy with (4), for the entire system as a whole we write

\[ \overline{W}_R=\Lambda_c(A_0t_p+B)+W_0, \tag{5} \]

where \(\Lambda_c=\sum_{i=1}^{a}\overline{\lambda_i}\) is the mean failure rate of the entire system, taking repair time into account;

\(t_p\) is the mean repair time per failure.

By analogy with (2), we obtain

\[ \Lambda_c=\frac{1}{T_p+t_p}, \tag{6} \]

where \(T_p\) is the mean operating time of the entire device between adjacent failures.

Taking (6) into account, expression (5) can be represented in the following form:

\[ \overline{W}_R=\frac{1}{T_p+t_p}(A_0+B)+W_0. \tag{7} \]

We shall represent the downtime for repair of the system \(t_p\) as two components,

\[ t_p=t_{ot}+t_y, \tag{8} \]

where \(t_{ot}\) is the mean time for locating the faulty element;

\(t_y\) is the mean time for eliminating the fault.

The mean time for eliminating a fault in a block-structured system is reduced mainly to replacing the faulty block with a serviceable one, provided that the stock of serviceable blocks is not limited. This quantity is more or less definite and constant.

As for the time expenditure for locating the faulty block, this quantity is random and depends on many factors: the type and class of failure, the adopted system for searching for and localizing the fault, the available test equipment and its reliability, the skill level of the maintenance personnel, etc.

It appears advisable, in order to assess the influence of this entire set of factors on downtime, to introduce a special coefficient \(K_{\mathrm{epn}}\)—the efficiency coefficient of the fault-search system. We define this coefficient as follows:

\[ K_{\mathrm{epn}}=\frac{t_y}{t_{ot}+t_y}=\frac{t_y}{t_p}. \tag{9} \]

If failures are detected instantaneously, then \(t_{ot}=0\) and \(K_{\mathrm{epn}}=1\). If \(t_{ot}>0\), then the value of \(K_{\mathrm{epn}}\) will decrease; thus, the permissible limits of variation of \(K_{\mathrm{epn}}\) are determined by the inequality

\[ 0<K_{\mathrm{epn}}\leq 1. \tag{10} \]

Using formula (9), we transform expression (7) to the form

\[ \overline{W}_R=\frac{1}{T_pK_{\mathrm{epn}}+t_y}\left(A_0t_y+BK_{\mathrm{epn}}\right)+W_0. \tag{11} \]

In practical cases

\[ B\ll T_pA_0, \]

and the quantity \(\overline{W}_R\) decreases monotonically with increasing \(K_{\mathrm{epn}}\).

In some practical cases one may assume \(B \cong 0\) and \(W_0 \cong 0\). Then from formula (7) we obtain the approximate relation

\[ \overline{W}_R \cong A_0 \frac{t_{\mathrm{p}}}{T_{\mathrm{p}} + t_{\mathrm{p}}}, \tag{12} \]

where the quantity

\[ \frac{t_{\mathrm{p}}}{T_{\mathrm{p}} + t_{\mathrm{p}}} = K_{\mathrm{p}} \tag{13} \]

is nothing other than the downtime coefficient, i.e., an operational characteristic widely used for the quantitative evaluation of various unproductive expenditures of time [2].

Using [9], it is not difficult to obtain

\[ K_{\mathrm{p}} = \frac{t_y}{K_{\mathrm{eff}}T_{\mathrm{p}} + t_y}. \tag{14} \]

As an example, let us consider the application of the proposed efficiency criterion for a fault-detection system to general-purpose electronic digital computers (EDCs). If an EDC provides monitoring of the correctness of computations, then any failure leads to an interruption of the continuous computation process. On these grounds, an EDC of this type may conventionally be classified as a system for processing discrete information of central importance [1].

Modern EDCs have very high productivity and, consequently, a large value of \(A_0\), whereas the parameters \(B\) and \(W_0\) are small. An examination of one computer of the “Minsk-1” type showed that during a year of operation \(T_{\mathrm{p}} = 21.7\) hours, \(t_{\mathrm{p}} = 2.7\) hours, \(t_y = 0.1\) hours, \(B = 5\) rubles, \(W_0 = 15\) rubles/hour. Further, if we assume that computations not performed because of downtime are carried out manually with the aid of keyboard calculating machines, then, in an approximate calculation, \(A_0 \cong 1000\) rubles/hour. On the basis of formulas (9) and (11), we obtain \(K_{\mathrm{eff}} = 0.037\) and

\[ \overline{W}_R = \frac{1}{21.7 \cdot 0.037 + 0.1} (1000 \cdot 0.1 + 5 \cdot 0.037) + 15 \cong 110 + \]

\[ + 0.2 + 15 = 125.2 \text{ rubles/hour}. \]

The use of EDCs of still greater productivity will increase the first term still more, and then the calculations may be carried out using formulas (12) and (14).

If an EDC is used in a system for the automatic control of production processes, then in this case the second term will have substantial significance, and it must be taken into account according to relation (11).

In conclusion, we note that, in order to evaluate the effectiveness of measures for locating faults in individual units of a device, the corresponding coefficients \(K_{\mathrm{eff}}\) may be introduced into equation (4) for the individual units of the system, in the same way as was done above for the entire system as a whole.

REFERENCES

1. T. A. Shastova. Criterion of average losses for evaluating the reliability of control systems. Automation and Remote Control, No. 6, 1962.

2. N. A. Shishonok, V. F. Repkin, L. L. Barvinskii. Fundamentals of the Theory of Reliability and Operation of Radioelectronic Equipment, Sovetskoe Radio, 1964.