CONDITIONS FOR DIVIDING COMPLEX NON-REDUNDANT RADIO-ELECTRONIC EQUIPMENT INTO UNITS WITH MONITORED OPERABILITY

V. M. RAZIN

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

When complex radio-electronic equipment is divided into units with monitored operability, there exist conditions under which the minimum value of the downtime coefficient is ensured [1].

These conditions are defined in [1] for equipment consisting of \(N\) elements identical in reliability. The equipment is divided into \(M\) monitored units, with \(N/M\) elements in each. The number of units of the monitoring system is assumed to be equal to the number of units of the monitored equipment [1, Ch. 13, § 3]. This circumstance makes it possible to monitor all units of the equipment simultaneously, and, naturally, the time required to locate the faulty element may be considered inversely proportional to the number of monitored units.

The indicated method of monitoring is not always rational, in view of the considerable volume of the monitoring equipment. In some cases, a substantial reduction in the volume of monitoring equipment can be achieved by using a sequential method of monitoring individual units. In this case the monitoring system, like the control, indication, and other circuits, is common to the entire monitoring device. Under these conditions, the number of units of the monitored equipment will mainly affect the number of switching elements in the monitoring-equipment system.

For the case under consideration, it is expedient to represent the mean time of one restoration \(t_{\mathrm{v}}\) as the sum of three terms

\[ t_{\mathrm{v}}=\frac{t_{\mathrm{ot}}}{M}+Mt_{\mathrm{p}}+t_{\mathrm{u}}, \tag{1} \]

where \(t_{\mathrm{ot}}\) is the mean time for locating a fault in the absence of a monitoring system;

\(t_{\mathrm{p}}\) is the mean time expended on carrying out an operability check in one unit when \(N/M\) elements are connected sequentially;

\(t_{\mathrm{u}}\) is the mean time for eliminating the fault.

The first term in relation (1) takes into account the circumstance that the time for locating a faulty element in a unit will be proportional to the number of elements in the unit. The second term is introduced under the assumption that, in order to reveal the faulty unit, a time-sequential check of all units of the monitored equipment is carried out.

Thus, the first two terms together constitute the resultant time spent on locating the faulty element when the units are checked sequentially.

It is obvious that the sequential search method should be introduced provided that the inequality

\[ M t_{\mathrm{p}} \le \frac{t_{\mathrm{ot}}}{M} \quad \text{or} \quad t_{\mathrm{p}} \le \frac{t_{\mathrm{ot}}}{M^{2}}, \]

is satisfied, so as not to increase substantially the time for fault location. Fulfillment of the latter inequality is possible in automated monitoring systems.

In what follows we shall rely on the same assumptions as in [1, Ch. 13, Sec. 3], except for the assumption that the time for locating a faulty element is inversely proportional to the number of monitored units. In our case, the time for locating a faulty element will be determined by the sum of the first two terms in relation (1). In accordance with [1], let us denote:

\[ \lambda_{\mathrm{k}} \]
— the failure rate of one unit of the monitoring system associated with each unit of the monitored equipment;

\[ \Lambda_{\mathrm{k}} \]
— the failure rate of the common devices of the monitoring system;

\[ \gamma_{1} \]
— the probability that failure of a unit of the monitoring system will lead to incorrect information about a faulty unit of the equipment;

\[ \gamma_{0} \]
— the probability that failure of the common devices of the monitoring system will lead to incorrect information about faulty equipment;

\[ \Lambda = \sum_{i=1}^{N} \lambda_i \]
— the failure rate of the equipment;

\[ t_{\mathrm{yk}} \]
— the mean time for locating and eliminating a fault in the monitoring system;

\[ t_{v\Sigma} \]
— the total downtime for restoration over a certain operating period \(T\), during which the equipment operated for a time \(T_{\mathrm{r}}\).

Following the method set forth in [1], we obtain

\[ \frac{t_{v\Sigma}}{T_{\mathrm{r}}} = M \left[\lambda_{\mathrm{k}}\left(t_{\mathrm{yk}}+\gamma_{1}t_{\mathrm{ot}}\right)+\Lambda t_{\mathrm{p}}\right] +\lambda_{\mathrm{k}}(1-\gamma_{1})t_{\mathrm{ot}} +\Lambda_{\mathrm{k}}t_{\mathrm{yk}} + \]

\[ +\Lambda_{\mathrm{k}}\gamma_{0}t_{\mathrm{ot}} +\Lambda t_{\mathrm{y}} +\frac{1}{M}\left[t_{\mathrm{ot}}\Lambda_{\mathrm{k}}(1-\gamma_{0})+\Lambda t_{\mathrm{ot}}\right], \tag{2} \]

The restoration downtime coefficient is determined, as usual, from the expression

\[ K_{\mathrm{p}} = \frac{t_{v\Sigma}}{t_{v\Sigma}+T_{\mathrm{r}}} = \frac{1}{1+\dfrac{T_{\mathrm{r}}}{t_{v\Sigma}}}. \tag{3} \]

Investigation of expression (2) for a minimum gives

\[ M_{\mathrm{opt}} = \sqrt{ \frac{ t_{\mathrm{ot}}\left[\Lambda_{\mathrm{k}}(1+\gamma_{0})+\Lambda\right] }{ \lambda_{\mathrm{k}}\left(t_{\mathrm{yk}}+\gamma_{1}t_{\mathrm{ot}}\right)+\Lambda t_{\mathrm{p}} } }. \tag{4} \]

When the number of monitored units is equal to \(M_{\mathrm{opt}}\), the minimum value of the downtime coefficient and the maximum value of the availability coefficient are ensured.

Substituting (4) into (2), after several simple transformations we find

\[ \left(\frac{t_{\mathrm{v}\Sigma}}{T_{\mathrm{p}}}\right)_{\min} = t_{\mathrm{ot}}\left[\Lambda_{\mathrm{k}}\gamma_{0}+\lambda_{\mathrm{k}}(1-\gamma)\right] + t_{\mathrm{uk}}\Lambda_{\mathrm{k}} + t_{\mathrm{u}}\Lambda + 2\sqrt{t_{\mathrm{ot}}\left[\Lambda_{\mathrm{k}}(1-\gamma_{0})+\Lambda\right]\left[\lambda_{\mathrm{k}}\left(t_{\mathrm{uk}}+\gamma_{1}t_{\mathrm{ot}}\right)+\Lambda t_{\mathrm{p}}\right]} . \tag{5} \]

and the minimum value of the downtime coefficient

\[ (K_{\mathrm{p}})_{\min} = \frac{1}{1+\left(\dfrac{T_{\mathrm{p}}}{t_{\mathrm{v}\Sigma}}\right)_{\max}} . \tag{6} \]

The downtime coefficient without division into monitored units is determined as follows:

\[ K'_{\mathrm{p}} = \frac{1}{1+\dfrac{1}{\Lambda\left(t_{\mathrm{ot}}+t_{\mathrm{p}}+t_{\mathrm{u}}\right)}} . \tag{7} \]

The gain from using the monitoring system with the optimal number of monitored units is expressed by the ratio

\[ K=\frac{K'_{\mathrm{p}}}{(K_{\mathrm{p}})_{\min}} . \]

The obtained calculation relations can be simplified if one takes into account that, as a rule, the reliability of the monitoring devices considerably exceeds the reliability of the equipment being monitored, i.e., the inequalities hold

\[ \lambda_{\mathrm{k}} < \Lambda_{\mathrm{k}} \ll \Lambda . \tag{8} \]

In this particular case, relations (2), (4), and (5) are written in the following form:

\[ \frac{t_{\mathrm{v}\Sigma}}{T_{\mathrm{p}}} \simeq \Lambda\left(t_{\mathrm{u}}+Mt_{\mathrm{p}}+\frac{t_{\mathrm{ot}}}{M}\right), \tag{9} \]

\[ M_{\mathrm{opt}} \simeq \sqrt{\frac{t_{\mathrm{ot}}}{t_{\mathrm{p}}}}, \tag{10} \]

\[ \left(\frac{t_{\mathrm{v}\Sigma}}{T_{\mathrm{p}}}\right)_{\min} \simeq \Lambda\left(t_{\mathrm{u}}+2\sqrt{t_{\mathrm{ot}}t_{\mathrm{p}}}\right). \tag{11} \]

The approximate formulas [9], [10], and [11] make it possible to calculate quickly, with a sufficient degree of accuracy, the corresponding quantities when dividing a complex radio-electronic system into monitored units.

References

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