ON THE QUESTION OF DETERMINING OPTIMAL MODULE DIMENSIONS IN COMPLEX RADIO-ELECTRONIC DEVICES

V. M. Razin

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

In modern complex radio-electronic devices, the modular principle of structural design of radio equipment has become widespread. Microminiaturization of radio equipment makes it possible to combine in a single module an ever increasing number of elements, with a corresponding expansion of the functional capabilities of the module. The use of micromodular designs and the development of technology for creating solid-state integrated circuits make it possible to pose and solve the problem of creating a complex radio-electronic device in a single unit. Such an approach to solving the problem of creating small-size radio equipment is justified only in the case when high reliability is simultaneously ensured both for the individual elements and for the device as a whole. The structural design of a device in the form of one integral unit excludes the possibility of repair by replacing the failed part, and by virtue of this circumstance the maintainability of such a device may prove insufficiently high. On the other hand, replacement of the entire device as a result of the failure of a single element will not always be economically justified.

For the purpose of increasing the maintainability of radio equipment and economy with respect to operating expenses, it appears advisable to preserve the modular principle of constructing complex radio-electronic devices and to establish, in a certain sense, the optimal dimensions of an individual module. The use of a modular design makes it possible, as is known, to reduce the time spent on repair, since all repair operations are reduced to replacing the failed module with a serviceable spare module. This leads to an increase in the maintainability of radio equipment in the case of its repeated use and to a reduction in operating expenses through savings in the cost of spare parts. In a particular case, the choice of optimal module dimensions in the sense of increasing maintainability and reducing operating expenses can be made on the basis of the calculations set forth below, as applied to complex logical or computational digital electronic devices.

Let us suppose that there is some discrete-action radio-electronic device consisting of \(M\) approximately identical elements with total volume \(V\). Let us divide this device into \(k\) modules identical in size and in number of parts, where the module will have volume \(v\) and a number of elements (parts) equal to \(m\).

Obviously,

\[ k=\frac{V}{v}=\frac{M}{m}, \tag{1} \]

where the number of modules \(k\) may take the values

\[ 1 \leqslant k \leqslant M \tag{2} \]

and \(V\) and \(M\) are, for the given device, constant quantities. When \(k=M\), \(m=1\), each individual element is a module; and when \(k=1\), the entire device is a single module. Let us denote by \(c_{\text{d}}\) the cost of one part (element), and by \(\lambda_{\text{d}}\) the failure rate of one part without taking dependent failures into account. In this case the cost of the total number of modules replaced per unit time will be

\[ S_1=k\lambda_{\text{d}}m(c_{\text{d}}m)=k\lambda_{\text{d}}c_{\text{d}}m^2, \tag{3} \]

where \(c_{\text{d}}m\) is the cost of one module;

\(\lambda_{\text{d}}m\) is the failure rate of one module without taking dependent failures into account [1];

\(k\lambda_{\text{d}}m\) is the failure rate of all \(k\) modules of the device.

Taking relation (1) into account, we rewrite (3) in the form

\[ S_1=\lambda_{\text{d}}c_{\text{d}}\frac{M^2}{k}=\frac{A}{k}, \tag{4} \]

where \(A=\lambda_{\text{d}}c_{\text{d}}M^2\) is a certain constant for the given radio-electronic device.

When the entire device is divided into modules, intermodule connections (links) inevitably appear, which somewhat increase the cost of the whole product. In addition, the links between modules, like the basic elements, may also be a cause of failures and, consequently, additional costs arise for repairing these links. We shall regard the links between elements inside a module as absolutely reliable. Let us introduce the following additional notation: \(C_{\text{sv}}\) is the cost of repairing one link, and \(\lambda_{\text{sv}}\) is the failure rate of one link. We shall take the total number of links into account approximately as follows. Suppose that our device has the form of a cube and is also divided into modules of cubic shape. Suppose further that the total number of links of a given module with all the others is proportional to the total external surface area of the module, for a constant value of the number of links \(\sigma_{\text{sv}}\) per unit surface area. The latter assumption to some extent reflects the fact that the number of links of a given module with the others will grow not in proportion to the number of parts in the module (i.e., to its volume), but somewhat more slowly, owing to the appearance of absolutely reliable links between elements inside the module. We shall subtract the external links of the entire device with the “external world” from the total number of external links of the modules, since these links are unchanged for any design configuration. Consequently, the total number \(N\) of intermodule links taken into account in our calculations will be equal to

\[ N=\sigma_{\text{sv}}\left(k6\sqrt[3]{v^2}-6\sqrt[3]{V^2}\right), \tag{5} \]

where \(6\sqrt[3]{v^2}\) is the surface area of one module;

\(6\sqrt[3]{V^2}\) is the external surface area of the entire device.

The cost \(S_2\) of the total number of repaired connections per unit time, taking into account the above notation and relation (1), will be

\[ S_2 = \lambda_{\mathrm{sv}} c_{\mathrm{sv}} N = B\left(\sqrt[3]{k}-1\right), \tag{6} \]

where \(B = 6\lambda_{\mathrm{sv}} c_{\mathrm{sv}} \sigma_{\mathrm{sv}} \sqrt[3]{V}\) is a certain constant of the radio-electronic device.

The resulting cost \(S\) per unit time, due to the replacement of modules and of connections between modules when repairing the entire device, will obviously be equal to

\[ S = S_1 + S_2, \tag{7} \]

or

\[ S = f(k) = \frac{A}{k} + B\left(\sqrt[3]{k}-1\right). \tag{8} \]

The obtained relation (8) can be investigated by known methods for a minimum, and thus the optimal module size can be determined, based on the conditions of reducing the part of operating costs associated with the replacement of modules and the repair of connections.

The graphs in Fig. 1 give an example of the function \(f(K)\) for a certain ratio of the constants \(A\) and \(B\), having a clearly expressed minimum.

Fig. 1. Dependence of the cost of replacing modules and its components on the number of modules.

From this example it is also evident that, for the given radio-electronic device, it is expedient to divide the entire volume into a number of modules on the order of 8. Thus, the calculation given above confirms the proposition that, when determining module dimensions, one should take into account the costs both of the modules themselves and of the connections between them, with allowance for the failure rates of both.

When solving these questions in specific cases as applied to quite definite devices, the division into modules must be carried out with allowance for a number of additional considerations, with the aid of electronic digital computers capable of taking into account all

specific interrelationships among modules in different variants and to select the optimal variant, taking into account the considerations set forth above.

It should be noted that, apparently, the increase in the cost of a module will depend on the volume of the module somewhat faster than to the first power, as was assumed above. This also applies to the increase in the cost of interconnections as a function of their number. The specific dependences are determined by the production technology and must be taken into account accordingly.

REFERENCES

1. V. A. Lutskii. Calculation of the Reliability and Effectiveness of Radio-Electronic Equipment. Publishing House of the Academy of Sciences of the Ukrainian SSR, 1963.