On the Reliability of Computer Calculations
Abstract Generated abstract
This paper analyzes the reliability of computer computations in terms of the probability that calculation results remain correct in the presence of equipment failures and malfunctions. It develops analytical formulas for the reliability level both without hardware control and with hardware checking, incorporating failure and malfunction intensities, the probability of initial serviceability, the effectiveness of test control, and the effectiveness of hardware control. The analysis shows that test-control effectiveness has the greatest influence over short operating intervals, while hardware-control effectiveness becomes dominant over longer intervals. The resulting expressions can be used to choose required control effectiveness values and maximum intervals between test checks for a specified computation reliability level.
Full Text
On the Question of the Reliability of Computer Computations
N. P. Baida, V. M. Razin, V. M. Tanaseichuk
(Presented by the scientific seminar of the Department of Computer Engineering)
For a computer, a quantitative assessment of reliability—the level of reliability—characterizes the truth of the result of computations and is one of the most important indicators of machine operation. To derive analytical dependencies that make it possible to compute the level of reliability, let us consider the following characteristics of a computer:
\[
\Lambda_{o}
\]
— the intensity of equipment failures;
\[
\Lambda_{c}
\]
— the intensity of equipment malfunctions \([1]\);
\[
x=\Lambda_{o}+\Lambda_{c}
\]
— the total intensity of equipment failures and malfunctions.
Let us suppose that, in the absence of hardware-control means, any malfunction or failure in the machine leads to an error in the computations. Then the reliability level of a computer without control is determined by the formula:
\[ D=P_{\text{и}}\cdot e^{-xt}, \tag{1} \]
where
\[
P_{\text{и}}
\]
— the probability that the computer will be serviceable at the moment computations begin;
\[
t
\]
— the time elapsed from the moment computations begin. The quantity \(P\) can be determined from the expression:
\[ P_{\text{и}}=e^{-\Lambda_{0}t^{*}}+\left(1-e^{-\Lambda_{0}t^{*}}\right)\cdot E_{\text{т.к.}}, \tag{2} \]
where
\[
E_{\text{т.к.}}
\]
— the effectiveness of test control of the computer (the probability that any failure in the machine will be detected during test control);
\[
t^{*}
\]
— the time elapsed from the moment of elimination of the last failure detected during previously conducted checks.
If no failures have been observed in the computer, then \(t^{*}\) must be counted from the moment the computer began operation.
If the computer has a hardware-control system that makes it possible to detect failures and random malfunctions, then the level of reliability will increase by the amount:
\[ D_{1}=\left(1-e^{-xt}\right)E_{a}, \tag{3} \]
where \(E_{a}\) is the effectiveness of hardware control (the conditional probability that an error will be detected if it has occurred);
\(t\) is the operating time of the computer from the moment of completion of the last test check;
\((1-e^{-\chi t})\) is the probability of the occurrence of a failure or malfunction in the machine during time \(t\).
In the presence of hardware checking, expression (1) is transformed as follows:
\[ D=P_{\mathrm{i}}e^{-\chi t}+(1-e^{-\chi t})\cdot E_{\mathrm{a}} =E_{\mathrm{a}}+e^{-\chi t}(P_{\mathrm{i}}-E_{\mathrm{a}}). \tag{4} \]
It follows from (4) that, as \(t\) increases, the level of reliability approaches the value \(E_{\mathrm{a}}\).
Let us differentiate expression (4) with respect to the parameters \(E_{\mathrm{a}}\) and \(E_{\mathrm{t.c.}}\):
\[ D'_{E_{\mathrm{a}}}=1-e^{-\chi t}; \tag{5} \]
\[ D'_{E_{\mathrm{t.c.}}}=(1-e^{-\Lambda_{0}t^{*}})e^{-\chi t}. \tag{6} \]
The corresponding graphs are shown in Fig. 1.
Fig. 1.
Analysis of expressions (5), (6) shows that the greatest influence on the level of reliability at small \(t\) is exerted by the quantity \(E_{\mathrm{t.c.}}\), and at large \(t\)—by \(E_{\mathrm{a}}\).
The optimal relations between the quantities \(E_a\) and \(E_{\text{t.c.}}\) are easy to determine on the basis of the required level of reliability of computations.
Conclusions
An analysis has been presented of one of the important characteristics of computer operation—the reliability of its functioning.
Analytical dependences have been obtained that make it possible to calculate the level of reliability of computations on a computer.
It has been shown that the greatest influence on the level of reliability is exerted by the quantities \(E_a\) and \(E_{\text{t.c.}}\). Using formula (4), for a specified level of reliability of computer operation one can determine the required values of the effectiveness of test and hardware control of a computer, as well as the maximum time between test checks.
References
- O. V. Shcherbakov. On the estimation of random failures in digital computing machines. Izv. AN SSSR. Tekhnicheskaya kibernetika, No. 3, 1964.