On the Calculation of the Effectiveness of a Hardware Control System on a Computer
Abstract Generated abstract
This paper formulates a probabilistic method for calculating the effectiveness of a computer hardware monitoring system, defined as the conditional probability that an error is detected after it occurs. It separates overall effectiveness into the effectiveness of the mathematical checking method and that of its circuit implementation, then develops formulas for modulo checking under assumptions of independent bit failures and binomially distributed error multiplicities. The authors extend the calculation to computers composed of multiple monitored blocks, weighting each block by its failure and malfunction intensity, and give simplified expressions for systems built from similar elements. The approach is presented as applicable to other coding based monitoring methods and to units employing multiple checking methods, supporting allocation of required effectiveness values across computer units.
Full Text
ON THE CALCULATION OF THE EFFECTIVENESS OF A HARDWARE MONITORING SYSTEM FOR A COMPUTER
N. P. BAIDA, V. M. RAZIN, V. M. TANASEICHUK
(Presented by the scientific seminar of the Department of Computer Engineering)
The effectiveness of a hardware monitoring system for a computer is understood as the conditional probability that the monitoring circuit detects an error, if it has occurred [1].
The effectiveness of hardware monitoring depends on the effectiveness of the mathematical method underlying the monitoring and on the circuit implementation of this method, i.e.,
\[ E = E_m \cdot E_c, \tag{1} \]
where \(E_m\) is the probability of detecting an error by the given monitoring method, if it has occurred; \(E_c\) is the probability of detecting errors in the monitoring circuit that implements the adopted monitoring method.
The effectiveness of the monitoring method in the general case can be calculated by the formula
\[ E_m = \frac{S_0}{N_0}, \tag{2} \]
where \(S_0\) is the number of different errors in the monitored circuit that are detected with an absolutely reliable monitoring circuit; \(N_0\) is the maximum possible number of different errors in the monitored circuit.
The effectiveness of the circuit implementation of the monitoring method can be calculated by the formula
\[ E_c = \frac{S_k}{N_k}, \tag{3} \]
where \(S_k\) is the number of different errors in the monitoring circuit that are detected during monitoring; \(N_k\) is the maximum possible number of different errors in the monitoring circuit.
At present, the problems of calculating monitoring effectiveness for various monitoring methods have been little developed.
Let us calculate the effectiveness for the most widely used method—modulo monitoring.
Suppose that the following conditions hold for the circuit under consideration:
— the occurrence of a 1 or a 0 in each digit is equiprobable;
— failures in different bits of the circuit are independent of one another;
— the probabilities of failures in zero and one are the same for all elements of the circuit of the same type and do not change over time.
Under these assumptions one may assume that errors are distributed according to the binomial law, i.e., the probability of occurrence of a \(j\)-fold error \(P_j\) is determined from the following relation:
\[ P_j=C_n^j P_t^j(1-P_t)^{n-j}, \tag{4} \]
where
\(P_t\) is the probability of occurrence of an error in one bit of the machine during the prediction time \(t\);
\(n\) is the number of independent bits in the computer;
\(j\) is the multiplicity of the error.
The calculation of the conditional probabilities of detecting errors of multiplicity \(j(\mathcal{E}_j)\) for various deductions is given in [1]. The total effectiveness of the modulo-checking method can be calculated as the weighted mean of the quantities \(\mathcal{E}_j\) (partial effectivenesses), taking into account the probabilities of occurrence of errors of multiplicity \(j\), i.e.,
\[ \mathcal{E}_{\mathrm{m}}= \frac{\displaystyle\sum_{j=1}^{m} P_j\mathcal{E}_j} {\displaystyle\sum_{j=1}^{m} P_j}. \tag{5} \]
Usually one restricts oneself to \(m=3\), since \(P_{m>3}\approx 0\). The effectiveness of modulo checking is
\[ \mathcal{E}=\mathcal{E}_{\mathrm{m}}\cdot \mathcal{E}_{\mathrm{c}}\approx \mathcal{E}_{\mathrm{m}}, \tag{6} \]
because almost all errors in the checking circuit are detected.
If the computer consists of \(n\) blocks and each block \(n_i\) has a hardware-monitoring circuit with checking effectiveness \(\mathcal{E}_i\), then the effectiveness of the hardware-monitoring system of the entire computer can be computed by the formula:
\[ \mathcal{E}= \frac{\displaystyle\sum_{i=1}^{n} P_i\mathcal{E}_i} {\displaystyle\sum_{i=1}^{n} P_i}, \tag{7} \]
where
\(P_i=1-e^{-x_i t}\) is the probability of occurrence of errors in the \(i\)-th block;
\(x_i\) is the total intensity of failures and malfunctions in the \(i\)-th block*;
\(t\) is the operating time of the machine after preventive maintenance. According to the formula of approximate calculations
\[ e^{-xt}\approx 1-xt, \tag{8} \]
and, if \(xt \leq 0.045\), then \(\delta \leq 0.1\%\).
Then formula (7) is transformed as follows:
\[ \mathcal{E}= \frac{\displaystyle\sum_{i=1}^{n} P_i\mathcal{E}_i} {\displaystyle\sum_{i=1}^{n} P_i} = \frac{\displaystyle\sum_{i=1}^{n} (1-e^{-x_i t})\cdot \mathcal{E}_i} {\displaystyle\sum_{i=1}^{n} (1-e^{-xt})} \approx \]
* As shown in [2], the intensity of malfunctions \((\lambda_{\mathrm{c}})\) in a computer obeys the Poisson law, just as does the intensity of failures \((\lambda_0)\). Therefore one may consider the quantity \(x=\lambda_0+\lambda_{\mathrm{c}}\) as the total intensity of failures and malfunctions.
\[ \simeq \frac{\sum_{i=1}^{n} \chi_i t \cdot \mathcal{E}_i}{\sum_{i=1}^{n} \chi_i t} = \frac{\sum_{i=1}^{n} \chi_i \mathcal{E}_i}{\sum_{i=1}^{n} \chi_i}. \tag{9} \]
If all the units of the computer are built from elements of the same type, having a total failure and malfunction rate \(\chi_0\), then
\[ \mathcal{E} = \frac{\sum_{i=1}^{n} \chi_i \mathcal{E}_i}{\sum_{i=1}^{n} \chi_i} = \frac{\sum_{i=1}^{n} l_i \chi_0 \mathcal{E}_i}{\sum_{i=1}^{n} l_i \chi_0} = \frac{\sum_{i=1}^{n} l_i \mathcal{E}_i}{\sum_{i=1}^{n} l_i}, \tag{10} \]
where \(l_i\) is the number of module elements in the \(i\)-th unit.
The proposed method can be applied to calculating the effectiveness of hardware monitoring with the use of various codes (Hamming, cyclic, \(AN+B\), etc.).
In the remaining cases the computations are performed according to formula (1).
If different monitoring methods are used to monitor the operation of a unit, then the overall effectiveness of the unit’s hardware-monitoring scheme can be determined, using methods of probability theory, by the formula:
\[ \mathcal{E}_{e i} = \mathcal{E}_{i1} + \mathcal{E}_{i2}\mid \overline{\mathcal{E}_{i1}} + \mathcal{E}_{i3}\mid \overline{\mathcal{E}_{i1}\cup \mathcal{E}_{i2}} +\ldots+ \mathcal{E}_{ip}\mid \overline{\mathcal{E}_{i1}\cup \mathcal{E}_{i,p-1}}. \tag{11} \]
where
\(m\) is the number of different monitoring methods used in the hardware-monitoring scheme of the \(i\)-th unit;
\(\mathcal{E}_{ip}\) are the effectivenesses of the different monitoring methods used to monitor the operation of the \(i\)-th unit;
\(\mathcal{E}_{ip}\mid \overline{\bigcup \mathcal{E}_{i1}, \mathcal{E}_{i,p-1}}\) is the partial effectiveness of the \(p\)-th method (the conditional probability of detecting an error by the \(p\)-th method, under the condition that it is not detected by the other methods with probability \(\overline{\bigcup \mathcal{E}_{i1}, \mathcal{E}_{i,p-1}}\)).
Using the proposed method for calculating monitoring effectiveness, one can, on the basis of a specified effectiveness of the hardware-monitoring system for the entire computer, assign such values of \(\mathcal{E}\) for the individual units that, when substituted into formula (7), the value obtained is
\[ \mathcal{E} \ge \mathcal{E}_{\mathrm{adm}}, \]
where \(\mathcal{E}_{\mathrm{adm}}\) is the effectiveness value of the computer hardware-monitoring system that is specified in the technical assignment for development or is selected on the basis of requirements for the reliability and trustworthiness of the computer’s computations.
REFERENCES
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N. D. Putintsev. Hardware Monitoring of Control Digital Computing Machines. “Sovetskoe radio,” Moscow, 1966.
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O. V. Shcherbakov. On the estimation of random malfunctions in digital computing machines. Publishing House of the Academy of Sciences of the USSR, Technical Cybernetics, No. 3, 1964.