ACCOUNTING FOR THE INFLUENCE OF HARDWARE MONITORING ON THE AVAILABILITY FACTOR OF A CLASS OF DIGITAL INFORMATION SYSTEMS

V. M. TANASEICHUK, V. M. RAZIN

(Presented by the scientific seminar of the Department of Computer Engineering)

The principal advantage of hardware monitoring (HM) is the possibility of using it for continuous diagnostics of an object during operation. For the class of digital information systems (DIS) considered here, hardware monitoring is the only type of monitoring that detects the appearance of errors in the process of information conversion.

However, it is not always possible to cover all devices of a DIS with HM. One of the features of the operating conditions of the DIS under consideration as part of an integrated object-control system is that the fact of failure of an unmonitored part of the DIS will, with some delay, be detected by the operator, since he has at his disposal sources of information that make it possible to monitor the correctness of the object-control process.

This article considers one possible approach to estimating the availability factor \(K_{\mathrm{g}}\) of a DIS, taking into account the depth of monitoring coverage, the unreliability of HM circuits, and the adopted maintenance procedure.

A general approach to determining the availability factor \(K_{\mathrm{g}}\) of a system can be found if the change of system states is regarded as a random process, continuous in time and discrete in space. Consider a device (a unit, a system) that consists of operating equipment (OE), which performs the main functions, and a monitoring unit (MU). By MU we mean the equipment of the hardware-monitoring circuits.

We assume that the operating equipment and the monitoring unit consist of a large number of elements, that element failures are independent, and that the failure of any element of the operating part and of the monitoring unit leads to failure of the device. The failure streams of the elements will be considered stationary. In the general case, the failure stream of the monitoring unit and of the operating equipment can be represented as the sum of a number of independent stationary failure streams, so that each component stream constitutes a very small fraction of the sum. Then, on the basis of the limit theorem [1], the failure streams of the OE and MU may be regarded as simple, with intensities \(\lambda\) and \(\lambda_k\), and the distribution of the time to equipment failure may be described by the exponential distribution law.

We shall make a similar assumption with respect to the recovery process as well. We shall assume that the main fraction of faults can be eliminated in a short time, while some units, which fail very rarely, require lengthy repair. Therefore, let us assume that the time

the recovery of the WE and the CB is distributed according to an exponential law. It should be noted that the hypothesis of exponentiality is well confirmed in practice [2].

We take the recovery rates of the CB and of the working part covered by monitoring to be equal, owing to the fact that:

— the CB circuits are built from the same elements as the WE;

— the WE unit and the CB have the same design;

— the methods of searching for and replacing failed units, followed by checking operability, coincide for the WE and the CB.

We shall denote the recovery rate of the WE covered by the monitoring scheme by \(\mu_1\), and the rate at which the operator establishes the fact of failure in the unmonitored part of the WE by \(\gamma\). We shall also assume that during recovery the device does not fail. Since the time required to establish the fact of failure of the monitored part of the WE and of the CB built on hardware monitoring circuits is negligibly small, we take it to be zero.

We establish the following procedure for servicing the system. After the fact of failure has been detected by the control block, the device immediately enters the state of operability restoration. When the fact of failure of the unmonitored part of the WE is established, the operability of the device as a whole is checked and the failure in the unmonitored part of the WE (NWE) is eliminated with rate \(\mu_2\).

The state diagram of the WE and CB, taking into account the adopted assumptions and the adopted servicing procedure, is shown in Fig. 1.

Fig. 1. State diagram of the working equipment and the control block under the established servicing procedure

Considering the change of states of the WE and CB as a random process \(X(t)\), continuous in time but discrete in space, it can be shown that, by virtue of the assumptions adopted, it is Markovian [3].

Let us explain the meaning of the graph states:

\(P_0\) — state of operability of the WE and CB;

\(P_1\) — state of localization and elimination of a detected failure in the monitored part of the working equipment (MWE) or CB;

\(P_2\) — state of an undetected failure in the unmonitored part of the WE;

\(P_3\) — state of localization and elimination of a failure in the MWE or CB; there is a failure in the NWE;

\(P_4\) — state of localization and elimination of a failure in the NWE and of a general check of operability.

The failure rate of the WE is equal to

\[ \lambda = \lambda_0 + \lambda_{\mathrm{n}}, \]

where \(\lambda_0\) and \(\lambda_{\mathrm{n}}\) are, respectively, the failure rates of the MWE and NWE. We introduce the following probabilities:

\[ P_i(t) = \{X(t) = P_i\}, \qquad i = [0,4]. \tag{1} \]

In accordance with the transition graph, we compose the following system of Kolmogorov differential equations for the introduced probabilities:

\[ \left\{ \begin{aligned} \frac{dP_0}{dt} &= \mu_1 P_1(t) + \mu_2 P_4(t) - (\lambda_0+\lambda_{\mathrm{k}}+\lambda_{\mathrm{n}})P_0(t);\\ \frac{dP_1}{dt} &= (\lambda_0+\lambda_{\mathrm{k}})P_0(t) - \mu_1 P_1(t);\\ \frac{dP_2}{dt} &= \lambda_{\mathrm{n}} P_0(t) + \mu_1 P_3(t) - (\lambda_0+\lambda_{\mathrm{k}}+\gamma)P_2(t);\\ \frac{dP_3}{dt} &= (\lambda_0+\lambda_{\mathrm{k}})P_2(t) - \mu_1 P_3(t);\\ \frac{dP_4}{dt} &= \gamma P_2(t) - \mu_2 P_4(t);\\ \sum_{i=0}^{4} P_i(t) &= 1 \quad \text{— the normalizing condition.} \end{aligned} \right. \tag{2} \]

The system of differential equations (2) can be solved by means of Laplace transforms. Here a multiple-action device is considered, and the behavior of the system only on time intervals far from the initial one is of practical interest. Also of interest are the asymptotic values of the probabilities of the various states of the PO and BK, i.e.,

\[ P_i=\lim_{t\to\infty} P_i(t), \qquad i=\overline{0,4}. \tag{3} \]

Equating the left-hand sides of equation (2) to zero, we find \(P_0\)—the probability of the system’s operable state.

\[ P_0= \frac{1}{ 1+\frac{\lambda_{\mathrm{n}}}{\gamma} +\frac{\lambda_{\mathrm{n}}}{\gamma\mu_1}(\lambda_0+\lambda_{\mathrm{k}}) +\frac{\lambda_{\mathrm{n}}}{\mu_2} +\frac{1}{\mu_1}(\lambda_0+\lambda_{\mathrm{k}}) }. \tag{4} \]

Introduce

\[ \alpha=\frac{V_0}{V_0+V_{\mathrm{n}}}=\frac{V_0}{V} \]

as the coefficient characterizing the depth of monitoring and equal to the ratio of the volume of the monitored part of the PO to the total volume of the PO,

\[ \beta=\frac{V_{\mathrm{k}}}{V_0} \]

as the coefficient of monitoring redundancy, equal to the ratio of the volume of the monitoring block to the volume of the monitored part of the PO. Since the PO and BK are made of identical elements, it is not difficult to see that

\[ \alpha=\frac{\lambda_0}{\lambda}, \qquad \beta=\frac{\lambda_{\mathrm{k}}}{\lambda_0}. \]

Express \(\lambda_{\mathrm{n}}, \lambda_0, \lambda_{\mathrm{k}}\) through \(\lambda\); then

\[ \lambda_{\mathrm{n}}=(1-\alpha)\lambda; \qquad \lambda_0=\alpha\lambda; \qquad \lambda_{\mathrm{k}}=\alpha\beta\lambda; \]

\[ 0\leq \alpha \leq 1; \qquad \beta \geq 0. \tag{5} \]

Substituting (5) into (4), we obtain

\[ P_0= \frac{1}{ 1+(1-\alpha)\lambda\left(\frac{1}{\gamma}+\frac{1}{\mu_2}\right) +\frac{\alpha(1+\beta)\lambda}{\mu_1} +\frac{\alpha(1-\alpha)(1+\beta)\lambda^2}{\gamma\mu_1} }. \tag{6} \]

Taking into account that the value of the term \(\dfrac{\alpha(1-\alpha)(1+\beta)\lambda^2}{\gamma\mu_1}\) is negligibly small, we obtain

\[ P_0 = \frac{1}{ 1+(1-\alpha)\lambda\left(\frac{1}{\gamma}+\frac{1}{\mu_2}\right) +\frac{\alpha(1+\beta)\lambda}{\mu_1} }. \tag{7} \]

Replacing in expression (7) the transition rates by the mean times, we may write

\[ P_0 = \frac{1}{ 1+\lambda(1-\alpha)(T_{00}+T_{\mathrm{vn}}) +\lambda\alpha(1+\beta)T_{\mathrm{vk}} } = \frac{1}{ 1+\lambda_{\mathrm{n}}(T_{00}+T_{\mathrm{vn}}) +(\lambda_0+\lambda_{\mathrm{k}})T_{\mathrm{vk}} }, \tag{8} \]

where \(T_{00}\) is the mean time for the operator to detect the fact of a failure in the NRO;

\(T_{\mathrm{vn}}\) is the mean time for locating and replacing the operability check;

\(T_{\mathrm{vk}}\) is the mean recovery time of the KRO and BK.

As is seen from the transition graph (Fig. 2, 1), states \(P_1, P_2, P_3\), and \(P_4\) are states of failure of the device, consisting of the operating equipment and the monitoring unit. Therefore \(K_{\mathrm{g}}\) is equal to the probability that the device is in state \(P_0\), i.e.

Fig. 2. Dependence of \(K_{\mathrm{g}}\) on \(\alpha\)
for \(\gamma=\mu_2=1\ \frac{1}{\text{hour}};\ \mu_1=10\ \frac{1}{\text{hour}}\), for different \(\beta\)

\[ K_{\mathrm{g}}=P_0= \frac{1}{ 1+\lambda(1-\alpha)\left(\frac{1}{\gamma}+\frac{1}{\mu_2}\right) +\frac{\lambda\alpha(1+\beta)}{\mu_1} } = \frac{1}{ 1+\lambda_{\mathrm{n}}(T_{00}+T_{\mathrm{vn}}) +(\lambda_0+\lambda_{\mathrm{k}})T_{\mathrm{vk}} }. \tag{9} \]

As follows from the expression (9) obtained by us, it is insufficient to evaluate system availability only by the time \(T_{\mathrm{vn}}\) for locating and replacing the failed element, as is usually done in analyzing \(K_{\mathrm{g}}\) for various kinds of multiple-action equipment. Here it is necessary to take into account the time \(T_{00}\) of an undetected failure in the unmonitored part of the operating equipment, since this quantity may have a substantial effect on the value of \(K_{\mathrm{g}}\). In addition, expression (9) takes into account the depth of monitoring and the influence of failures of the AK circuits on the availability factor.

Formula (9) is simple and convenient for calculations. It shows that in the absence of monitoring \((0)\) the availability factor is determined by the expression

\[ K_{\mathrm{g}}= \frac{1}{ 1+\lambda\left(\frac{1}{\gamma}+\frac{1}{\mu_2}\right) } = \frac{1}{1+\lambda(T_{00}+T_{\mathrm{vn}})}, \tag{10} \]

and with fully monitored working equipment ($\alpha=1$)

\[ K_{\mathrm{r}}= \frac{1}{1+\dfrac{\lambda(1+\beta)}{\mu_1}} = \frac{1}{1+(\lambda+\lambda_{\mathrm{k}})T_{\mathrm{vk}}} = \]

\[ = \frac{1}{1+\lambda T_{\mathrm{vk}}+\lambda T_{\mathrm{vk}}}. \tag{11} \]

From Figs. 2, 3, and 4 the following conclusions can be drawn:

1. $K_{\mathrm{r}}$ increases as the depth of monitoring increases. The reliability of the monitoring circuits reduces the rate of increase of $K_{\mathrm{r}}$ (Fig. 2).

Fig. 3. Dependence of $K_{\mathrm{r}}$ on $\alpha\beta$ for $\gamma=\mu_2=1\ \dfrac{1}{\text{hour}}$;
$\mu_1=10\ \dfrac{1}{\text{hour}}$; $\lambda=10^{-2}\ \dfrac{1}{\text{hour}}$.

1. As the volume of monitoring equipment increases, $K_{\mathrm{r}}$ first increases and then decreases (Figs. 2, 3), i.e., there exists an optimum value of the availability factor corresponding in each case to a certain value of the volume of monitoring equipment (i.e., to a particular variant of the monitoring system). The value of $K_{\mathrm{r}}$ increases monotonically as the time $T_{\mathrm{oo}}$ for detecting a failure in the unmonitored part of the working equipment decreases (Fig. 4).

Fig. 4. Dependence of $K_{\mathrm{r}}$ on $\gamma$
for $\mu_2=\beta=1\ \dfrac{1}{\text{hour}}$; $\mu_1=10\ \dfrac{1}{\text{hour}}$;
$\lambda=10^{-2}\ \dfrac{1}{\text{hour}}$ for different $\alpha$.

From the analysis presented it follows that the depth of monitoring, the reliability of the MC circuits, and the time for which the working equipment remains in a state of undetected failure have a substantial influence on the readiness of the apparatus for operation and must be taken into account when comparing different variants of constructing the hardware monitoring system. Since the mean time $T_{\mathrm{oo}}$ for detection by the operator of a failure in the unmonitored part of the working equipment does not depend on the principle by which the MC circuits are constructed, but is determined by the conditions of use of the system, and the mean operating time to failure of the working equipment on

at the design stage is given; in order to obtain \(K_{\Gamma}\) close to unity, it is necessary to strive for the maximum coverage of RO by hardware monitoring with minimal costs for its implementation.

REFERENCES

1. B. K. Grigelionis. Limit theorems for sums of recovery processes. In: Cybernetics in the Service of Communism. Vol. 2, Moscow, “Energiya,” 1964.

2. O. V. Shcherbakov. Mathematical problems in estimating the reliability of digital computers. In: Cybernetics in the Service of Communism. Vol. 2, Moscow, “Energiya,” 1964.

3. B. V. Gnedenko, I. N. Kovalenko. Introduction to Queueing Theory. Moscow, “Nauka,” 1966.