On the Reliability of a Computing Center's Operation
Abstract Generated abstract
This paper examines reliability estimation for a computing center composed of multiple homogeneous digital computers, reserve machines, and repair crews, where center failure is defined by the number of failed computers exceeding the reserve capacity. It reviews an existing exact probability formula and a simplified approximation, showing through a numerical example that the earlier approximation can substantially overestimate failure probability. The author derives a more accurate approximate expression by separating the probability terms into two sums and using the practical condition that the tail term is small relative to the main sum. The resulting formula closely matches the exact calculation in the example and provides simpler estimates for common parameter regimes.
Full Text
ON THE RELIABILITY OF A COMPUTING CENTER’S OPERATION
V. M. Razin
A characteristic feature of modern computing centers (CCs) is the presence of several homogeneous general-purpose digital computers (DCs) operating in a multiprocessor mode. The operation of these DCs is coordinated by means of a central DC—the dispatcher. To increase the reliability of the CC’s operation, the presence of a certain number of reserve DCs is provided. Maintenance of all DCs is performed by a certain number of repair crews. In [1] the situation in which the number \(k\) of failed DCs exceeds the number \(n\) of reserve ones is regarded as failure of the CC. The probability of CC failure \(P(k>n)\) is then calculated by the formula
\[ P(k>n)=1-P(k\leq n)= \]
\[ =1-\left\{ \sum_{k=0}^{q}\frac{m!}{k!(m-k)!}\left(\frac{\lambda}{\mu}\right)^k + \sum_{k=q+1}^{n} \frac{m!}{q^{\,k-q}\cdot q!(m-k)!} \left(\frac{\lambda}{\mu}\right)^k \right\}P_0, \tag{1} \]
where \(m\) is the total number of DCs in the CC;
\[ \begin{aligned} n&\text{ is the number of reserve DCs;}\\ k&\text{ is the number of faulty machines;}\\ q&\text{ is the number of repair crews;} \end{aligned} \]
\[ \lambda=\frac{1}{T_0}; \quad T_0 \text{ is the mean time of failure-free operation of a DC;} \]
\[ \mu=\frac{1}{T_B}; \quad T_B \text{ is the mean restoration time of a DC.} \]
The probability \(P_0\) that all DCs are in working order is determined by the relation
\[ P_0= \left\{ 1+ \sum_{k=1}^{q} \frac{m!}{k!(m-k)!} \left(\frac{\lambda}{\mu}\right)^k + \sum_{k=q+1}^{m} \frac{m!}{q^{\,k-q}\cdot q!(m-k)!} \left(\frac{\lambda}{\mu}\right)^k \right\}^{-1}. \tag{2} \]
Under the assumption
\[ \frac{\lambda}{\mu}\ll 1 \]
an approximate relation was obtained in [1]:
\[ P(k>n)\simeq 1- \left[ 1+ m\left(\frac{\lambda}{\mu}\right) + \frac{m(m-1)}{2} \left(\frac{\lambda}{\mu}\right)^2 \right]P_0. \tag{3} \]
Calculations carried out for a specific case \(\left(\dfrac{\lambda}{\mu}=0.1,\ m=7,\ n=3,\ q=2\right)\) showed that formula (3) gives a large error. The failure probability calculated by formula (3) turned out to be 7 times higher than the same quantity computed by the exact formula (1), taking into account the exact relation (2). In this connection, it seems advisable to obtain a more accurate approximation for estimating \(P(k>n)\).
For \(n>q\), relation (1), taking (2) into account, can be reduced to the form
\[ P(k>n)=1-\frac{1}{1+\dfrac{1}{1+\Delta}\cdot\delta}, \tag{4} \]
where
\[ \Delta=\sum_{k=1}^{q}\frac{m!}{k!(m-k)!}\left(\frac{\lambda}{\mu}\right)^k +\sum_{k=q+1}^{n}\frac{m!}{q^{k-q}\cdot q!(m-k)!}\left(\frac{\lambda}{\mu}\right)^k, \tag{5} \]
\[ \delta=\sum_{k=n+1}^{m}\frac{m!}{q^{k-q}\cdot q!(m-k)!}\left(\frac{\lambda}{\mu}\right)^k. \tag{6} \]
Approximate estimates show that in most practical cases \(\delta\ll\Delta<1\), and the approximate formula takes the form
\[ P(k>n)\simeq \frac{\delta}{1+\Delta}. \tag{7} \]
In the example considered above, \(\Delta\simeq 0.96\), \(\delta\simeq 0.0121\), and \(P(k>n)\simeq 0.00615\). Calculation by the exact formula (1) gives \(P(k>n)\simeq 0.0062\), which indicates a good approximation. For a certain ratio of the parameters, the case \(\Delta\ll 1\) may occur (especially when \(\dfrac{\lambda}{\mu}<0.05\)); then the calculations should be carried out using the approximate formula
\[ P(k>n)\simeq \delta. \tag{8} \]
By an analogous method, approximate calculation formulas can be obtained for estimating the reliability of the computing center in the case \(n=q\) or \(n<q\).
REFERENCES
- Yu. S. Golubev-Novozhilov. Multi-Machine Complexes of Computing Facilities. Moscow, “Sovetskoe radio,” 1967.