On the Distribution Law of Requests at the Input of a Universal System for Monitoring Complex Objects
Abstract Generated abstract
The paper considers how to specify probability distributions for heterogeneous request parameters arriving at the input of a universal automated monitoring system, treated as a multichannel queueing system. Because empirical data on monitored objects and parameter variation are often insufficient, the authors compare conditional assumptions for statistical simulation, including logarithmically uniform, uniform, and normal distributions. They argue that a logarithmically uniform law is appropriate in the general case, a uniform law may be used for narrow parameter ranges, and a normal law may be justified when parameter variation results from many independent or weakly dependent factors. These distributional assumptions are presented as a basis for computer simulation used to optimize device characteristics in a system developed with the Dnepr-1 control computer.
Full Text
ON THE DISTRIBUTION LAW OF REQUESTS AT THE INPUT OF A UNIVERSAL SYSTEM FOR MONITORING COMPLEX OBJECTS
G. P. PAVLENKO, V. M. RAZIN
(Presented by the scientific seminar of the Department of Computer Engineering)
A universal system of automated monitoring (USAM) may be represented as a multichannel mass-service system [3], at whose input there arrives a heterogeneous flow of requests; each request of this flow is characterized, in addition to the time of arrival \((t_j)\), by a set of parameters
\[ \alpha_0,\ \alpha_1,\ \alpha_2,\ldots,\ \alpha_k . \]
One of the tasks in the operation of a USAM in this case is the optimization of the characteristics of the system’s servicing devices \((\beta_i^k)\), depending on the application of the values of the request parameters \((\alpha_j^k)\).
Such a problem can be solved by the method of statistical modeling on a digital computer. In doing so, it becomes necessary to determine the values of the request parameters.
Obtaining statistical data on the distribution laws \((\alpha_j^k)\) is extremely difficult, since:
— the values of request parameters are influenced by many factors;
— in most cases there is no sufficient sample of monitoring objects.
In this connection, under certain assumptions these laws may be specified conditionally.
There are several opinions regarding the form of the distribution law for the values of analogous quantities. The first of them, expressed by C. Shannon [4], consists in the fact that, in the general case, the most characteristic law of variation of the distribution density for most quantities is the logarithmically uniform distribution law, which for our case may be written in the following form:
\[ P(\ln \alpha)= \frac{1}{\ln \alpha_j^{(k)'}-\ln \alpha_j^{(k)''}} =\mathrm{const}, \]
where \(\alpha_j^{(k)'}\) and \(\alpha_j^{(k)''}\) are the boundaries of the interval, and the quantities \(\alpha_j^k\) obey the given law.
Such a law means that if the entire range of possible values of \(\alpha_j^k\) is divided into equal intervals on a logarithmic scale, then the probability that \(\alpha_j^k\) will fall into each such segment will be the same.
On a linear scale of the quantities \(a_j^x\), such a law corresponds to a change in the distribution density according to a hyperbolic law of the form
\[ P(a)=\frac{1}{a_j^x \ln\left(a_j^{(k)''}/a_j^{(k)'}\right)}, \]
i.e., the current value of the probability density \(P(a)\) is inversely proportional to the current value of the quantity \(a\) itself, where by \(P(a)\) in this case we mean the number of gradations falling per unit of the quantity \(a\).
Along with the foregoing, there is also another conception of the law of distribution of the density of the quantities under study. It may be assumed that the values \(a_j^x\) are a priori equiprobable, i.e., uniformly distributed with a density independent of \(a\) and constant,
\[ P(a)=\frac{1}{a_j^{(k)''}-a_j^{(k)'}}=\mathrm{const}. \]
However, in [2] it is asserted that the use of such a distribution law is expedient only in general cases of solving research problems.
The assertion that \(P(a)\) is independent of the value of the quantity \(a\) itself, when applied problems in the theory of information-measuring systems are considered, cannot be regarded as usual, especially for a wide range of variation of the values of the quantities under study.
Thus, in the general case the general regularity of variation of the probability density may be taken to be the law of hyperbolic variation of \(P(a)\), i.e., the law of logarithmic uniform distribution.
As shown in [2], the principal feature of Shannon’s distribution is its sharp asymmetry. When the limits of variation of the quantities \(a_j^{(k)'}\) and \(a_j^{(k)''}\) approach each other, this asymmetry becomes smaller and the form of the distribution \(P(a)\) approaches the uniform one. In this connection, for cases of a small range of variation, the approximate replacement of Shannon’s distribution by a uniform one may be considered entirely legitimate.
Still another approach is possible for specifying the distribution law of the values of the request parameters.
Thus, it can be proved [1] that the sum of a sufficiently large number of independent (or weakly dependent) random variables subject to arbitrary distribution laws is approximately subject to the normal law.
Most of the values \(a_j^x\), as random variables, may be represented as sums of a very large number of relatively small terms, each of which is caused by the action of an individual cause independent of the others. In this connection it may be assumed that, in our case, the conditions of the well-known limit theorem of A. M. Lyapunov are satisfied, and, for independent factors in the variation of the values \(a_j^x\), the conditions of the theorem of S. N. Bernstein [1]. Then the distribution law may be regarded as normal, characterized by the probability density
\[ P(a)=\frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{(x-m)^2}{2\sigma^2}}, \]
where \(m\) is the mathematical expectation of the quantity \(a_j^x\);
\(\sigma\) is the root-mean-square deviation of this quantity.
Thus, on the basis of the foregoing, one may conclude that the most acceptable distributions for modeling the UACS are two extreme distribution laws for \(\alpha_j^k\):
— the uniform-density law, when a relatively narrow range of variation of \(\alpha_j^k\) is in effect;
— the normal distribution law, under the assumption of independence and relative uniformity of the influence of various factors on changes in the quantities.
In the general case, it may be considered that the values of the request parameters are distributed according to a logarithmically uniform law.
Such representations of the distribution law of the quantities were specified when solving the problem of optimizing the characteristics of devices for communication with the object by the method of statistical modeling on a digital computer.
The modeling results were used in the development of a UACS based on the “Dnepr-1” control computer.
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