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THE SPECTRAL DECOMPOSITION METHOD FOR SOLVING THE LINDLEY INTEGRAL EQUATON AND RELATED NUMERICAL METHODS
Lyudmila V. Lipilina, Povolzhskiy State University of Telecommunications and Informatics, Samara, Russia, mila199113@gmail.com
Abstract
In researching the traffic of modern computer networks and telecommunication networks, the methods of queuing theory are widely used. In turn, in studies of queuing systems (QS) of type G/G/1 with arbitrary laws of the distribution of intervals between adjacent requirements in the input flow and service time, the spectral decomposition method (SDM) of solving the Lindley integral equation is used [1-3]. The basis of this method is the search for zeros and poles of the constructed spectral decomposition in the form of some fractional rational function involving numerical methods for determining the roots of polynomials. Moreover, the coefficients of the polynomial in the decomposition numerator are usually expressed in terms of the unknown parameters of the distribution laws used to describe the QS. Typically, these unknown parameters of the laws of distribution can be determined through the numerical characteristics of the analyzed traffic by the known method of moments. The purpose of this article is to illustrate in detail the spectral decomposition method as applied to QS H2/H2/1 of type G/G/1 with hyperexponential laws of second-order distributions and its relation to problems of numerical analysis. A characteristic feature of this distribution law is the possibility of its unambiguous description both at the level of the first two initial moments of time intervals, and at the level of three moments. The second-order hyper-exponential distribution law of Í2 provides the coefficient of variation of time intervals , and starting from four, the Í2 distribution law has a heavy tail, which is well suited for describing traffic with a heavy-tail distribution. The use of this law of higher order distribution in the method of spectral decomposition leads to an increase in the computational complexity of the problem. The proposed approach to the use of the spectral decomposition method allows us to determine, in addition to the average waiting time, other moments of waiting time. In the telecommunications standard, the concept of jitter is defined through the spread of waiting time around its average value. Then, the presented approach of applying the spectral decomposition method allows one to determine jitter through the second initial moment of waiting time. This refers to the practical applicability of queuing systems to the study of delays in telecommunication networks.
Keywords: queuing system, average queue waiting time, Lindley integral equation, Laplace transform, hyperexponential distribution.
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Information about author:
Lyudmila V. Lipilina, Povolzhskiy State University of Telecommunications and Informatics, graduate student of Department of Software and Management in Technical Systems, Samara, Russia