scholarly journals Derivation of Kolmogorov-Chapman type equations with Fokker-Planck operator

2020 ◽  
pp. 90-107
Author(s):  
D.B. Prokopieva ◽  
◽  
T.A Zhuk ◽  
N.I. Golovko ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Service Systems ◽  
Storage Device ◽  
Queuing Systems ◽  
Input Flow ◽  
Random Diffusion ◽  
Global Computer ◽  
Drift Coefficient ◽  
Diffusion Intensity ◽  
Fokker Planck

In this paper we obtain the differential equation of the type Kolmogorov-Chapman with differential operator of the Fokker-Planck, having theoretical and practical value in the differential equations theory. Equations concerning non-stationary and stationary characteristics of the number of applications obtained for a class of Queuing systems (QS) with an infinite storage device, one service device with exponential service, the input of which is supplied twice stochastic a Poisson flow whose intensity is a random diffusion process with springy boundaries and a non-zero drift coefficient. Service systems with diffusion intensity of the input flow are used for modeling of global computer networks nodes.

OPTIMIZATION OF MARKOV SYSTEMS OF QUEUEING WITH WAITING TIME IN THE MATLAB SYSTEM

2017 ◽  
pp. 39-47
Author(s):  
Viktor Afonin ◽  
Vladimir Valer'evich Nikulin
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Random Variable ◽  
Model Systems ◽  
Queuing Systems ◽  
Input Flow ◽  
Continuous Random Variable ◽  
Input Stream ◽  
Time Intervals ◽  
Markov Systems

The article focuses on attempt to optimize two well-known Markov systems of queueing: a multichannel queueing system with finite storage, and a multichannel queueing system with limited queue time. In the Markov queuing systems, the intensity of the input stream of requests (requirements, calls, customers, demands) is subject to the Poisson law of the probability distribution of the number of applications in the stream; the intensity of service, as well as the intensity of leaving the application queue is subject to exponential distribution. In a Poisson flow, the time intervals between requirements are subject to the exponential law of a continuous random variable. In the context of Markov queueing systems, there have been obtained significant results, which are expressed in the form of analytical dependencies. These dependencies are used for setting up and numerical solution of the problem stated. The probability of failure in service is taken as a task function; it should be minimized and depends on the intensity of input flow of requests, on the intensity of service, and on the intensity of requests leaving the queue. This, in turn, allows to calculate the maximum relative throughput of a given queuing system. The mentioned algorithm was realized in MATLAB system. The results obtained in the form of descriptive algorithms can be used for testing queueing model systems during peak (unchanged) loads.

Smooth first-passage densities for one-dimensional diffusions

1987 ◽  
Vol 24 (02) ◽  
pp. 370-377 ◽  
Author(s):  
E. J. Pauwels
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Time Scale ◽  
First Passage ◽  
One Dimensional ◽  
Drift Coefficient

The purpose of this paper is to show that smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic differential equation imply the existence and smoothness of a first-passage density. In order to be able to prove this, we shall show that Brownian motion conditioned to first hit a point at a specified time has the same distribution as a Bessel (3)-process with changed time scale.

scholarly journals Supplement of differential equations of fraction order for forecasting of financial markets

2018 ◽  
Vol 170 ◽  
pp. 01075
Author(s):  
Sergey Erokhin ◽  
Olga Roshka
Keyword(s):  
Differential Equation ◽  
Diffusion Equation ◽  
Fractional Order ◽  
Capital Markets ◽  
Equity Markets ◽  
Planck Equation ◽  
Theoretical Physics ◽  
Advection Diffusion Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck

In this paper, the analysis of capital markets takes place using the advection-diffusion equation. It should be noted that the methods used in modern theoretical physics have long been used in the analysis of capital markets. In particular, the Fokker-Planck equation has long been used in finding the probability density function of the return on equity. Throughout the study, a number of authors have considered the supplement of the Fokker-Planck equation in the forecasting of equity markets, as a differential equation of second order. In this paper, the first time capital markets analysis is performed using the fractional diffusion equation. The rationale is determined solely by the application nature, which consists in generation of trading strategy in equity markets with the supplement of differential equation of fractional order. As the subject for studies, the differential operator of fractional order in partial derivatives was chosen – the Fokker-Planck equation. The general solutions of equation are the basis for the forecast on the exchange rate of equities included in the Dow Jones Index Average (DJIA).

Effect of scale on solute dispersion in saturated porous media

1984 ◽  
Vol 16 (1) ◽  
pp. 18-18 ◽  
Author(s):  
Vijay K. Gupta ◽  
R. N. Bhatiacharya
Keyword(s):  
Differential Equation ◽  
Porous Media ◽  
Time Scale ◽  
Space Time ◽  
Saturated Porous Media ◽  
Solute Dispersion ◽  
Image Position ◽  
Steady Velocity ◽  
Fokker Planck

Consider a saturated porous m edium in which water is flowing slowly with a steady velocity. Suppose at some space-time scale the concentration C(x, r) of a non-reactive dilute solute is governed by the following Fokker-Planck differential equation:

scholarly journals OPTIMIZATION OF QUEUING SYSTEMS UNDER SIGNIFICANT LOADS

2020 ◽  
Vol 2020 (3) ◽  
pp. 105-115
Author(s):  
Viktor Afonin ◽  
Vladimir Nikulin
Keyword(s):  
Denial Of Service ◽  
Transition Process ◽  
Analytical Models ◽  
System Throughput ◽  
Queuing Systems ◽  
Input Flow ◽  
Internet Applications ◽  
Conflict Situations ◽  
System States ◽  
Service Channels

The article deals with analytical models of Markov Queuing systems with service failures, incoming requests and requirements. The systems are analyzed in the conflict situations, for example, under significant loads, when the input flow intensity is high relative to the service intensity, which is important for extreme situations, both in technical applications, Internet applications, and social ones. There occurs a problem of optimization – minimization of the number of channels, provided the Queuing system has a guaranteed throughput. There is considered the approach to solving the optimization problem, when the relative system throughput is maximized while minimizing the number of service channels. Given the fact that the analytical formulas of Markov Queuing systems contain factorials, the analytical analysis of systems encounters the computational limitations. In the conducted research, in order to resolve computational difficulties it was decided to apply the approximation of the probabilities of the system states using the Laplace probability integral. Its use is justified precisely at high system load rates and a large number of service channels. There are described the features of applying the Laplace integral in conjunction with the numerical optimization for a conditional extremum. There is given the method of determining the number of service channels, when the probability of denial of service is minimized, respectively, maximizing the relative throughput of the system. There is given a graphical interpretation of the proposed method for optimizing Queuing systems with failures at the significant load. It is shown that during the search for the optimum there is a transition process in which there take place the significant changes in the system parameters: the intensity of the input flow and the intensity of service.

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