Mathematical Modeling of Electrodialysis Demineralization Using a Stochastic Model

Yustratov, V. P.; Pavskii, V. A.; Krasnova, T. A.; Иванова, Светлана

doi:10.1007/s11236-005-0073-9

Cited by 9 publications

(11 citation statements)

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“…The assumption of the exponential law of the distribution of the time of residence of the system in each of the states is equivalent to the fulfillment of the condition of absence of aftereffect and ordinariness, and if the parameter of the exponential law is constant, then the stationarity [2,4,21,24]. The fulfillment of these conditions makes it possible to construct a system of linear differential equations for the QS transition from a state to a state determined by a marked state graph, whose structure depends on the object under study [2][3][4][5][6][7][8][9]14]. Let us denote by ) , ( i t P k the probability that at the time…”

Section: Markov Processmentioning

confidence: 99%

“…Two groups of models can be distinguished -some characterize objects from general positions, in the first approximation and reflect the common, most important trend of development and analysis of real processes, others reveal their hidden essence, help to explore the internal properties of objects. Models of the first group, usually prognostic, are not very parametric, they have good statistics, and analytical solutions, representable by formulas convenient in engineering calculations, or easily computable are preferable for them [1][2][3][4][5][6][7][8]. Models of the second group are multiparameter, heavy, statistics are not always reliablealgorithmic methods of investigation are usually effective for them, usually approximate, simulation modeling [9][10][11][12][13][14], etc.…”

Section: Introductionmentioning

confidence: 99%

“…This often leads to more effective results than traditional mathematical models of the exact sciences can provide. Many random phenomena and objects representing dynamical systems of a special type are effectively described by methods of queuing theory (QT) [2][3][4][5][6][7][8]. In the QT, the queuing system (model) (QS) is first formulated, which, in the continuous case, is formalized by a system of differential equations.…”

Section: Introductionmentioning

confidence: 99%

“…In the QT, the queuing system (model) (QS) is first formulated, which, in the continuous case, is formalized by a system of differential equations. The unknown time functions in the systems of equations are the probability distributions usually associated with the Markov random process [2][3][4][5][6][7][13][14][15][16]. Analytical solutions, although exist, but are found, basically, by approximate methods (in real time in the presence of high-speed computing tools).…”

Section: Introductionmentioning

confidence: 99%

“…The model is formulated in the terminology of QT. The solution obtained has direct practical applications, for example, to the analysis of the efficiency of the functioning of computer systems [2-5, 11, 12], processes arising in chemistry and the chemical industry, biotechnology [5][6][7][8]17], etc.…”

Section: Introductionmentioning

confidence: 99%

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The Continuous-Time Time Bernulli Process

Pavsky¹,

Pavsky²,

Иванова³

2017

Science Evolution

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A random Bernoulli process with continuous time and a finite number of states (random events) is proposed. The process is obtained by two mutually complementary methods - directly from the Poisson process with an intensity parameter that depends on time and methods of queuing theory, from a queuing system with two parameters. In the first case, the process was formalized on a probability space with measure, as a measurable function of time. The intensity of the Poisson process was considered as a measure. The Bernoulli process for each fixed time was obtained as a conditional distribution from a suitable Poisson distribution. The parameter of the Poisson distribution was determined from the differential equation, in the formulation of which the approximation of the Bernoulli formula by the Poisson formula was essentially used. In the second method, standard methods of queuing theory were used. A two- parameter queuing model was formulated in which for all customer flows the time between occurrence of neighboring customers was a random value satisfying the exponential law. The model was formalized by a system of differential equations, whose analytical solution represented the continuous-time Bernoulli process. In finding solutions, the method of generating functions was used. It is of interest to derive the Bernoulli process both from the probability space constructed for the Poisson process and from the queuing theory model. The authors believe that the proposed process can be generalized to a wider class of functions than that used in the work, down to measurable ones. The possibilities for the practical application of the continuous-time Bernoulli process will undoubtedly be expanded, since its discrete analog is well known in many fields of science and technology.

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