scholarly journals DEVELOPMENT OF A MODEL FOR THE DYNAMICS OF PROBABILITIES OF STATES OF SEMI-MARKOV SYSTEMS

2021 ◽  
pp. 62-68
Author(s):  
Lev Raskin ◽  
Oksana Sira ◽  
Larysa Sukhomlyn ◽  
Roman Korsun
Keyword(s):  
Probability Distribution ◽  
Stationary Distribution ◽  
Transition Process ◽  
Initial Distribution ◽  
Markov Systems ◽  
Markov System ◽  
Original Distribution ◽  
Distribution Laws ◽  
Analytical Relations ◽  
Erlang Distributions

The subject is the study of the dynamics of probability distribution of the states of the semi-Markov system during the transition process before establishing a stationary distribution. The goal is to develop a technology for finding analytical relationships that describe the dynamics of the probabilities of states of a semi-Markov system. The task is to develop a mathematical model that adequately describes the dynamics of the probabilities of the states of the system. The initial data for solving the problem is a matrix of conditional distribution laws of the random duration of the system's stay in each of its possible states before the transition to some other state. Method. The traditional method for analyzing semi-Markov systems is limited to obtaining a stationary distribution of the probabilities of its states, which does not solve the problem. A well-known approach to solving this problem is based on the formation and solution of a system of integral equations. However, in the general case, for arbitrary laws of distribution of the durations of the stay of the system in its possible states, this approach is not realizable. The desired result can only be obtained numerically, which does not satisfy the needs of practice. To obtain the required analytical relationships, the Erlang approximation of the original distribution laws is used. This technique significantly increases the adequacy of the resulting mathematical models of the functioning of the system, since it allows one to move away from overly obligatory exponential descriptions of the original distribution laws. The formal basis of the proposed method for constructing a model of the dynamics of state probabilities is the Kolmogorov system of differential equations for the desired probabilities. The solution of the system of equations is achieved using the Laplace transform, which is easily performed for Erlang distributions of arbitrary order. Results. Analytical relations are obtained that specify the desired distribution of the probabilities of the states of the system at any moment of time. The method is based on the approximation of the distribution laws for the durations of the stay of the system in each of its possible states by Erlang distributions of the proper order. A fundamental motivating factor for choosing distributions of this type for approximation is the ease of their use to obtain adequate models of the functioning of probabilistic systems. Conclusions. A solution is given to the problem of analyzing a semi-Markov system for a specific particular case, when the initial distribution laws for the duration of its sojourn in possible states are approximated by second-order Erlang distributions. Analytical relations are obtained for calculating the probability distribution at any time.

Some exact results for dams with Markovian inputs

1976 ◽  
Vol 13 (02) ◽  
pp. 329-337
Author(s):  
Pyke Tin ◽  
R. M. Phatarfod
Keyword(s):  
Probability Distribution ◽  
Stationary Distribution ◽  
Special Form ◽  
Geometric Distribution ◽  
Transition Probabilities ◽  
Exact Results ◽  
Input Process ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Finite Dam

In the theory of dams with Markovian inputs explicit results are not usually obtained, as the theory depends very heavily on the largest eigenvalue of the matrix (pijzj ) where p ij are the transition probabilities of the input process. In this paper we show that explicit results can be obtained if one considers an input process of a special form. The probability distribution of the time to first emptiness is obtained for both the finite and the infinite dam, as well as the stationary distribution of the dam content for the finite dam. Explicit results are given for the case where the stationary distribution of the input process has a geometric distribution.

On certain aspects of non-homogeneous Markov systems in continuous time

1990 ◽  
Vol 27 (3) ◽  
pp. 530-544 ◽  
Author(s):  
Ioannis I. Gerontidis
Keyword(s):  
Continuous Time ◽  
Initial Distribution ◽  
Markov Systems ◽  
Numerical Examples ◽  
Stationary Structure ◽  
Periodic Environment ◽  
Time Formulation ◽  
Relative Structure ◽  
The Relationship ◽  
Theoretical Results

In the present paper we study three aspects in the theory of non-homogeneous Markov systems under the continuous-time formulation. Firstly, the relationship between stability and quasi-stationarity is investigated and conditions are provided for a quasi-stationary structure to be stable. Secondly, the concept of asymptotic attainability is studied and the possible regions of asymptotically attainable structures are determined. Finally, the cyclic case is considered, where it is shown that for a system in a periodic environment, the relative structure converges to a periodic vector, independently of the initial distribution. Two numerical examples illustrate the above theoretical results.

Non-homogeneous semi-Markov systems and maintainability of the state sizes

1992 ◽  
Vol 29 (3) ◽  
pp. 519-534 ◽  
Author(s):  
P.-C. G. Vassiliou ◽  
A. A. Papadopoulou
Keyword(s):  
Convex Set ◽  
Analytic Form ◽  
The State ◽  
Markov Systems ◽  
Markov System ◽  
Basic Parameters ◽  
Manpower System ◽  
Closed Analytic Form ◽  
First Time ◽  
Expected Duration

In this paper we introduce and define for the first time the concept of a non-homogeneous semi-Markov system (NHSMS). The problem of finding the expected population stucture is studied and a method is provided in order to find it in closed analytic form with the basic parameters of the system. Moreover, the problem of the expected duration structure in the state is studied. It is also proved that all maintainable expected duration structures by recruitment control belong to a convex set the vertices of which are specified. Finally an illustration is provided of the present results in a manpower system.

The size order of the state vector of discrete-time homogeneous Markov systems

2001 ◽  
Vol 38 (2) ◽  
pp. 357-368 ◽  
Author(s):  
I. Kipouridis ◽  
G. Tsaklidis
Keyword(s):  
Lower Bound ◽  
Discrete Time ◽  
State Vector ◽  
The State ◽  
Order Problem ◽  
Markov Systems ◽  
Markov System

The size order problem of the probability state vector elements of a homogeneous Markov system is examined. The time t0 is evaluated, after which the order of the state vector probabilities remains unchanged, and a formula to quickly find a lower bound for t0 is given. A formula for approximating the mode of the state sizes ni(t) as a function of the means Eni(t), and a relation to evaluate P(ni(t) = x+1) by means of certain terms which constitute P(ni(t) = x) are derived.

Some exact results for dams with Markovian inputs

1976 ◽  
Vol 13 (2) ◽  
pp. 329-337 ◽  
Author(s):  
Pyke Tin ◽  
R. M. Phatarfod
Keyword(s):  
Probability Distribution ◽  
Stationary Distribution ◽  
Special Form ◽  
Geometric Distribution ◽  
Transition Probabilities ◽  
Exact Results ◽  
Input Process ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Finite Dam

In the theory of dams with Markovian inputs explicit results are not usually obtained, as the theory depends very heavily on the largest eigenvalue of the matrix (pijzj) where pij are the transition probabilities of the input process. In this paper we show that explicit results can be obtained if one considers an input process of a special form. The probability distribution of the time to first emptiness is obtained for both the finite and the infinite dam, as well as the stationary distribution of the dam content for the finite dam. Explicit results are given for the case where the stationary distribution of the input process has a geometric distribution.

The methodological apparatus for planning and regulating trends in high-tech production costs under risk

2020 ◽  
Vol 16 (3) ◽  
pp. 400-414
Author(s):  
A.V. Leonov ◽  
A.Yu. Pronin
Keyword(s):  
Mathematical Modeling ◽  
Probability Theory ◽  
Economic Evaluation ◽  
Probability Distribution ◽  
Comprehensive Approach ◽  
Production Costs ◽  
Financial Resources ◽  
High Tech ◽  
Distribution Laws ◽  
The Given

Subject. The article discusses the methodological apparatus for assessing how many financial resources are needed, and optimize their gradual consumption in high-tech production projects. Results of the projects can be forecasted only as a probable estimate, since there is great uncertainty due to a multitude of random factors. Objectives. The study aims to form the methodological apparatus to assess the amount of financial resources needed under risk and substantiate the reasonable strategy for consuming them in high-tech production projects within a given period of time. The apparatus is to allow for quick adjustments of the financing plan and estimation of project expenditures. Methods. We applied the comprehensive approach to planning and regulating trends in high-tech production costs, methods of economic and mathematical modeling and the probability theory. Results. We reviewed methods used to assess how many financial resources are needed through demand probability distribution laws. Based on them, we devised the interval technique for regulating cost trends so as to substantiate the reasonable strategy for the performance of projects with the desired probability within the given period of time. What distinguishes the interval technique is that it provides the overall vision of the period, during which the project will be performed, and the probability of the advanced prediction of a shortage or excess of financial resources. Conclusions and Relevance. The methodological apparatus proposed herein will facilitate the technological and economic evaluation of various options of high-tech production and choose those ones which ensure the best use of financial resources, quickly regulate the economic dynamism throughout the high-tech production phases in line with a variety of factors, which randomly emerge at certain phases.

scholarly journals P Systems Computing the Period of Irreducible Markov Chains

2009 ◽  
Vol 4 (3) ◽  
pp. 291
Author(s):  
Mónica Cardona-Roca ◽  
M. Ángels Colomer-Cugat ◽  
Agustín Riscos-Núñez ◽  
Miquel Rius-Font
Keyword(s):  
Markov Chain ◽  
Comparative Analysis ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Initial Distribution ◽  
P Systems ◽  
P System

<p>It is well known that any irreducible and aperiodic Markov chain has exactly one stationary distribution, and for any arbitrary initial distribution, the se- quence of distributions at time n converges to the stationary distribution, that is, the Markov chain is approaching equilibrium as n→∞.<br /> In this paper, a characterization of the aperiodicity in existential terms of some state is given. At the same time, a P system with external output is associated with any irre- ducible Markov chain. The designed system provides the aperiodicity of that Markov chain and spends a polynomial amount of resources with respect to the size of the in- put. A comparative analysis with respect to another known solution is described.</p>

scholarly journals Obtaining Probability Distribution Laws of Power System Steady-State Mode Parameters

2020 ◽  
Vol 20 (3) ◽  
pp. 41-51
Keyword(s):  
Renewable Energy ◽  
Probability Distribution ◽  
Power Systems ◽  
Power System ◽  
Renewable Energy Sources ◽  
Current Trend ◽  
System Stability ◽  
Energy Sources ◽  
Stable Operation ◽  
Distribution Laws

Stable operation of electrical power systems is one of the crucial issues in the power industry. Current vo­lumes of electricity consumption cause the need to constantly increase the generated capacity, repeatedly modifying and complicating the original circuit. In addition to this, given the current trend towards the use of digital power systems and renewable energy sources, more and more uncertainties difficult to predict by standard mathematical methods appear. Events in the power system are deterministic, i.e. random. Thus, it is difficult to fully assess the system stability, voltage levels, currents, or possible power losses. Finding the probability distribution laws can give us an understanding of all the possible states in which an object can exist. Obtaining them is complicated by the difficulty of accounting for all the correlations between the random arguments of the source data. These laws are necessary to determine the optimal operating modes, the possibility of solving the problem of determining the optimal renewable energy sources installation locations and the required amount of generated energy in a non-deterministic way. The purpose of this article is to test the developed SIBD method for obtaining the full probabilistic characteristics. This method, unlike the Monte Carlo methods, does not use a random sample of initial data, but completely covers the studied functional dependence. The problem was solved using the provisions of probability theory and mathematical statistics, numerical optimization methods in particular. The MATLAB Matpower application package was also used to solve technical computing problems.

scholarly journals Composition and superposition failures probability distribution laws for assessing military electrical products reliability

2021 ◽  
Vol 25 ◽  
pp. 131-137
Author(s):  
V.V. Alekseev ◽  
◽  
Yu.P. Batyrev ◽  
M.A. Boldyrev ◽  
P.S. Vorontsov ◽  
...  
Keyword(s):  
Probability Distribution ◽  
Standard Test ◽  
Structural Method ◽  
Reliability Indices ◽  
Transformer Type ◽  
Distribution Laws ◽  
Laws Of Distribution

An implementation of the developed computational structural method for assessing the reliability of complex electrical products. The basis of this method is the use of combinations of composition and superposition laws of distribution of probabilities of failures. The estimation of reliability of a rotating transformer type ВT-5. The calculated reliability indices are compared with results of standard test ВT-5 for reliability. Received high convergence of the results.

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