Begell House Inc.
Journal of Automation and Information Sciences
JAI(S)
1064-2315
51
8
2019
Markov Models of Queuing-Inventory Systems with Different Types of Retrial Customers
1-15
10.1615/JAutomatInfScien.v51.i8.10
Agasi Zarbali ogly
Melikov
Institute of Control Systems of National
Academy of Sciences of Azerbaijan, Baku
Leonid A.
Ponomarenko
International Research and Training Center of Information Technologies and Systems of National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine, Kiev, Ukraine
Ismail Alakper ogly
Aliyev
Baku State University, Baku (Azerbaijan)
queuing-inventory systems
Markov model
retrial customer
inventory replenishment policy
Consideration is given to queuing-inventory system with two types of retrial customers and instantaneous service time. It is assumed that if at the time of high priority customer arrival the inventory level is above zero then it
receives inventory and leaves the system. A low priority customer receives inventory if at the time of its arrival the inventory level is above a certain critical level, otherwise this customer according to Bernoulli scheme either
goes into orbit or does not receive an inventory and leaves the system. The sojourn time of customers in an infinite orbit is a random variable with an exponential distribution function. If at the time of retrial customer arrival the
inventory level is above the critical one, then it instantly receives the required inventory and leaves the orbit; otherwise according to Bernoulli scheme it either leaves the orbit or remains in it. Consideration is given to three
inventory replenishment policies: two-level policy, a variable replenishment size policy and the policy in which an order for inventory supply is made after each inventory release act. The main characteristics of the system are
the average inventory level, the average intensity of orders, the probability of failure of servicing customer of each type when entering the system, the average number of customers in orbit, the average intensities of successful
and unsuccessful repetition of customers from orbit. For the mathematical analysis of the system under study there was constructed the corresponding two-dimensional Markov chain and the algorithm was given for finding its generating matrix. Joint distribution of the system inventory level and the number of customers in orbit as well as the formulas for calculating the averaged characteristics of the studied models were developed.
Bank − Complex System
16-30
10.1615/JAutomatInfScien.v51.i8.20
Valeriy A.
Velichkin
University of Customs and Finance, Dnipro
Marina V.
Timoshenko
University of Customs and Finance, Dnipro
bank
complex system
state vector
transition matrix
modeling complex system behavior
The bank is represented as a complex financial and economic system with a certain state column vector. The parameters of an arbitrary state as a column vectors are defined as the product of the transition matrix from one state to another. This representation of the bank allows one to make the transition to statistical modeling of the bank's activities via the statistics of the transition matrix. When analyzing the activities of a bank, the most common tool used by researchers is the method of indicators, that is, the relation of financial and economic aggregates. With this approach, the evaluation of the bank's activities is carried out using various absolute and relative indicators. A comprehensive study of these indicators allows one to conclude about the effectiveness of the financial and economic activities of the bank. One should mention the method of comparing the actual state of the values of the studied indicators with the standard values, values of the past period, and values of the average level. The purpose of the research is to study the relationship between the parameters describing the state of the bank at a control instant of time from the parameters of the initial state and, on the basis of this dependence to develop methods for managing financial resources under the conditions of specified constraints. The concept of a transition matrix from one bank state to an arbitrary state is introduced. The transition of a bank as a complex FES (financial economic system) from one state to another occurs under the influence of the flow of finance consisting of elements − payments, the size of which may be equal to the minimum monetary unit. In general, each flow from payments, regardless of the value, transfers the system from one state to another. One banking (operational) day is selected as the time sampling. Consequently, a bank is considered as a complex financial and economic system with a discrete number of states, and one banking (operational) day is selected as a time unit. This corresponds to the current approach to financial management in the banking system. The practice of applying the method to the problem of asset liquidity indicators and the Monte Carlo method showed the effectiveness of the proposed model. The general conclusions of the work are as follows: the bank is represented as a complex dynamic financial and economic system with an numerous number of states; further development of the application of a system approach to financial management, the bank is presented in the form of a system consisting of a specific set of parameters, financial management is carried out through managing these parameters; the dependence of the parameters describing the bank factors under study at the control instant of time from the parameters of the initial state has been revealed; systematized payment flows, which are represented by the transition matrix, by breaking them into components, the quantitative influence of these components on the system parameters has been determined; it is proved that the relationship between the parameters of the control and initial states with a sufficient degree of probability can be expressed by a linear operator, the proposed formulas allow one to calculate all the components of a linear operator. On the basis of the obtained dependence of the parameters of the control state on the parameters of the initial state, methodical recommendations were developed for managing the parameters (assets and liabilities) of the banking structure.
Method of Solving Problem of Conditional Optimization on Combinatorial Set of Arrangements
31-42
10.1615/JAutomatInfScien.v51.i8.30
Victor V.
Semenov
V.M. Glushkov Institute of Cybernetics of National Academy of Sciences of Ukraine, Kiev
Alla N.
Nagornaya
National University of "Kyiv-Mohyla
Academy", Kiev
Lyudmila N.
Kolechkina
University of Lodz, Lodz (Poland)
conditional optimization problem
combinatorial set of arrangements
function extremum
normalization matrix
The paper considered a formulated optimization problem on a combinatorial set of arrangements and suggested a method for its solution taking into account satisfaction of conditions imposed on gains of restrictions and objective function. The method consists of three steps. The first step constructs normalization and compliance matrices which provide transformation of arrangements set elements to a necessary form for the objective function and the given restrictions. The second step consists in finding the first support solution taking into account the arrangements set property. It is worth noting that to find the first support solution it is sufficient to calculate gains of restrictions. If the feasible solution satisfies presented inequalities, the initial data are fixed to be the verification conditions for the next improved solution. The value of objective function is determined by calculating objective function gains without the need to calculate the entire previous function. The third step provides finding the optimal solution at direct improvement of the obtained support solution. This step formulated sufficient and necessary conditions to search for the optimal solution, considered numerical examples of searching for externa of functions on the arrangements set and also presented numerical experiment for the case of
|Ak3|
with growing number of sample units
of the arrangements set (k). It is also worth noting that the number of steps of searching for the optimal solution does not significantly increase at a sharply increased number of elements of arrangements set. Analyzing the indicator of percentage ratio of the considered points number when searching for the optimal solution and the number of elements of arrangements set it should be noted its considerable reduction that indicates to efficiency of the proposed method. So this method application allows us to find the function extremum on the set of arrangements over the finite number of steps.
Walsh Functions in Linear-Quadratic Optimization Problems of Linear Nonstationary Systems
43-57
10.1615/JAutomatInfScien.v51.i8.40
Alexander A.
Stenin
National Technical University of Ukraine "Igor Sikorsky Kiev Polytechnic Institute", Kiev
Yuriy A.
Timoshin
National Technical University of Ukraine
"Igor Sikorsky Kiev Polytechnic Institute",
Kiev
Irina G.
Drozdovich
Institute of Telecommunication and Global Information Space of National Academy of Sciences of Ukraine, Kiev
linear nonstationary systems
linear-quadratic optimization problems
Walsh functions
fundamental matrix
closed optimal control
Currently the solution of problems of analytical design of an optimal controller (ADOC) for stationary dynamic objects has been well studied, and a number of works has been devoted to them. At the same time the synthesis of optimal control laws of nonstationary dynamic objects in a general case is quite complex task which often cannot be solved in an analytical form. This is primarily due to difficulty of solving a nonstationary nonlinear vector-matrix Riccati equation. This article deals with linear-quadratic problems of synthesis of a closed optimal control law for one class of linear nonstationary systems. Determination of the optimal control law within the framework of ADOC problem is based on the Pontryagin maximum principle. The fundamental matrix of the system of simplified canonical equations is used to establish the connection between the auxiliary vector and the state vector. It is worth noting that in a general case it is not possible to obtain an analytical expression of the fundamental matrix for linear nonstationary systems. This article proposed to find the fundamental matrix of the system of simplified canonical equations by means of approximate integration of the linear matrix differential equation of state, which it satisfies, using the mathematical apparatus of Walsh functions. In this case the elements of the matrix of the optimal control law are also determined in the form of Walsh series the constant coefficients of which are found from the system of algebraic equations. Since elements of the matrix of the optimal control law are piecewise constant functions this essentially simplified their practical implementation in comparison with nonstationary matrices of optimal control obtained based on the solution of Riccati equation. The accuracy of the obtained approximate optimal solution is achieved by choosing the appropriate number of terms of Walsh series expansion.
Approximation in the Mean of Classes of the Functions with Fractional Derivatives by their Abel-Poisson Integrals
58-69
10.1615/JAutomatInfScien.v51.i8.50
Tatyana V.
Zhyhallo
Lesya Ukrainka Eastern European National University, Lutsk
fractional derivatives; boundary problem; resolving functions; asymptotic equality; approximation in the mean
The constant development of the applied mathematics is due to its close connection with the fundamental directions of research in the related fields of natural sciences. One of the most important areas of modern science is the study of linear and nonlinear mathematical game models of various phenomena and processes of nature. The emergence of such models is due to the use in modern physics and techniques of influence on matter of electric fields of high intensity, beams of high-energy particles, powerful laser coherent radiation of shock waves of high intensity and powerful heat fluxes. The differential equations in partial derivatives, one of which is the equation of the elliptic type, describing the stationary processes of different physical nature, are the basis of such models. The simplest and most widespread equation of the elliptic type is the Laplace equation whose solution, under given conditions, on the boundary of the considered region, is the well-known Abel-Poisson integral. Approximate properties of the solution of an elliptic boundary value problem with the given boundary conditions at the boundary of the domain on classes of functions with fractional derivatives have been investigated. The solution to this problem finds its application in the study and further application of methods of resolving functions for game dynamics problems. Here we found the asymptotic equalities for the exact upper bounds of the deviations of classes of functions with fractional derivatives from their Abel-Poisson integrals in the integral metric. We establish the equivalence of the approximation characteristics of solutions of an elliptic boundary value problem with the given boundary conditions at the boundary of domain both in the uniform and in integral metrics for classes of functions with fractional derivatives.
Estimate of Time Series Similarity Based on Models
70-80
10.1615/JAutomatInfScien.v51.i8.60
Tatyana V.
Knignitskaya
Yuriy Fedkovych Chernovtsy National
University, Chernovtsy
distance between time series by models
clustering
cluster
time series
time series model
DTW
ERP
Determining the measure as a distance between time series is a starting point for many data mining tasks such as clustering and classification. Clustering is a main method of teaching without a teacher, which is used to divide data into groups based on the internal and a priori unknown characteristics inherent in the data. When dividing data into clusters, the need arises to select the similarity metric between objects. The paper describes the main existing algorithms for the "distance" searching between time series, which describe well this problem for small time series and under the absence of outliers. Outliers inherent in real processes lead to improper clustering, and, consequently, to wrong decisions making. It is proposed to consider the distance between time series in the form of the distance between models (ARIMA) of these time series. In the presence of a large number of outliers, classical methods linearly increase the distances between time series, while the distance proposed in the article according to the models behaves as a logarithmic function. It is shown that with an increase in the number of measurements, the relative errors for all classical methods remain almost unchanged. At the same time, the relative error for estimating the distance by the models is much smaller and decreases with an increase in the number of measurements. The main achievement of the article is the determination of the distance between time series, based on the concept of a model, and the comparison of this distance with the corresponding classical methods most commonly used. Using the Monte Carlo method, it has been shown that the proposed distance is more resistant to outliers and gives more accurate results for time series with a large number of observations. In addition, the complexity of the algorithm for calculating distances based on models is less than the analogous computational complexity of existing algorithms (DTW, ERP, Euclidean distance). There is no doubt that the use of models is one of the most convenient tools for studying the similarity of processes. In addition, for analysis taking into account this algorithm, it is convenient to use the averaged evolutions and the limiting evolutions in the diffusion approximation scheme. Also, due to the resistance to outliers of limiting evolutions, the entered distance can be used in clustering to build more noise-resistant clusters.