Chapman-Kolmogorov Equation Research Articles

Lorentz covariant physical Brownian motion: Classical and quantum

In this work, we re-examine the Goldstein-Kaç (also called Poisson-Kaç) velocity switching model from two points of view. On the one hand, we prove that the forward and backward Chapman–Kolmogorov equations of the stochastic process are Lorentz covariant when the trajectories are parameterized by their proper time. On the other hand, to recast the model as a quantum random evolution, we restate the Goldstein-Kaç model as a Hamiltonian system, which can then be quantized using the standard correspondence rules. It turns out that the density matrix for the random quantum evolution satisfies a Chapman–Kolmogorov equation similar to that of the classical case, and therefore, it is also Lorentz covariant. To finish, we verify that the quantum model is also consistent with special relativity and that transitions outside the light cone, that is, transitions between states with disjoint supports in space–time, cannot occur.

Devising an approach to the construction of an adapted model of the reconnaissance-fire system functioning

The object of this study is the operational model of a reconnaissance-fire system. The problem that was solved is the lack of an approach to building a model of the functioning of a combat system, in particular a reconnaissance-fire system, which would take into account the influence of all subsystems and include the necessary number of system states. An improved procedure for building an adapted operational model of the reconnaissance-fire system has been proposed. The essence of the improved methodology is the formalization of processes through the definition of system states and intensities of transitions from state to state. The improved procedure is based on the Kolmogorov-Chapman equations and the goal tree construction method. A feature of the improved methodology is the breakdown of the states of the reconnaissance-fire system by hierarchy levels, which allows taking into account more necessary states of the system. The field of practical use of the improved methodology is planning and management processes during the development of action algorithms during combat operations. An adapted operational model of the reconnaissance-fire system has been built. The essence of the model is to determine the probability of the reconnaissance-fire system being in a certain state based on the Chapman-Kolmogorov equations, taking into account the necessary level of detail in the process of its operation. Special feature of the proposed model is that it makes it possible to model by taking into account 39 states of the system with the necessary accuracy both for the system as a whole and separately for subsystems. This is explained by the fact that the test of the adequacy of the model showed that the discrepancy of the results is within the statistical error from 2 to 9 %. The field of application of the adapted operational model of the reconnaissance-fire system is the processes of making a decision on the application of the operation of the intelligence-fire system during hostilities and their management during combat operations.

Probability Bracket Notation for Probability Modeling

Following Dirac’s notation in Quantum Mechanics (QM), we propose the Probability Bracket Notation (PBN), by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc. Using the PBN, many formulae, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, which are easy to expand by inserting a proper P-identity. The time evolution of homogeneous Markov processes can also be formatted in such a way. Our system P-kets are identified with probability vectors and our P-bra system is comparable with Doi’s state function or Peliti’s standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process, and the Chapman–Kolmogorov equations are obtained by inserting a time-dependent P-identity. Also, some QM expressions in Dirac notation are naturally transformed to probability expressions in PBN by a special Wick rotation. Potential applications show the usefulness of the PBN beyond the constrained domain and range of Hermitian operators on Hilbert Spaces in QM all the way to IT.

On positively divisible non-Markovian processes

There are some positively divisible non-Markovian processes whose transition matrices satisfy the Chapman–Kolmogorov equation. These processes should also satisfy the Kolmogorov consistency conditions, an essential requirement for a process to be classified as a stochastic process. Combining the Kolmogorov consistency conditions with the Chapman–Kolmogorov equation, we derive a necessary condition for positively divisible stochastic processes on a finite sample space. This necessary condition enables a systematic approach to the manipulation of certain Markov processes in order to obtain a positively divisible non-Markovian process. We illustrate this idea by an example and, in addition, analyze a classic example given by Feller in the light of our approach.

Probability density of the solution to nonlinear systems driven by Gaussian and Poisson white noises

A new method is proposed to compute the probability density of the multi-dimensional nonlinear dynamical system perturbed by a combined excitation of Gaussian and Poisson white noises. We first deduce a probability-density solver from the Euler–Maruyama scheme of the stochastic system and the corresponding Chapman–Kolmogorov equation. This solver actually is an explicit numerical formula of the probability density of the solution to this stochastic system. To compute the probability density, we propose an efficient algorithm for this solver, which actually is the implementation of a numerical integration. Furthermore, we prove this solver is an approximated solution of the corresponding forward Kolmogorov equation. Numerical examples are conducted to illustrate our probability-density solver.

Efficient path integral approach via analytical asymptotic expansion for nonlinear systems under Gaussian white noise

In this paper an efficient formulation of the Path integral (PI) approach is developed for determining the response probability density functions (PDFs) and first-passage statistics of nonlinear oscillators subject to stationary and time-modulated external Gaussian white noise excitations. Specifically, the evolution of the response PDF is obtained in short time steps, by using a discrete version of the Chapman-Kolmogorov equation and assuming a Gaussian form for the conditional response PDF. Next, the technique involves proceeding to treating the problem via an analytical asymptotic expansion procedure, namely the Laplace’s method of integration. In this manner, the repetitive double integrals involved in the standard implementation of the PI approach are evaluated in a closed form, while the response and first-passage PDFs are obtained by mundane step-by-step application of the derived approximate analytical expression. It is shown that the herein proposed formulation can drastically decrease the associated computational cost by several orders of magnitude, as compared to both the standard PI technique and Monte Carlo solution (MCS) approach. A number of nonlinear oscillators are considered in the numerical examples. Notably, for these systems both response PDFs and first-passage probabilities are presented, whereas comparisons with pertinent MCS data demonstrate the efficiency and accuracy of the technique.

Retaining bus riders: A lifecycle longitudinal analysis of behavioral status transitions from entry to exit

Amidst a global decline in bus ridership, this study pioneers a longitudinal approach to understanding individual-level transitions and churning in urban bus systems. Utilizing a novel framework that leverages smart card data, we construct and analyze user behavior transition matrices over time, employing Markov processes and the Chapman-Kolmogorov Equation. Our analysis, derived from a 22-month dataset from Shenzhen, reveals a two-stage churning process: users first decrease travel frequency before transitioning to irregular travel patterns. Crucially, this study introduces targeted retention policies, including tiered usage incentives and personalized communication strategies, aimed at different stages of the user lifecycle. By offering free subsequent trips to irregular travelers and combining policy approaches for users at high risk of churning, we provide actionable insights for transit operators to counter the trend of declining ridership.

ОЦІНЮВАННЯ ВПЛИВУ ЗАСОБІВ ПІДКЛЮЧЕННЯ РЕЗЕРВУ НА ПОКАЗНИКИ НАДІЙНОСТІ СИСТЕМИ ПЕРЕДАВАННЯ ДАНИХ

Structural redundancy is included in modern data transmission systems to ensure the specified reliability indicators. Structural redundancy gives the system fault tolerance and significantly increases its reliability. To manage the redundant resource, it is necessary to have additional tools that will connect the redundant equipment to the main configuration of the data transmission system. The redundant equipment is a separate technical means and, from the point of view of reliability, it is connected in series with other modules of the data transmission system. Thus, the inclusion of redundant connection facilities in the data transmission system reduces its reliability. In addition, even with perfect reliability, redundant connections may fail or malfunction and, under certain conditions, cause the transmission system to fail. To determine the impact of reserve connection facilities on the reliability of the transmission system, the article applies an original technology for building models of discrete-continuous stochastic systems. ASNA software was used as a software tool for modeling the reliable behavior of the data transmission system. This software provides automated construction of the model in the form of a graph of states and transitions based on a structural automaton model. Based on the graph of states and transitions, an analytical model is automatically generated in the form of a system of linear differential Kolmogorov-Chapman equations. Using the developed model, the article quantitatively analyzes the impact of redundant connection means (switches) on the reliability indicators of the data transmission system. In the study, the proposed models, unlike the existing ones, take into account both reliability indicators (probability of failure) and functionality indicators (probability of switching, speed) of the reserve connection means. On the basis of the developed model, the limit values of reliability and functionality of the redundant connection means (switches) were determined, at which they will minimize the reliability of the fault-tolerant data transmission system. Based on the analysis of the obtained results, practical recommendations for the selection of reliability indicators of the backup connection means, their speed and probability of switching are formed.

Do traffic flow states follow Markov properties? A high-order spatiotemporal traffic state reconstruction approach for traffic prediction and imputation

Assessing traffic states accurately is challenging due to the complex, high-dimensional, and nonlinear nature of traffic systems. This study introduces the innovative High-Order Spatiotemporal Traffic State Reconstruction (HOSTSR) algorithm, designed to track and predict traffic flow dynamics effectively. It combines phase space reconstruction with time delays and high-order neighborhood concepts from graph theory to improve traffic state assessments' accuracy. The algorithm's effectiveness is validated using chi-square tests and the Chapman-Kolmogorov equation to confirm the Markovian properties of traffic flows. A lean autoencoder, informed by prior Markov knowledge of traffic states, is developed for mapping traffic states to real traffic data, proving highly effective for traffic data imputation due to the Markov model's memoryless property. Experimental results from the PeMSD04 and PeMSD08 datasets show that HOSTSR outperforms traditional state reconstruction methods based on delayed coordinate embedding in predicting future traffic flow state based on four key metrics. The autoencoder framework, guided by prior Markov knowledge, shows significant advantages in addressing traffic data gaps in different cases over six baseline models. Gradient sensitivity analysis further evaluates the impact of prior knowledge on improving the autoencoder's interpretability for interpolation efforts.

Stochastic dynamics of mechanical systems with impacts via the Step Matrix multiplication based Path Integration method

AbstractIn this work we propose the Step Matrix Multiplication based Path Integration method (SMM-PI) for nonlinear vibro-impact oscillator systems. This method allows the efficient and accurate deterministic computation of the time-dependent response probability density function by transforming the corresponding Chapman–Kolmogorov equation to a matrix–vector multiplication using high-order numerical time-stepping and interpolation methods. Additionally, the SMM-PI approach yields the computation of the joint probability distribution for response and impact velocity, as well as the time between impacts and other important characteristics. The method is applied to a nonlinear oscillator with a pair of impact barriers, and to a linear oscillator with a single barrier, providing relevant densities and analysing energy accumulation and absorption properties. We validate the results with the help of stochastic Monte-Carlo simulations and show the superior ability of the introduced formulation to compute accurate response statistics.

The dynamic model of lubricating layer formation/destruction processes in boundary and mixed friction modes

The article is dedicated to studies of self-organization processes evolving in time in boundary lubricant layers. The essence of the article consists in use of the model of the connected processes of formation and destruction of a shielding delayed lubricant layer based on differential equations similar to that which are used for description in time of dynamics of probabilities of states of a single channel system of queueing theory with Poisson stream of queries and exponential service (Kolmogorov-Chapman equations). Proposed method involves introducing concepts: the intensity of delayed layer growth - l and the intensity of delayed layer destruction - m. This kind of concepts, in author’s opinion, is organically inherent in atomic-scale molecular systems consisting of large numbers of particles with many significant connections between them, all of which are impossible to describe and simulate. The studies of lubricating layers formation/destruction processes and structure of boundary lubricant layers in the medium of paraffin oil as a lubricant at different sliding speeds and loads were made. It was established that minimal values of friction coefficient were characteristic for layers with high intensity of delayed layer growth and high values of limit thickness. Proposed approach to modeling phenomena of self-organization in boundary lubricant layer can be useful for understanding the mechanism of lubricating action in conditions of boundary and mixed friction and applied for solving some practical problems related to the selection of effective lubricants for tribo couplings of the specified configuration and loading parameters.

Data-Augmented Numerical Integration in State Prediction: Rule Selection

This paper deals with the state prediction of nonlinear stochastic dynamic systems. The emphasis is laid on a solution to the integral Chapman-Kolmogorov equation by a deterministic-integration-rule-based point-mass method. A novel concept of reliable data-augmented, i.e., mathematics- and data-informed, integration rule is developed to enhance the point-mass state predictor, where the trained neural network (representing data contribution) is used for the selection of the best integration rule from a set of available rules (representing mathematics contribution). The proposed approach combining the best properties of the standard mathematics-informed and novel data-informed rules is thoroughly discussed.

Dual-channel call center model with exchange of requests and repeat calls

Computer simulation of stationary distribution of states in a call center for a two-channel RQ system with exchange of requests is presented. Such systems are becoming increasingly relevant due to the parallel use of both human dispatchers and voice intelligent bots for servicing calls. The simulation is based on solving the system of stationary Chapman-Kolmogorov equations for the Markov process describing the RQ-system. The system of equations proposed in the work differs from previous models by the presence of an exchange of requests between service channels in accordance with customer preferences. It takes into account the possibility of random repeated calls within a given average time. Service time in channels and delay time of requests in orbits have exponential distribution laws. Requests in orbit have the property of impatience, i.e. leave the system after some random time. To find the stationary distribution of states in orbits, the iterative numerical Gauss-Seidel method is used, which ensures fast convergence of calculations. Each channel has its own orbit of requests. The accuracy of the solution is controlled by increasing the maximum number of requests in orbit until the result stabilizes. The model demonstrates the sensitivity of system performance to the asymmetry of customer preferences when changing channels. Numerical simulation was carried out for the call center of the housing management company "StroyTekhnika" in the city of Voronezh. Application flows and repeat call parameters were calculated based on data from the company website and social network analysis. Accounting for repeat calls reduces system throughput compared to the option of completely impatient customers who do not use repeat calls. The case of completely impatient customers describes the limit state of a service system. At the same time, redistributing calls in favor of a more productive channel improves the overall performance of the system. The results obtained show the feasibility of using high-performance multi-channel voice bots while simultaneously stimulating a shift in customer preferences in favor of intelligent automata.

Methodology for Assessing the Risks of Regional Competitiveness Based on the Kolmogorov–Chapman Equations

The relevance of research on competitiveness at the meso level is related to the contemporary views of a region as an essential element of the economic space. The development of forecasting and analytical methods at the regional level of the economy is a key task in the process of strategic decision making. This article proposes a method of quantitative assessment of the risks of regional competitiveness. The novelty of this approach is based on both a fixed-point risk assessment and scenario-based predictive analysis. A hierarchical structure of indicators of competitiveness of regions is offered. A method based on the Kolmogorov–Chapman equations was used for the predictive estimation of risks of regional competitiveness. The integrated risk assessment is performed using the modified fuzzy ELECTRE II method. A web application has been implemented to assess the risks of competitiveness of Russian regions. The functionality of this application provides the use of multi-criteria decision-making methods based on a fuzzy logic approach to estimate risks at a specified time, calculating the probability of risk events and their combinations in the following periods and visualizing the results. Approbation of the technique was carried out for 78 Russian regions for various scenarios. The analysis of the results obtained provides an opportunity to identify the riskiest factors of regional competitiveness and to distinguish regions with different risk levels.

Research on and Assessment of the Reliability of Railway Transport Systems with Induction Motors

Increasing the efficiency and reliability of modern railway transport is accompanied by an increase in monitoring and diagnostic systems for the current state of electric drives. Modern railway transport contains a large number of induction motors to ensure the operation of the drives of various mechanisms. In the article, based on the operational statistics of engine failures and the proposed scheme for diagnosing them, studies were carried out and a model was developed for assessing the reliability of a transport system equipped with an on-board diagnostic system for the current state. When building the models, the Markov method was used, including the construction of graphs for the five most relevant states of the induction electric motor during operation. The results obtained are relevant for evaluating the effectiveness of using the built-in diagnostic system and scheduling routine maintenance, which will affect the efficiency of railway transport. Based on the process of the diagnosis of railway transport systems with induction motors, five operating states of the object studied were interpreted: the state of full operation, state “S0”; the state of incomplete serviceability, state “S1”; critical serviceability, state “S2”; the state of the pre-damage condition, state “S3”; the state of unserviceability (defect), state “S4”. Subsequently, a five-state model of the operation process of railway transport systems with induction motors was developed. This model is also described by equations of state: Kolmogorov–Chapman equations. The reliability quantities determined form the basis for simulation reliability studies. The effect of the simulation study is the reliability quantities determined in the form of reliability functions and probabilities of the occurrences of the operating states of railway transport systems with induction motors; an important part of the reliability study of the system examined is to estimate the times of the occurrences in the object studied of the operating states in the future.

A flow of quantum genetic Lotka-Volterra algebras on [formula omitted

In the present paper, we define a quantum analog of Lotka-Volterra algebras using a coalgebra scheme. This new framework provides a fresh perspective for the treatment of generic algebras. Additionally, a flow of quantum analogs of Lotka-Volterra genetic algebras is investigated. It's worth mentioning that such types of algebras are first introduced in this work. We observe that a flow of algebras is a particular type of continuous-time dynamical system, with states that are algebras and a structural constant matrix that depends on time and satisfies certain analogs of the Kolmogorov-Chapman equations. Using quantum quadratic operators, it is constructed a flow of quantum Lotka-Volterra algebras for the given multiplication. Furthermore, time-dependent behavior properties of these flow algebras are examined. The algebraic properties of the introduced flows are also studied, such as finding idempotents and examining an algebra generated by a pair of idempotents. It is shown that the later one is associative, while the flow is not associative. Additionally, derivations of the flow of the algebras are described.

Magnetospheric Multiscale Observations of Markov Turbulence on Kinetic Scales

In our previous studies we have examined solar wind and magnetospheric plasma turbulence, including Markovian character on large inertial magnetohydrodynamic scales. Here we present the results of the statistical analysis of magnetic field fluctuations in the Earth’s magnetosheath, based on the Magnetospheric Multiscale mission at much smaller kinetic scales. Following our results on spectral analysis with very large slopes of about −16/3, we apply a Markov-process approach to turbulence in this kinetic regime. It is shown that the Chapman–Kolmogorov equation is satisfied and that the lowest-order Kramers–Moyal coefficients describing drift and diffusion with a power-law dependence are consistent with a generalized Ornstein–Uhlenbeck process. The solutions of the Fokker–Planck equation agree with experimental probability density functions, which exhibit a universal global scale invariance through the kinetic domain. In particular, for moderate scales we have the kappa distribution described by various peaked shapes with heavy tails, which, with large values of the kappa parameter, are reduced to the Gaussian distribution for large inertial scales. This shows that the turbulence cascade can be described by the Markov processes also on very small scales. The obtained results on kinetic scales may be useful for a better understanding of the physical mechanisms governing turbulence.

Assessment of the Reliability of Wind Farm Device on the Basis of Modeling Its Operation Process

The evaluation and analysis of the procedures for determining the dependability of WF wind farm equipment employed in a few publications are this article’s main problems. The publications chosen for review specifically mention investigations into the dependability of WF wind farm machinery. The following topics were the authors’ main areas of analysis: description and review of the techniques used to represent how technical items operate and the selection of the weight of the theoretical ideas of reliability that were used to gauge the dependability of the wind farm equipment under study. The authors of the studied works set out to address a number of significant problems pertaining to the modernization of the management of the WF equipment renewal process. The subjects of the studied works suggest that the established models of the technical object’s operational process are particularly significant in both the theory and practice of the reliability of technical objects. Using Kolmogorov–Chapman equations, models of the WFD operating process that are based on the idea of Markov processes are very helpful for simulation studies.

Hydrodynamics of a class of [formula omitted]-urn linear systems

In this paper we are concerned with hydrodynamics of a class of N-urn linear systems, which include voter models, pair-symmetric exclusion processes and binary contact path processes on N urns as special cases. We show that the hydrodynamic limit of our process is driven by a C[0,1)′-valued linear ordinary differential equation and the fluctuation of our process, i.e, central limit theorem from the hydrodynamic limit, is driven by a C[0,1)′-valued Ornstein–Uhlenbeck process. To derive above main results, we need several replacement lemmas. An extension in linear systems of Chapman–Kolmogorov equation plays key role in proofs of these replacement lemmas.

Transient analysis and reliability modeling of fault-tolerant system operating under admission control policy with double retrial features and working vacation

Maintaining a high-level reliability and efficiency without interruption are the key concerns for many real-time machining systems. Using the redundancy and repair facility features, we develop a double retrial orbit queueing model for the fault-tolerant machining system (FTMS) operating under the restriction of admission of repair jobs based on threshold policy and working vacation. The provision of primary and secondary orbits is made so that the failed units can wait there based on the facility available in case the repairman is occupied. From the orbits, the failed units retry to get the repairman free so that the repair job can be accomplished. Chapman–Kolmogorov equations for the system states of FTMS have been constructed to evaluate the transient reliability and queueing indices using the spectral technique. The sensitivity along with the relative sensitivity analysis of crucial system parameters, have facilitated. The impacts of parameter variability on the system metrics and total expected cost are examined for illustrative examples.