Chapman-Kolmogorov Equation Research Articles

Systematic matrix formulation for efficient computational path integration

In this work we introduce a novel methodological treatment of the numerical path integration method, used for computing the response probability density function of stochastic dynamical systems. The method is greatly accelerated by transforming the corresponding Chapman-Kolmogorov equation to a matrix multiplication. With a systematic formulation we split the numerical solution of the Chapman-Kolmogorov equation into three separate parts: we interpolate the probability density function, we approximate the transitional probability density function of the process and evaluate the integral in the Chapman-Kolmogorov equation. We provide a thorough error and efficiency analysis through numerical experiments on a one, two, three and four dimensional problem. By comparing the results obtained through the Path Integration method with analytical solutions and with previous formulations of the path integration method, we demonstrate the superior ability of this formulation to provide accurate results. Potential bottlenecks are identified and a discussion is provided on how to address them.

Organization and Reliability Testing of a Wind Farm Device in Its Operational Process

This article deals with the importance of simulation studies for the reliability of wind farm (WF) equipment during the operation process. Improvements, upgrades, and the introduction of new solutions that change the reliability, quality, and conditions of use and operation of wind farm equipment present a research problem during study. Based on this research, it is possible to continuously evaluate the reliability of WF equipment. The topic of reliability testing of complex technical facilities is constantly being developed in the literature. The article assumes that the operation of wind farm equipment is described and modeled based on Markov processes. This assumption justified the use of Kolmogorov–Chapman equations to describe the developed research model. Based on these equations, an analytical model of the wind farm operation process was created and described. As a result of the simulation analysis, the reliability of the wind farm was determined in the form of a probability function (R0(t)) for the WPPs system.

A multi-state model for sick leave and its impact on partial early retirement incentives: the case of the Netherlands

We investigate the effect of part-time and full-time work on health using a Markov framework and generalized linear models to smooth the resulting crude rates. The Chapman-Kolmogorov equations are used for a general solution. We apply this model to assess a partial early retirement incentive in the Netherlands, known as ‘the generation pact’. The smoothed rates imply that working part time does not necessarily mean a better health for the elderly. In fact, men are healthier when working full time, while women fall sick more often when working full time but recover more often as well. However, when comparing the future rates of a person currently aged 50 working full time and using the generation pact, both the recovery and the morbidity rates drop when starting the generation pact. A stylized assessment of the costs associated yields that costs drop by half when using the generation pact.

Assessment of the Reliability of Wind Farm Devices in the Operation Process

The article deals with simulation tests on the reliability of the equipment of the wind farm WF in the operation process. The improvement, modernization, and introduction of new solutions that change the reliability, as well as the quality and conditions of use and operation of wind farm equipment, require testing. Based on these tests, it is possible to continuously evaluate the reliability of the equipment of WF. The issue of reliability assessment of wind farm equipment, for which intelligent systems, diagnostic systems DIAG, and Wind Power Plant Expert System (WPPES) are used to modernize the operation process, can only be tested in a simulative way. The topic of testing the reliability of complex technical objects is constantly developing in the literature. In this paper, it is assumed that the operation of wind farm equipment is described and modeled based on Markov processes. The adoption of this assumption justified the use of the Kolmogorov–Chapman equations to describe the developed model. Based on this equation, an analytically developed model of the wind farm operation process was described. The simulation analysis determines the reliability of the wind farm in terms of the availability factor Kg(t). The simulation tests are performed in two phases using the computer program LabView. In the first stage, the reliability value in the form of the readiness factor Kg(t) as a function of changes in the mean repair time value ranging {from 0.3 to 1.0} was investigated. In the second stage, the reliability value of WF devices was examined as a function of changes in the value of the average time between successive failures, ranging from 1000 to 3000 (h)}.

Flow selections for (nonlinear) Fokker–Planck–Kolmogorov equations

We provide a method to select flows of solutions to the Cauchy problem for linear and nonlinear Fokker–Planck–Kolmogorov equations (FPK equations) for measures on Euclidean space. In the linear case, our method improves similar results of a previous work of the author. Our consideration of flow selections for nonlinear equations, including the particularly interesting case of Nemytskii-type coefficients, seems to be new. We also characterize the (restricted) well-posedness of FPK equations by the uniqueness of such (restricted) flows. Moreover, we show that under suitable assumptions in the linear case such flows are Markovian, i.e. they fulfill the Chapman-Kolmogorov equations.

Response Statistics of Single-Degree-of-Freedom Systems with Lévy Noise by Improved Path Integral Method

In this paper, we put forward an improved version of the path integral (PI) method for the response statistics of single-degree-of-freedom (SDOF) system excited by Lévy noise. To overcome the problem of large amount of calculation and storage, the PI method is simplified and parallelized, which makes the PI method for SDOF system with Lévy noise feasible and efficient. As the key to the PI method, the short-time transition probability density function (PDF) of the SDOF system is derived and verified by proving that the PI solution satisfies the corresponding fractional Fokker–Planck–Kolmogorov (FPK) equation. The fractional FPK equation, which is the governing equation of the SDOF system, is derived through the characteristic function and the Chapman–Kolmogorov equation. To solve the problem of large storage and calculations in the PI method, we simplify the one-step iteration formula and perform parallel calculations on the simplified formula. The simplification of the one-step iteration formula reduces one-fold integration, thereby reducing the storage capacity of the one-step transition matrix. Parallel calculation by domain decomposition can effectively reduce the calculation time, which can be seen from the running time of two prototypical examples. Besides, to show the effectiveness of the improved PI method, Monte Carlo solutions and analytical solutions are used as reference solutions.

A method for converting microdosimetric spectra in diamond to tissue in proton therapy.

Diamond has been regarded as a promising microdosimeter in radiation protection and radiotherapy due to its excellent properties. However, as the diamond is not tissue equivalent, a conversion of the measured spectra in the diamond microdosimeter to the tissue site is needed. In this work, we intend to deduce a method for converting the microdosimetric spectra from diamond to tissue in the proton therapy application based on the Chapman-Kolmogorov equation and investigate the validity of this method in spectral conversion. The comparison of stopping power and energy deposition distribution of diamond and tissue shows that the conversion of the spectra in diamond to tissue can be performed by a simple scaling factor. Therefore, the equivalence of the energy deposition spectra in the diamond microdosimeter and a tissue site of the same size in the same radiation field was studied first to obtain the scaling factor. Then, the spectra conversion method was derived from the Chapman-Kolmogorov equation and the scaling factor. The Geant4 simulation was employed to settle this study. Theoretical and Geant4 simulation results indicate that the linear stopping power ratio of diamond to tissue is adequate to convert the microdosimetric spectra in diamond to tissue of identical dimension. The conversion results indicate that the energy deposition spectra converted from diamond to bone agree well with the spectra calculated by Geant4 along the Bragg curve. As for water, a good agreement of the converted and calculated spectra was found at the plateau of the Bragg curve and the distal part of the Bragg peak. At the proximal part of the Bragg peak, the converted and calculated spectrum is poorly coincident. The method of the energy deposition spectra conversion from the diamond microdosimeter to the tissue site of equal size shows relatively good results in most situations. But the deficiency was also found. Therefore, further investigation is needed to improve the reliability of this method.

Mathematical analysis of a Markovian multi-server feedback queue with a variant of multiple vacations, balking and reneging

In this paper, we analyze a multi-server queue with customers impatience and Bernoulli feedback under a variant of multiple vacations. On arrival, a customer decides whether to join or balk the system, based on the observation of the system size as well as the status of the servers. It is supposed that customer impatience can arise both during busy and vacation period because of the long wait already experienced in the system. The latter can be retained via certain mechanism used by the system. The feedback occurs as returning a part of serviced customers to get a new service. The queue under consideration can be used to model the processes of information transmission in telecommunication networks. We develop the Chapman-Kolmogorov equations for the steady-state probabilities and solve the differential equations by using the probability generating function method. In addition, we obtain explicit expressions of some important system characteristics. Different queueing indices are derived such as the probabilities when the servers are in different states, the mean number of customers served per unit of time, and the average rates of balking and reneging.

Reliability analysis of PV generating systems in the islanded DC microgrid under dynamic and transient operation

Dynamic fluctuations in the PV-source power and different DC-fault scenarios in the off-grid PV-battery-based DC microgrid could lead to a rapid decrease in the reliability of the PV-generating system. To make this viewpoint clearer, this paper proposes a novel reliability-evaluating methodology for PV-generating systems in the islanded DC microgrid under dynamic and transient operation conditions. Firstly, the dynamic-voltage-varying failure rate (DVVFR) and the fault-current-varying failure rate (FCVFR) of a PV-generating system in the off-grid DC microgrid are formulated. The DVVFR mainly depends on dynamic fluctuations in the PV-source power and the load power, whereas the FCVFR mostly represents the failure probability due to pole-to-pole and pole-to-ground faults in the DC microgrid. Then, a possible combination of the used-time-varying failure rate (TVFR), the power-loss and temperature-dependent failure rate, the DVVFR, and the FCVFR is proposed for evaluating the system-level and component-level reliability of PV-generating sources. Finally, Markov-state transition diagrams and Chapman–Kolmogorov equations are derived and applied for the PV-system reliability assessment in the off-grid PV-battery-based DC microgrid. Experimental results are analysed to reveal that the operation of DC-DC power converters dominates the PV-system reliability under the dynamic and transient operation. The DVVFR of the PV system is mostly smaller than its FCVFR, however, the system-level reliability of the PV-generating source is significantly reduced by dynamic-voltage-fluctuating cases due to the more frequent repetition of these dynamic cases. Moreover, the MTTF and MTBF of the PV-generating system could also be dramatically decreased by the dynamic and transient operation of the DC microgrid.

Classification in chains of three-dimensional real evolution algebras

A chain of evolution algebras (CEA) is an uncountable family (depending on time) of evolution algebras on the field of real numbers. The matrix of structural constants of a CEA satisfies the Chapman-Kolmogorov equation. In this paper, we consider three CEAs of three-dimensional real evolution algebras. These CEAs depend on several (non-zero) functions defined on the set of time. For each chain we give a full classification (up to isomorphism) of the algebras depending on the time-parameter. We find concrete functions ensuring that the corresponding CEA contains all possible three-dimensional evolution algebras.

Phase Type Zero Truncated Poisson Lindley Distributions and their application in modeling Secondary Cancer Cases

Insurance of chronic illness is slowly gaining ground in Kenya which has lead to insurance firms introducing insurance products of chronic illness among them being cancer insurance policies. However, unlike other chronic illnesses, cancer can move from the organ of origin to another which will consequently lead to increased cost of treatment. This can not be modeled using ordinary distributions hence it has become an area of interest for many researchers. Zero-truncated phase type distributions are used to solve this drawback of ordinary distributions as it can in-cooperate these transitions while modeling claim count data. They further improve modeling of claim count data as they only consider positive values of claim count excluding zeros. This is the nature of real claim count data as zero claim frequency can not attract any claim severity amount. In this paper aggregate claim losses of secondary cancers in Kenya are estimated using Zero-truncated Poisson Lindley distributions. Zero-truncated one parameter as well as Zero-truncated two parameter Poisson Lindley distributions are derived. Their compound probability generating functions are also constructed. The transitions states of secondary cancer states are estimated using continuous Chapman Kolmogorov equation and used as the matrix parameters for the claim count distributions. Pareto, Generalized Pareto, Weibull, OPPL and TPPL distributions are the distributions considered in this research in modeling claim numbers. This study concludes that aggregate losses of secondary cancer cases using Kenyan data are best modeled by PH-ZTOPPL Generalized Pareto model for PH-ZTOPPL distribution models while for PH-ZTTPPL distribution models the best model was PH-ZTTPPL-Generalized Pareto model. The two best models were compared and PH-ZTTPPL-Generalized Pareto model was proven to be the best model. Comparing this model with PH-TPPL Generalized Pareto model from earlier research PH-TPPL Generalized Pareto model proved to be a better model implying that zero claim count data should be considered in estimation of aggregate losses

Theory of Heterogeneous Circuits With Stochastic Memristive Devices

We introduce an approach based on the Chapman-Kolmogorov equation to model heterogeneous stochastic circuits; namely, the circuits combining binary or multi-state stochastic memristive devices and continuum reactive components (capacitors and/or inductors). These circuits are described in terms of occupation probabilities of memristive states that are functions of reactive variables. As an illustrative example, the series circuit of a binary memristor and capacitor is considered in detail. Some analytical solutions are found. Our work offers a novel analytical/numerical tool for modeling complex stochastic networks, which may find a broad range of applications.

Probability-conservative simulation for Lévy financial model by a mixed finite element method

We propose an expanded mixed finite element method (EMFEM) with sharp recognition for such intrinsic natures of a Markov process as the probability conservation, the Chapman-Kolmogorov equation, skewness and excess kurtosis for the fractional-order advection diffusion equation (FADE) oriented in financial engineering. We improve the result in [11] for purely fractional diffusion model by modifying the domain of one bilinear form so that it is well defined for the appearance of the advection term, and the result in [13] for second order elliptic models by using lower-order finite element spaces to reduce computation costs. The solvability and optimal convergence rates of the EMFEM are proved by showing the coerciveness and the inf-sup condition for the modified bilinear forms over the lower-order finite element spaces, and thus the systematically mathematical theory of the EMFEM for general FADEs are developed. Numerical experiments are conducted to confirm these theoretical findings.

Continuous rate modeling of bacterial stochastic size dynamics.

Bacterial division is an inherently stochastic process with effects on fluctuations of protein concentration and phenotype variability. Current modeling tools for the stochastic short-term cell-size dynamics are scarce and mainly phenomenological. Here we present a general theoretical approach based on the Chapman-Kolmogorov equation incorporating continuous growth and division events as jump processes. This approach allows us to include different division strategies, noisy growth, and noisy cell splitting. Considering bacteria synchronized from their last division, we predict oscillations in both the central moments of the size distribution and its autocorrelation function. These oscillations, barely discussed in past studies, can arise as a consequence of the discrete time displacement invariance of the system with a period of one doubling time, and they do not disappear when including stochasticity on either division times or size heterogeneity on the starting population but only after inclusion of noise in either growth rate or septum position. This result illustrates the usefulness of having a solid mathematical description that explicitly incorporates the inherent stochasticity in various biological processes, both to understand the process in detail and to evaluate the effect of various sources of variability when creating simplified descriptions.

Reliability evaluation of a system with active redundancy strategy and load-sharing time-dependent failure rate components using Markov process

Due to the high sensitivity in applying electronic and mechanical equipment in functional systems, creating conditions to increase a system's reliability is always critical for system designers. In most of the studies in the reliability area, it is assumed that the failure rates of the system's components are constant but considering time-dependent failure rates for the components is more realistic and draws the models near to real conditions. This paper presents a framework to use Markov process to obtain the system's reliability under some assumptions. In this regard, we worked on a load-sharing system with identical time-dependent failure rate components and used Markov process to calculate the system reliability. First, we define the requirements for using Markov process, and it is shown that it is possible to use it to calculate the considered system's reliability. Then, a formula is presented to calculate the reliability of the considered system using the Markov process through solving the Chapman Kolmogorov equation, when the components' life has Weibull Distribution. Next, we validate the results of the presented formula using the simulation technique. Finally, we compare the computational time of the proposed model for solving large-size problems with the simulation technique. The results show the superiority of the proposed model in terms of computational time.

Estimation of aggregate losses of secondary cancer cases using PH Panjer class (a,b,1) distributions,

This research in-cooperates phase type distributions of Panjer class $(a,b,1)$ in estimation of aggregate loss probabilities of secondary cancer. Matrices of the phase type distributions are derived using Chapman-Kolmogorov equation and transition probabilities estimated using modified Kaplan-Meir and consequently the transition intensities and transition probabilities. Stationary probabilities of the Markov chains represents $\vec{\gamma}$. Claim amount are modeled using OPPL, TPPL, Pareto, Generalized Pareto and Wei-bull distributions. PH ZT Poisson with Generalized Pareto distribution provided the best fit.

Probability of an Intermediate (Reduced Operational) State

This work examines with the form of the well-known sum: p + q = 1 – which is the sum of the probabilities of opposite events, in particular: the sum of the probabilities of the operational and non-operational (failure) states of a single element (a creation characterised by one output and any number of inputs). It was found that without significantly compromising the accuracy of the previous analyses, it was possible to introduce an additional component to the sum: iiipq3, a component that embodies the probability of an intermediate state, or a reduced operational state. With a constant value of the sum of the components in question, their variation as a function of probability q was determined, following which in the function of the same variable the variation of the entropy of an element's i state was examined using Chapman-Kolmogorov equations; here the focus was on investigating the intensity of the transition from the operational state to the non-operational state or an intermediate state, and from an intermediate state to the non-operational state. The meaning of intermediate probability was also referenced to the object: its diagnostic program, the entropy of structure, the full set of discriminable states, and the relevant transition intensities. It became indispensable in this respect to describe the object using the language of graph theory, in which the basic concepts are layers and an availability matrix. It should be noted that the subject object is an entity that comprises a set of individual elements, with a number and structure of connections that are consistent with the purpose of this entity.

An Availability Evaluation Method for Desalination Process with Three-States Equipment under the Specific Repair Queuing Policy

Oil waste, is one of the most important pollutants in the oil and gas industry. Since the wells' oil has significant saltwater, the effluent amount increases with increasing oil reservoir extraction. Separating the saltwater from the extracted oil before starting the refinery process plays an essential role in reducing the oil costs as well as the useful transfer capacity. This paper presents a new Chapman-Kolmogorov Equation-based (CKEB) method to evaluate a desalination system's availability with three-state equipment and weighted-k-out-of-n configuration. In this system, the equipment is repairable, and each repair facility can repair all equipment types of different sub-systems (pump stations). We consider all failures and repairs to have a constant rate (with Exponential distribution) and use the Chapman-Kolmogorov Equation to drive the system’s availability. Then we validate the presented method using a simulation technique. Finally, the elapsed time of both solving techniques is compared. The results show the superiority of the CKEB technique in terms of computational time. Compared with the simulation technique, the computational time ratio for the CKEB method is between 0.0002% to 0.0058% for the small-size problems, between 0.05% to 0.94% for the medium-size problems, and between 1.31% to 5.39% for the large-size problems.

Quantized noncommutative Riemann manifolds and stochastic processes: The theoretical foundations of the square root of Brownian motion

We lay the theoretical and mathematical foundations of the square root of Brownian motion and we prove the existence of such a process. In doing so, we consider Brownian motion on quantized noncommutative Riemannian manifolds and show how a set of stochastic processes on sets of complex numbers can be devised. This class of stochastic processes are shown to yield at the outset a Chapman–Kolmogorov equation with a complex diffusion coefficient that can be straightforwardly reduced to the Schrödinger equation. The existence of these processes has been recently shown numerically. In this work we provide an analogous support for the existence of the Chapman–Kolmogorov–Schrödinger equation for them, performing a Monte Carlo study. It is numerically seen as a Wick rotation can turn the heat kernel into the Schrödinger one, mapping such kernels through the corresponding stochastic processes. In this way, we introduce a new kind of improper complex stochastic process. This permits a reformulation of quantum mechanics using purely geometrical concepts that are strongly linked to stochastic processes. Applications to economics are also entailed.

Mathematical Modeling and Time Dependent Availability Analysis of Poly - Ethylene Terephthalate Bottle Hot Drink Filling System: A Case Study

The current research aims to analyze the Time Dependent System Availability (TDSA) and identify the most sensitive component/unit of Poly Ethylene Terephthalate (PET) Bottle Hot Drink Filling System of a beverage Industry with Markovian approach. TDSA and Mean time between failures (MTBF) is computed for system concerned. The industrial system chosen includes five components namely Blow Moulding Machine, Filling Machine, Cooling Machine, Labeling Machine and Coding Machine. Transition diagram of the system has been developed and respective Chapman-Kolmogorov equations are originated. The derived equations are further simplified by an advance and receptive method acknowledged as adjustive step size control Runge-Kutta method. On basis of TDSA analysis most sensitive unit/component has been identified and maintenance priorities to other units/components have been assigned.