Chapman-Kolmogorov Equation Research Articles

The scenario modeling of regional competitiveness risks based on the Chapman-Kolmogorov equations

The paper is devoted to the development of analytical tools for assessing the dynamics of risks to the competitiveness of Russian regions within the conditions of incomplete information. The choice of analysis methods for assessing the risk dynamics was substantiated. A solution based on the implementation of homogeneous Markov models with a discrete set of states and continuous time was proposed, it allowed to assess the probability of risks affecting regional competitiveness. A hierarchical structure of factors that determine the risks of regional competitiveness was formed as a cause-and-effect graph. Corresponding systems of Kolmogorov – Chapman differential equations were obtained and solved. An example for analyzing a 3-element minimal cut set of a cause-effect graph was presented. A set of scenarios for the study of risks and their combinations in various conditions was proposed. The solution was obtained for three leading Russian regions. For a set of scenarios, numerical assessments of risk factors that significantly affect the decline in the competitiveness of regions have been fulfilled. The study results made it possible to rank the effects of critical event occurrence, to identify the most significant risk factors for the loss of sustainable development and competitiveness for the Russian Federation regions.

The research of cognitive methods of infocommunication management under conditions of uncertainty

The methods of management of the fifth generation infocommunication networks under conditions of uncertainty are shown. The research of super-complex systems into which modern infocommunications with social networks superimposed on them turn is based on nonlinear cognitive methods, which currently claim the role of the basic paradigm in the management of intercommunications of the future. The scientific research is to ensure efficient and sustainable management of infocommunication networks under conditions of higher network capacity, multiservices, meeting the appropriate requirements for control systems for new heterogeneous superdense networks. The probabilistic-temporal characteristics of infocommunications management are used to carry out the research. This article offers a multi-agent management model that consists of a large number of agents, which own limited intellectuality and submit requests for services; this model is described by the Chapman–Kolmogorov equation. The analytical relations of the total number of intelligent agents in the system on receipt and service intensity in telecommunication networks are received. The relations that enable the telecommunication service management with different target functions: minimization of the number of downtime of intelligent agents or, visa versa, maximization of their number when there is the need to reserve resources for the expected burst of requests are described. It should be noted that control can be performed by specifying the probability value of transactions of intelligent agents, and the multi-agent system will perform the operational management independently.

Stochastic Processes Occurring during the Transition of Technical State of the Structure

The possibility of applying the theory of stochastic processes for evaluating the dynamic pattern of different states is studied for critical-duty structures. Heterogeneous Markovian processes of technical state transition for metallurgical overhead crane structure, Markovian theorem and Kolmogorov-Chapman equation are analyzed. Markovian chain is reviewed at t →∞, i.e. under marginal steady-state (stabilized) condition. Real values of limit probabilities are obtained for the structure of the metallurgical overhead crane under review. The proposed approach redefines and elaborates the existing methods and procedures for evaluating the technical state of structures and reduces the level of ambiguity associated with such kind of problems.

РАСЧЕТ СВОЕВРЕМЕННОСТИ ПЕРЕДАЧИ ЗАПРОСА В РЕЖИМЕ ДВОЙНОГО ИСПОЛЬЗОВАНИЯ КАНАЛА СИГНАЛИЗАЦИИ

Introduction: it is shown that when organizing the collection of short digital messages in a trunking radio communication network, service request communication channels can be used, transmitting packets-requests to establish a connection. With this approach, the question arises about the size of the bandwidth of the request communication channel, allocated for collecting information, at which the fulfillment of the requirements for establishing a connection will be preserved. Purpose: the purpose of the work is to develop a methodology for calculating the timeliness of transmission of requests for establishing a connection in the context of transmission of short digital messages This technique is the basis of the technique for determining the bandwidth of the request communication channel, allocated for solving the problem of collecting short digital messages. Results: A mathematical description of the revealed feature was carried out, which is the scientific novelty of the research results. An analytical rule for calculating the matrix of transition probabilities and the procedure for calculating the vector of initial probabilities of the request transmission process are presented, which allow, based on the Kolmogorov-Chapman equation, to mathematically describe the request transmission process based on the random multiple access algorithm and calculate the probabilistic-temporal characteristics of the request transmission. The method of calculating the probabilistic-temporal characteristics of the request transmission based on the compiled mathematical model of the request transmission and the mathematical model of the operation of the request channel is presented. The effect of taking into account the revealed features is shown, expressed in increasing the timeliness of the request transmission. Practical relevance: Formed a methodology, which is the basis of the methodology for determining the optimal values of the parameters of the double use of the request channel, which makes it possible to organize the transmission of deterministic traffic of short digital messages over the request channel with a given level of timeliness of transmission of requests to establish a connection. Discussion: the use of the developed methodology will allow organizing a periodic collection of short digital messages in the trunking radio access network without using traffic channels.

Estimation of Aggregate Losses of Secondary Cancer Using PH-OPPL and PH-TPPL Distributions

Kenyan insurance firms have introduced insurance policies of chronic illnesses like cancer; however, they have faced a huge challenge in the pricing of these policies as cancer can transit into different stages, which consequently leads to variation in the cost of treatment. This has made the estimation of aggregate losses of diseases which have multiple stages of transitions such as cancer, an area of interest of many insurance firms. Mixture phase type distributions can be used to solve this setback as they can in-cooperate the transition in the estimation of claim frequency while also in-cooperating the heterogeneity aspect of claim data. In this paper, we estimate the aggregate losses of secondary cancer cases in Kenya using mixture phase type Poisson Lindley distributions. Phase type (PH) distributions for one and two parameter Poisson Lindley are developed as well their compound distributions. The matrix parameters of the PH distributions are estimated using continuous Chapman Kolmogorov equations as the disease process of cancer is continuous while severity is modeled using Pareto, Generalized Pareto and Weibull distributions. This study shows that aggregate losses for Kenyan data are best estimated using PH-OPPL-Weibull model in the case of PH-OPPL distribution models and PH-TPPL-Generalized Pareto model in the case of PH-TPPL distribution models. Comparing the two best models, PH-OPPL-Weibull model provided the best fit for secondary cancer cases in Kenya. This model is also recommended for different diseases which are dynamic in nature like cancer.

ПРОЕКТИРОВАНИЕ ИНТЕЛЛЕКТУАЛЬНОЙ ИНФОРМАЦИОННОЙ СИСТЕМЫ ДЛЯ АНАЛИЗА И ПРОГНОЗИРОВАНИЯ ДИНАМИКИ РИСКОВ КОНКУРЕНТОСПОСОБНОСТИ РЕГИОНОВ РФ

It was revealed that there are no generally accepted methods for assessing competitiveness; modern research is mainly a theoretical analysis of regional competitiveness. It was substantiated that the use of mathematical models and quantitative methods will improve the objectivity of the assessment. A methodology for an information system design for analysis and prediction of the dynamics of risks of competitiveness of regions of the Russian Federation was proposed. The prospects and directions of intellectualization of that system were shown. Based on the analysis of officially published statistical data and information from the regions and the proposed methodology, methods of systematization of socio-economic indicators were designed. An information system module was developed to implement the model based on the Kolmogorov-Chapman equations for assessing the risks of regional competitiveness and its validation was carried out

Anomalous Diffusion in Systems with Concentration-Dependent Diffusivity: Exact Solutions and Particle Simulations

We explore the anomalous diffusion that may arise as a result of a concentration dependent diffusivity. The diffusivity is taken to be a power law in the concentration, and from exact analytical solutions we show that the diffusion may be anomalous, or not, depending on the nature of the initial condition. The diffusion exponent has the value of normal diffusion when the initial condition is a step profile, but takes on anomalous values when the initial condition is a spike. Depending on the sign of the exponent in the diffusivity the diffusive behavior will then be either sub-diffusive or super-diffusive. We introduce a particle model that behaves according to the non-linear diffusion equation in the macroscopic limit. This correspondence is demonstrated via kinetic theory, i.e. by means of Chapman-Kolmogorov equation, as well as by direct simulations.

Convolution algebra for extended Feller convolution

We apply the recently introduced framework of admissible homomorphisms in the form of a convolution algebra of $$\mathbb{C}^2$$ -valued admissible homomorphisms to handle two-dimensional uni-directional homogeneous stochastic kernels. The algebra product is a non-commutative extension of the Feller convolution needed for an adequate operator representation of such kernels: a pair of homogeneous transition functions uni-directionally intertwined by the extended Chapman–Kolmogorov equation is a convolution empathy; the associated Fokker–Planck equations are re-written as an implicit Cauchy equation expressed in terms of admissible homomorphisms. The conditions of solvability of such implicit evolution equations follow from the consideration of generators of a convolution empathy.

Taboo rate and hitting time distribution of continuous-time reversible Markov chains

The taboo rate is first defined, which satisfies with the Chapman–Kolmogorov equation. Then the differentials of hitting time distribution are expressed by many different taboo rates, which deeply reveal the intrinsic relationship between the transition rate matrix and the hitting time distribution in continuous-time reversible Markov chains. As an example, the explicit expressions of the differentials of the hitting time distribution at a single state are provided for the birth and death chain, hence the transition rate matrix can be identified. Such differentials improve the theory of statistical identification of continuous-time reversible Markov chains.

Modeling and availability analysis of data center: a fuzzy approach

Data Centers are the backbone of any industry that provides a specialized environment to safeguard the company's valuable equipment and intellectual property. The successful operation of company's Data Center is typically shared among multiple departments and personnel. But, the involvement of multiple personal and departments make its configuration complex. In such situations, reliability of data center become a necessary factor. But, the traditional theory of reliability is based on the Bernoulli trials, i.e., either operative or failure. But this situation is unrealistic in case of complex systems like data center. To rectify this problem, here a mathematical model has been developed using the concept of fuzzy reliability. All the failure and repair rates are exponential distributed along with coverage factor. Chapman-Kolmogorov differential equations have been developed for the fuzzy system using Markov birth–death process. A new methodology Runge–Kutta method of order four has been used to solve Chapman-Kolmogorov differential equations using MATLAB (Ode 45 function).

Deep Teams: Decentralized Decision Making With Finite and Infinite Number of Agents

Inspired by the concepts of deep learning in artificial intelligence and fairness in behavioural economics, we introduce deep teams in this paper. In such systems, agents are partitioned into a few sub-populations so that the dynamics and cost of agents in each sub-population is invariant to the indexing of agents. The goal of agents is to minimize a common cost function in such a manner that the agents in each sub-population are not discriminated or privileged by the way they are indexed. Two non-classical information structures are studied. In the first one, each agent observes its local state as well as the empirical distribution of the states of agents in each sub-population, called deep state, whereas in the second one, the deep states of a subset (possibly all) of sub-populations are not observed. Novel dynamic programs are developed to identify globally optimal and sub-optimal solutions for the first and second information structures, respectively. The computational complexity of finding the optimal solution in both space and time is polynomial (rather than exponential) with respect to the number of agents in each sub-population and is linear (rather than exponential) with respect to the control horizon. This complexity is further reduced in time by introducing a forward equation, that we call deep Chapman-Kolmogorov equation, described by multiple convolutional layers of Binomial probability distributions. Two different prices are defined for computation and communication, and it is shown that under mild assumptions they converge to zero as the quantization level and the number of agents tend to infinity. In addition, the main results are extended to the infinite-horizon discounted model and arbitrarily asymmetric cost function. Finally, a service-management example with 200 users is presented.

Path Integral Methods for the Probabilistic Analysis of Nonlinear Systems Under a White-Noise Process

Abstract In this paper, the widely known path integral method, derived from the application of the Chapman–Kolmogorov equation, is described in details and discussed with reference to the main results available in literature in several decades of contributions. The most simple application of the method is related to the solution of Fokker–Planck type equations. In this paper, the solution in the presence of normal, α-stable, and Poissonian white noises is first discussed. Then, application to barrier problems, such as first passage problems and vibroimpact problems is described. Further, the extension of the path integral method to problems involving multi-degrees-of-freedom systems is analyzed. Lastly, an alternative approach to the path integration method, that is the Wiener Path integration (WPI), also based on the Chapman–Komogorov equation, is discussed. The main advantages and the drawbacks in using these two methods are deeply analyzed and the main results available in literature are highlighted.

Guaranteed management of the prevention, detection, and proof of offenses in socio-technical construction using a geoinformation system

In the conditions of narrow departmental management, the administrative practice integrity in socio-technical construction is being destroyed by a lack of completeness and authenticity of administrative offense proceedings within a reasonable time. The timely application of the geoinformation system in order to prompt prevention, reliable revealing and proper proving of offenses ensures a preventively guaranteed efficiency and security of socio-technical construction. This paper proposed a synthesis of the administrative practice management model in construction, based on the natural-scientific approach to the adoption of managerial decisions. The condition for the existence of guaranteed management, which relates the characteristics of processes and performance indicators, is concretized by the Kolmogorov-Chapman equations system. Network models of interacting processes of the revealing and proving of offenses, the origin and identification of violations, their neutralization through the managerial impact on the limited resources of administrative practice in construction and of geoinformation system are built. Based on the simulation results, the requirements for the geoinformation system that ensure guaranteed management and maximum security to the socio-technical construction within a reasonable time are determined.

Reliability analysis of a multi-component machining system with service interruption, imperfect coverage, and reboot

In this study, the reliability and queueing characteristics of the fault-tolerant system (FTS) with unreliable server, vacation, and working breakdown are investigated. The performance metrics of FTS having L on-line and W warm standby machines are derived by incorporating the impact of recovery failure along with reboot process. If the failures/faults in the system are not covered successfully due to imperfect switching, the system reconfigures automatically by rebooting to clear the faults. It is assumed that when there is no failed machine in the system, the idle server can take a vacation and return as soon as a unit fails and requires repair. The failed units of FTS are repaired by a single failure-prone operator who can provide the repair with a slower rate in the partially failed state also. To develop Markov model, Chapman-Kolmogorov equations are framed for the system states, and transient probabilities are obtained by computing eigenvalues of transition matrix and using a spectral method. The queueing and reliability metrics are established to explore the redundancy and maintainability issues for the up-gradation of system design. By taking an illustration, numerical results for performance metrics and cost optimization are presented to examine the variation in system parameters.

A new approach for time-variant probability density function of the maximal value of stochastic dynamical systems

The extreme value distribution (EVD) of stochastic processes is an important but still challenging problem for the determination of reliability function and distribution of first excursion time in various science and engineering fields. In the present paper, a new method to evaluate the time-variant probability density function (PDF) of the maximal value of a Markov process or Markov vector process is proposed. In this method, a joint maximum-state vector process is constructed by combining the maximal value process (MVP) and its underlying Markov process. The Markov property of the joint maximum-state vector process is rigorously proved. Incorporated with the Itô stochastic differential equation (SDE) governing the underlying Markov process, a numerical method is developed based on the Chapman-Kolmogorov equation. In particular, the short-time transition probability density function of the joint maximum-state process in the path integral solution (PIS) is derived. The detailed algorithms for the proposed method in two- and three-dimensional cases, respectively, are elaborated. Several examples are illustrated, demonstrating the effectiveness of the proposed method. Problems to be further studied are also discussed.

Markovian Model of Unreliable Server Retrial Queue with Discouragement

In this paper, we present the performance analysis of Markovian retrial queue with unreliable server by incorporating the features of balking and reneging. The customers may abandon the queue at the moment of busy server or at the departure epoch from the orbit zone. When the customers notice the server in busy state, they will join to the orbit zone and wait there to avail the service. After a random period, the customers from the orbit either may try for the re-attempts for the service or may renege from the system without getting the service. Markov model is developed by considering the inter-arrival, retrial and service times governed by the exponential distributions. The probabilities of the system states are determined by implementing recursive approach to solve the governing Chapman–Kolmogorov equations. The cost function is constructed which is further used to find the total expected cost and associated optimal service rate of the system. A numerical illustration is provided to explore the system metrics with respect to various system descriptors.

Rainfall analysis in the Indian Ocean by using 6-States Markov Chain Model

In this study, we used Markov chain approach to analyze rainfall dataset from one buoy (4N90E) in the eastern Equatorial Indian Ocean. This study aims to determine the opportunity for transition (displacement) of daily rainfall intensity, where there are six states or conditions of rainfall intensity, i.e. no rain, very weak, weak rain, moderate rain, heavy rain, and very heavy rain. The Markov Chain method used is the Chapman-Kolmogorov equation and the steady state equation. The investigation of the 6-states in Markov chain model show that dynamic probability of transition state for rainfall data is reflexive properties majority. By using the model, it is concluded that the transition rate matrix of the largest transition probability in the area of 4N90E occurs at the transition from state-1 to state-1 as much is 0.72 and state-2 to state-2 is 0.60. The transition probability value becomes 0.5184 and 0.36 for the same state dominant of two periods P2. The use of the 1st order Markov chain is better than 2nd order.

Chains of three-dimensional evolution algebras: A description

In this paper we describe locally all the chains of three-dimensional evolution algebras (3-dimensional CEAs). These are families of evolution algebras with the property that their structure matrices with respect to a certain natural basis satisfy the Chapman-Kolmogorov equation. We do it by describing all 3-dimensional CEAs whose structure matrices have a fixed rank equal to 3, 2 and 1, respectively. We show that arbitrary CEAs are locally CEAs of fixed rank. Since every evolution algebra can be regarded as a weighted digraph, this allows us to understand and visualize time-dependent weighted digraphs with 3 nodes.

Optimal Control F-policy for M/M/R/K Queue with an Additional Server and Balking

The multi-server finite queueing model with balking under admission control of customers based upon F-policy is investigated. In order to reduce the balking behavior of the customers, there is the provision of one additional server so as to shorten the queue length formed by the customers in rush hour. The noble realistic feature of the proposed Markov model is to control the admission of customers based on the threshold level ‘F’ of the queue size. According to ‘F’ policy, no more customers can be accommodated once the system is full and then after the further admission of the customers is not acceptable until the queue size drops to the predefined threshold level ‘F’. Chapman–Kolmogorov equations have been framed and are solved by using the recursive technique to establish the steady-state queue size probability distribution of the customers present in the system. A numerical example is taken to explore the various system indices. Adaptive neuro-fuzzy inference system (ANFIS) approach is applied to compare the soft computing results obtained by ANFIS with the results computed by the analytical method. The cost function is framed to determine the optimal control parameters namely threshold value ‘F’, the capacity of the system, the number of servers and service rate.

Analyzing dynamic decision-making models using Chapman-Kolmogorov equations.

Decision-making in dynamic environments typically requires adaptive evidence accumulation that weights new evidence more heavily than old observations. Recent experimental studies of dynamic decision tasks require subjects to make decisions for which the correct choice switches stochastically throughout a single trial. In such cases, an ideal observer's belief is described by an evolution equation that is doubly stochastic, reflecting stochasticity in the both observations and environmental changes. In these contexts, we show that the probability density of the belief can be represented using differential Chapman-Kolmogorov equations, allowing efficient computation of ensemble statistics. This allows us to reliably compare normative models to near-normative approximations using, as model performance metrics, decision response accuracy and Kullback-Leibler divergence of the belief distributions. Such belief distributions could be obtained empirically from subjects by asking them to report their decision confidence. We also study how response accuracy is affected by additional internal noise, showing optimality requires longer integration timescales as more noise is added. Lastly, we demonstrate that our method can be applied to tasks in which evidence arrives in a discrete, pulsatile fashion, rather than continuously.