Stationary Probability Distribution Research Articles

Method of calculating stationary probability distribution in Markov chain in modeling social and economic processes

Method of calculating stationary probability distribution in Markov chain in modeling social and economic processes

Assessing extreme significant wave height in China’s coastal waters under climate change

Accurately estimating the return values of significant wave height is essential for marine and coastal infrastructure, particularly as climate change intensifies the frequency and intensity of extreme wave events. Traditional models, which assume stationarity in wave data, often underestimate future risks by neglecting the impacts of climate change on wave dynamics. Combining time series decomposition and recurrence analysis, the research develops a nonstationary framework to predict significant wave height. The stochastic component is modelled using a stationary probability distribution, while the deterministic component is predicted based on sea surface temperature projections from CMIP6 climate scenarios. The model evaluation demonstrates strong predictive capability for both stochastic and deterministic components. Application of the model to China’s coastal waters reveals significant discrepancies between stationary and nonstationary return value estimates. Compared to conventional distribution models, the nonstationary model predicts substantial increases in extreme wave heights. These findings underscore the importance of adopting nonstationary models to more accurately assess future risks posed by extreme wave events in a changing climate.

Multi-objective simulated annealing-based quantum circuit cutting for distributed quantum computation

In the current noisy intermediate-scale quantum (NISQ) era, the number of qubits and the depth of quantum circuits in a quantum computer are limited because of complex operation among increasing number of qubits, low-fidelity quantum gates under noise, and short coherence time of physical qubits. However, with distributed quantum computation (DQC) in which multiple small-scale quantum computers cooperate, large-scale quantum circuits can be implemented. In DQC, it is a key step to decompose large-scale quantum circuits into several small-scale subcircuits equivalently. In this paper, we propose a quantum circuit cutting scheme for the circuits consisting of only single-qubit gates and two-qubit gates. In the scheme, the number of non-local gates and the rounds of subcircuits operation are minimized by using the multi-objective simulated annealing (MOSA) algorithm to cluster the gates and to choose the cutting positions whilst using non-local gates. A reconstruction process is also proposed to calculate the probability distribution of output states of the original circuit. As an example, the 7-qubit circuit of Shor algorithm factoring 15 is used to verify the algorithm. Five cutting schemes are recommended, which can be selected according to practical requirements. Compared with the results of the mixing integer programming (MIP) algorithm, the number of execution rounds is efficiently reduced by slightly increasing the number of nonlocal gates.

Stochastic modeling of multiple-server charging stations for electric vehicle networks using feedback strategies: A queueing-theoretic approach

Nowadays, electric vehicles (EVs) significantly affect transportation as they provide a more environmentally friendly alternative to traditional fossil-fueled automobiles. Electric vehicles, which depend on energy stored in batteries, significantly contribute to environmental preservation and comply with worldwide efforts to tackle climate change. However, the growing demand for electric vehicles causes traditional power grids under pressure emphasizing the necessity of establishing a suitable infrastructure for charging electric vehicles. Charging stations are becoming increasingly critical since they allow for the recharging of electric vehicles and play a significant role in stabilizing the power system. In order to optimize charging station infrastructure with multiple servers, the current research incorporates a Markovian queueing modeling approach. The primary objective of the study is to address queue management concerns and boost overall productivity. Considering the real-world challenges, a queue-based stochastic model for multi-server EV systems and individual feedback strategies is developed. Subsequently, a transition state diagram is provided by balancing the input-output rates between the adjacent states. Next, the system of Chapman-Kolmogorov differential-difference equations is formulated to help understand mathematical modeling better. The matrix method is employed to demonstrate the state probability distribution in equilibrium. The infographics are utilized and incorporated for better visualization of the research findings. For a better understanding from an individual's point of view, numerous managerial insights are provided. Lastly, several concluding remarks and future perspectives are provided that can help decision-makers and practitioners to construct and analyze economic strategies based on EV management systems.

Stationary distribution and mean extinction time in a generalist prey–predator model driven by Lévy noises

In this study, a delayed generalist prey–predator model incorporating the logistic-like growth and Holling type-III functional response for predator subjected to two Lévy noises is established. Such environmental fluctuations can describe abrupt change in population density. First, we conduct stability analysis and determine the critical value of time delay when Hopf bifurcation takes place. Then, the existence of the global positive solution and stability in distribution of stochastic model are studied. Numerical results are given to illustrate the analytical findings, and mainly dedicated to the consequences of time delay and Lévy noises on the stationary distribution, mean extinction time and reproduction rate of prey–predator populations, via stationary probability distribution function (PDF), mean first passage time (MFPT) and relaxation time, respectively. Our findings indicate that the decreases in time delay, noise intensities (Di,i=1,2), and stability indexes within the range of αi∈[1,2] are more likely to maintain the population in a high density state. Phase transition occurs when the system is in the suitable parameter regime. Additionally, there exist optimal noise intensities that slow down switching of prey species from a stable state to an extinction state. Smaller time delay τ and stability indexes αi inhibit population extinction, whereas smaller skewness parameter β speed up prey extinction. More importantly, two noises can tune the enhanced stability characteristic of the system.

Cauchy universality and random billiards

Motion in bounded domains represents a paradigm in several settings: from billiard dynamics, to random walks in a finite lattice, with applications to relevant physical, ecological, and biological problems. A remarkable universal property, involving the average of return times to the boundary, has been theoretically proposed and experimentally verified in quite different contexts. We discuss here mechanisms that lead to violations of universality, induced by boundary effects and we also emphasize the role played by replacing straight lines with random walks in this framework. We suggest that our analysis should be relevant where nonhomogeneity appears in the stationary probability distribution in bounded domain. Published by the American Physical Society 2024

Growth, poverty trap and escape

The well-known Solow growth model is the workhorse model of the theory of economic growth, which studies capital accumulation in a model economy as a function of time with capital stock, labour and technology-based production as the basic ingredients. The capital is assumed to be in the form of manufacturing equipment and materials. Two important parameters of the model are: the saving fraction s of the output of a production function and the technology efficiency parameter A , appearing in the production function. The saved fraction of the output is fully invested in the generation of new capital and the rest is consumed. The capital stock also depreciates as a function of time due to the wearing out of old capital and the increase in the size of the labour population. We propose a stochastic Solow growth model assuming the saving fraction to be a sigmoidal function of the per capita capital kp . We derive analytically the steady state probability distribution P(kp) and demonstrate the existence of a poverty trap, of central concern in development economics. In a parameter regime, P(kp) is bimodal with the twin peaks corresponding to states of poverty and well-being, respectively. The associated potential landscape has two valleys with fluctuation-driven transitions between them. The mean exit times from the valleys are computed and one finds that the escape from a poverty trap is more favourable at higher values of A. We identify a critical value of Ac below (above) which the state of poverty (well-being) dominates and propose two early signatures of the regime shift occurring at Ac . The economic model, with conceptual foundations in nonlinear dynamics and statistical mechanics, shares universal features with dynamical models from diverse disciplines like ecology and cell biology.

A parallel algorithm for reliability assessment of multi-state flow networks based on simultaneous finding of all multi-state minimal paths and performing state space decomposition

This article presents a new efficient method for reliability evaluation of two-terminal Multi-state Flow Networks (MFNs). Indirect State Space Decomposition (SSD) methods are usually more efficient than direct SSD methods. However, indirect SSD methods require searching for all d-MPs/d-MCs. The presented method requires only an MFN, a system demand requirement d, and the probability distributions of states of components as an input. The presented algorithm uses a partition technique to decompose a state space into smaller and distinct subspaces. The algorithm tries to determine a d-flow with a lower bound for each subspace. If a flow exists and it is a d-MP, the algorithm computes the probability that a capacity vector belongs to this subspace and is bounded below by the d-MP and decomposes this subspace into smaller subspaces. For all subspaces without d-MP, the presented algorithm uses parallel computations and the SSD method to compute the probability that a capacity vector belongs to this subspace and is bounded below by a d-MP. The numerical experiments show that the efficiency of the presented method improves in comparison with other indirect SSD methods if the number of available logical processes and the number of d-MPs increase.

Selection of 110 kV Prefabricated Steel Substations Considering Seismic Vulnerability in China

Prefabricated modular substations are expected to become the mainstream construction type for substations in China. However, there is a lack of scientific basis for structural selection and seismic performance evaluation. Taking a 110 kV substation as an example, this study compares the construction cost of cast-in situ reinforced concrete (RC) substations and prefabricated steel (PS) ones. Two types of PS structures are considered: one with H-section steel columns and the other with box-section steel columns. A seismic vulnerability analysis is performed to compare the probability distribution of various damage states of substation building structures under different seismic damage levels. Results indicate that the construction cost of PS structures is approximately 27.9% higher than that of cast-in situ concrete. When using H-section steel columns, there is a significant difference in the flexural stiffness in two horizontal directions, resulting in reduced seismic performance in the weak-axis direction. The construction cost of using box-section steel columns is slightly higher than that of the H-section steel case, but its seismic performance is significantly improved. Although the probability of slight and moderate damage states for the box-section steel column scheme is generally higher than that of the cast-in situ RC scheme, the probability of collapse is reduced. Thus, box-section steel columns are recommended for prefabricated modular substation building structures.

Quantitative analysis of aeolian dust incidence and diagnosis using a dynamic Bayesian network model: A case study of estuary area in central Taiwan

The presence of high concentrations of PM10 in aeolian dust within the estuary area of central Taiwan has been a focal point for air quality protection and remediation efforts over the past 15 years. This study explores the quantitative analysis of aeolian dust occurrences concerning various levels of PM10 concentration, aiming to offer early warning information to local communities and government agencies. Employing Dynamic Bayesian network (DBN) models, the research interrelates conditional probability dependencies among air quality and meteorological factors monitored in the downstream area of the Jhuoshuei River, predicting the probability distribution of aeolian dust events. The constructed DBNs demonstrate the inference of probability distributions of meteorological factors' states based on PM10 concentrations, enabling diagnostic analysis of meteorological conditions during high PM10 concentration dust events. The DBN models exhibit robust predictive ability, achieving an overall accuracy of 85.72% in validation and 85.8% in testing, with high discriminatory capacity that the area under the receiver operating characteristic curve (AUC) values of 0.924 and 0.915 during validation and testing, respectively. Sensitivity analysis highlights wind direction, wind speed, relative humidity as significant factors affecting PM10 concentrations. The probability of low PM10 concentrations was notably higher compared to higher concentrations, indicating effective aeolian dust remediation efforts. However, attention is warranted toward the 10% incidence of aeolian dust events deemed unhealthy or hazardous to human health. Diagnosis of meteorological conditions during high PM10 concentration events suggests low temperature, high wind speeds, and dry conditions, particularly with northeastern or southwestern winds, as conducive factors. This study underscores the physical relationship between meteorological factors and PM10 concentrations, employing DBNs to consider temporal relationships and uncertainties in data, aligning with human health impact standards proposed by the Ministry of Environment. Future research should incorporate soil and land surface conditions to provide a more comprehensive understanding of aeolian dust dynamics and mitigation strategies.

Topological Phase and Tunable Quantum State Transfer of Su–Schrieffer–Heeger Chain with an Embedded Quantum Ring

AbstractAn extended Su–Schrieffer–Heeger (SSH) model consisting of an SSH chain with an embedded quantum ring (QR) is investigated. In the case of two SSH chains with a symmetric distribution with respect to QR, by calculating the real‐space winding number and energy spectrum, the hopping amplitude‐induced topological phase is discovered. The probability distributions of gap states and the relationship between energy and magnetic flux prove the existence of the Aharonov–Bohm effect in the QR. Moreover, in the case of asymmetric distribution, the model possesses a zero‐energy mode within the energy gap, and it is found that it can realize quantum state transfer. By adjusting the connection site between the right SSH chain and QR, the direction of the output port can be flexibly engineered. Furthermore, it is shown that high‐fidelity quantum state transfer can still be achieved with the increasing of the system size. The tunable quantum state transfer based on the zero‐energy mode can be equivalent to a topological tunable directional switch with properties of non‐directional transfer. This work provides an approach for studying topological phase transition and tunable topological devices in an SSH chain with an embedded QR.

Meta-state–space learning: An identification approach for stochastic dynamical systems

Available methods for identification of stochastic dynamical systems from input–output data generally impose restricting structural assumptions on either the noise structure in the data-generating system or the possible state probability distributions. In this paper, we introduce a novel identification method of such systems, which results in a dynamical model that is able to produce the time-varying output distribution accurately without taking restrictive assumptions on the data-generating process. The method is formulated by first deriving a novel and exact representation of a wide class of nonlinear stochastic systems in a so-called meta-state–space form, where the meta-state can be interpreted as a parameter vector of a state probability function space parameterization. As the resulting representation of the meta-state dynamics is deterministic, we can capture the stochastic system based on a deterministic model, which is highly attractive for identification. The meta-state–space representation often involves unknown and heavily nonlinear functions, hence, we propose an artificial neural network (ANN)-based identification method capable of efficiently learning nonlinear meta-state–space models. We demonstrate that the proposed identification method can obtain models with a log-likelihood close to the theoretical limit even for highly nonlinear, highly stochastic systems.

Group consensus of nonlinear semi-Markov multi-agent systems by a quantity limited controller

This paper solves the group consensus problem of incremental quadratic constraints nonlinear multi-agent systems with semi-Markov switching parameters via a quantity limited controller. The quantity limited controller is designed according to the stationary probability distribution of the system modes, and the number of controllers can be selected from 1 to the number of system modes. Therefore, the controller can be applied in cases where there is a mismatch between the number of system modes and the number of controllers. Next, the more general incremental quadratic constraint is used to handle the nonlinear terms. Finally, in the simulation section, the method of adding topology links is used to implement external equitable partitions of arbitrary topologies, and the validity of the proposed method is verified by a spring damping system.

Cost optimization for system stability and orbital search under working vacation and starting failure by using the ANFIS soft computing

This article deals with the concepts of a single server retrial queueing model with balking, orbital search under working vacation and starting failure. If the server is busy, the prospective consumer may either enter the pool or leave the system; if the server is idle, the arriving client begins providing its services instantaneously. We assume the server is unstable since it did not start during the usual busy period. Instead of being completely unavailable at some point during the working vacation, the server will operate at a slow rate. Following the completion of the vacation period, the server either searches the orbit for clients or returns to idle mode, during which it resumes normal operations and continues to serve customers. The essential and adequate conditions for system stability have been discovered. The supplementary variable approach has been utilised to generate the stationary probability distribution and a few performances metrices. To justify the suggested paradigm, several numerical examples and special cases are presented. We have used the ANFIS soft computing strategy to assess the outcomes of the analytical and neural fuzzy methods.

Multiserver Retrial Queue with Two-Way Communication and Synchronous Working Vacation

This work investigates a two-way communication retrial queue with synchronous working vacation and a constant retrial policy. During the idle time, a server makes an outgoing call after a random length. The service time of the incoming call and outgoing call obeys exponential distribution with different rates. If the incoming call finds all servers to be unavailable, it may or may not enter orbit. All servers immediately go on vacation simultaneously as soon as they find an empty system after the service finishes. During vacation, the servers can provide a service to those incoming calls, but this is at a lower-speed rate. The stationary probability distribution and the ergodic condition are obtained utilizing the matrix geometric technique. Some system characteristics are developed. Using MATLAB software, the variation in average orbit length, idle ratio, and the average number of servers in different server states is plotted for different values of the incoming/outgoing call rate and retrial rate. We further propose a multi-objective optimization model from which the optimal rate of outgoing calls and optimal vacation rate are explicitly obtained.

A novel resampling-free update framework-based cubature Kalman filter for robust estimation

The resampling-free update (RFU) framework avoids discarding the higher-order moment information of the state probability distribution in Gaussian approximation filters. Still, it suffers from the problem of numerical instability and estimation optimality being corrupted caused by non-closed mapping without Gaussian reconstruction. This study proposes a novel robust RFU framework-based cubature Kalman filter. The maximum correntropy criterion is adopted as the optimization cost to exploit the non-Gaussian moments caused by non-closed mapping in RFU. An RFU update is reconstructed based on the square root of a posterior error matrix to improve the numerical stability. In addition, the periodic resampling operation is implemented to mitigate the non-Gaussianity while keeping higher-order moments. The illustrative example demonstrates that the proposed method can improve the estimation robustness and consistency of the RFU framework compared to other state-of-the-art RFU-based filters.

Performance analysis and ANFIS computing of a Markovian queuing model with intermittently accessible server under a hybrid vacation policy

In this study, we investigate a heterogeneous queueing model with intermittent server availability, server catastrophes, and a hybrid vacation policy. Our focus is on a specific scenario: server 1 is always available, while server 2 may experience breakdowns or vacations, making it intermittently accessible. Using the matrix-geometric approach (MGA), we derive matrix-based expressions for the stationary probability distribution of the number of customers in the system and various system performance measures. Additionally, we evaluate the cost function per unit of time to determine optimal values for the system’s decision variables. Furthermore, we employ an adaptive neural fuzzy inference system (ANFIS) based on soft computing technology to compare and analyze the numerical results obtained. Through this comprehensive analysis, our study contributes to the understanding and optimization of this complex queueing system, attracting the attention of researchers in the field and offering practical insights for real-world applications.

Fault Diagnostics Based on the Analysis of Probability Distributions Estimated Using a Particle Filter

This paper proposes a monitoring procedure based on characterizing state probability distributions estimated using particle filters. The work highlights what types of information can be obtained during state estimation and how the revealed information helps to solve fault diagnosis tasks. If a failure is present in the system, the output predicted by the model is inconsistent with the actual output, which affects the operation of the estimator. The heterogeneity of the probability distribution of states increases, and a large proportion of the particles lose their information content. The correlation structure of the posterior probability density can also be altered by failures. The proposed method uses various indicators that characterize the heterogeneity and correlation structure of the state distribution, as well as the consistency between model predictions and observed behavior, to identify the effects of failures.The applicability of the utilized measures is demonstrated through a dynamic vehicle model, where actuator and sensor failure scenarios are investigated.

Model order reduction and stochastic averaging for the analysis and design of micro-electro-mechanical systems

Electro-mechanical systems are key elements in engineering. They are designed to convert electrical signals and power into mechanical motion and vice-versa. As the number of networked systems grows, the corresponding mathematical models become more and more complex, and novel sophisticated techniques for their analysis and design are required. We present a novel methodology for the analysis and design of electro-mechanical systems subject to random external inputs. The method is based on the joint application of a model order reduction technique, by which the original electro-mechanical variables are projected onto a lower dimensional space, and of a stochastic averaging technique, which allows the determination of the stationary probability distribution of the system mechanical energy. The probability distribution can be exploited to assess the system performance and for system optimization and design. As examples of application, we apply the method to power factor correction for the optimization of a vibration energy harvester, and to analyse a system composed by two coupled electro-mechanical resonators for sensing applications.

A Kinetic Finite Volume Discretization of the Multidimensional PIDE Model for Gene Regulatory Networks

In this paper, a finite volume discretization scheme for partial integro-differential equations (PIDEs) describing the temporal evolution of protein distribution in gene regulatory networks is proposed. It is shown that the obtained set of ODEs can be formally represented as a compartmental kinetic system with a strongly connected reaction graph. This allows the application of the theory of nonnegative and compartmental systems for the qualitative analysis of the approximating dynamics. In this framework, it is straightforward to show the existence, uniqueness and stability of equilibria. Moreover, the computation of the stationary probability distribution can be traced back to the solution of linear equations. The discretization scheme is presented for one and multiple dimensional models separately. Illustrative computational examples show the precision of the approach, and good agreement with previous results in the literature.