Discrete Markov Research Articles

Approximate dynamic programming approach to efficient metro train timetabling and passenger flow control strategy with stop-skipping

Urban metro systems continuously face high travel demand during rush hours, which brings excessive energy waste and high risk to passengers. In order to alleviate passenger congestion, improve train service levels and reduce energy consumption, a nonlinear dynamic programming (DP) model of efficient metro train timetabling and passenger flow control strategy with stop-skipping is presented, which consists of state transition equations concerning train traffic and passenger load. To overcome the curse of dimensionality, the formulated nonlinear DP problem is transformed into a discrete Markov decision process, and a novel approximate dynamic programming (ADP) approach is designed based on the lookahead policy and linear parametric value function approximation. Finally, the effectiveness of this method is verified by three groups of numerical experiments. Compared with Particle Swarm Optimization (PSO) and Simulated Annealing (SA), the designed ADP approach could obtain high-quality solutions quickly, which makes it applicable to the practical implementation of metro operations.

Markov decision processes approximation with coupled dynamics via Markov deterministic control systems

Abstract This article presents an approximation of discrete Markov decision processes with small noise on Borel spaces with an infinite horizon and an expected total discounted cost by the corresponding deterministic Markov process. In both cases, the dynamics evolve through a system consisting of two coupled difference equations. It is assumed that the difference equations of the system are perturbed by a small noise. Under our assumptions, a bound for the stability index is given, and the optimal cost convergence rate is estimated using a small perturbation parameter. Moreover, the convergence of the optimal policy on compact subsets is verified. Finally, two examples are presented to illustrate the developed theory.

Digital core reconstruction based on discrete element and Markov chain-Monte Carlo methods

The digital core is an essential platform for pore-scale flow simulation in geoenergy exploitation, CO2 storage, and hydrogen storage. However, the existing reconstruction methods, such as the Markov chain-Monte Carlo (MCMC) method, can not obtain the digital core reflecting the influence of complex stress. Moreover, the discrete element method (DEM)-based reconstruction, which can consider the effect of stress, has yet to be reported to consider the actual particle shape. To this end, a hybrid reconstruction method based on DEM and MCMC methods is proposed in this study. Firstly, watershed and level set algorithms are used to segment and extract the contours of rock particles, and clump in PFC2D is used to fill the contours to establish a clump template library. Then, the initial condition (ground condition) digital core slice is established with clump packing based on porosity and particle size distribution. Subsequently, the stress loading simulation is carried out to obtain 2D digital core slices with different stress combinations. Finally, the MCMC method obtains 3D digital cores. This proposed method is used to reconstruct the digital core of Belgian fieldstone with different stresses and effective moduli, and the evolution of pore structure characteristics is analyzed. Results show that the stress compresses the range of the pore radius distribution curve, resulting in the shortening of the throat length distribution; the conductivity of the model is poor. The maximum reduction of porosity and permeability is 6.56% and 24.81%, respectively. The model with a large effective modulus has a stronger resistance to deformation and can protect the pore structure. These results prove that the method proposed can obtain the digital core reflecting the stress effects and provide a reference for accurately describing the in-situ rocks' pore structure and the pore-scale flow simulation.

Integrated robust navigation and guidance for the kinetic impact of near-earth asteroids based on deep reinforcement learning

The defense against near-Earth asteroids (NEAs) using kinetic impact is faced with various challenges, including limited maneuverability of the impactor, inaccurate dynamic models, poor observability of relative navigation, and control execution errors. To address these challenges, an integrated robust navigation and guidance method for the kinetic impact of NEAs based on deep reinforcement learning (DRL) is proposed in this paper, which can directly map angle measurements by a monocular camera to guidance maneuvers. Firstly, a discrete partial observable Markov decision process (POMDP) is modeled for the integrated navigation and guidance problem of NEA interception. To address the situation where impactors are usually only equipped with aiming cameras and only the line of sight (LOS) angle is measured, past and current LOS measurements are concatenated into a one-dimensional state observation vector to directly introduce the memory of historical state information. A shaping reward function design based on potential energy has also been proposed to solve the problem of sparse reward as the main goal. Subsequently, proximal policy optimization (PPO) is used to solve the established POMDP model, obtaining an integrated navigation and guidance policy that directly maps from the original output of the navigation sensor to the guidance command. The potentially hazard asteroid (PHA) Bennu is used as the target and a typical kinetic impact defense scenario is designed to simulate and verify the model and method proposed in this paper, considering the influence of multiple factors. Numerical simulation results show that the proposed method achieves an interception accuracy of 203.58 m (mean value). The proposed method abandons the traditional separate design of navigation and guidance algorithms, and its robustness has been tested and verified in a wide range of uncertain environments.

The entropy production of stationary diffusions

The entropy production rate is a central quantity in non-equilibrium statistical physics, scoring how far a stochastic process is from being time-reversible. In this paper, we compute the entropy production of diffusion processes at non-equilibrium steady-state under the condition that the time-reversal of the diffusion remains a diffusion. We start by characterising the entropy production of both discrete and continuous-time Markov processes. We investigate the time-reversal of time-homogeneous stationary diffusions and recall the most general conditions for the reversibility of the diffusion property, which includes hypoelliptic and degenerate diffusions, and locally Lipschitz vector fields. We decompose the drift into its time-reversible and irreversible parts, or equivalently, the generator into symmetric and antisymmetric operators. We show the equivalence with a decomposition of the backward Kolmogorov equation considered in hypocoercivity theory, and a decomposition of the Fokker-Planck equation in GENERIC form. The main result shows that when the time-irreversible part of the drift is in the range of the volatility matrix (almost everywhere) the forward and time-reversed path space measures of the process are mutually equivalent, and evaluates the entropy production. When this does not hold, the measures are mutually singular and the entropy production is infinite. We verify these results using exact numerical simulations of linear diffusions. We illustrate the discrepancy between the entropy production of non-linear diffusions and their numerical simulations in several examples and illustrate how the entropy production can be used for accurate numerical simulation. Finally, we discuss the relationship between time-irreversibility and sampling efficiency, and how we can modify the definition of entropy production to score how far a process is from being generalised reversible.

Structure recovery for partially observed discrete Markov random fields on graphs under not necessarily positive distributions

AbstractWe propose a penalized conditional likelihood criterion to estimate the basic neighborhood of each node in a discrete Markov random field that can be partially observed. We prove the convergence of the estimator in the case of a finite or countable infinite set of nodes. The estimated neighborhoods can be combined to estimate the underlying graph. In the finite case, the graph can be recovered with probability one. In contrast, we can recover any finite subgraph with probability one in the countable infinite case by allowing the candidate neighborhoods to grow as a function , with the sample size. Our method requires minimal assumptions on the probability distribution, and contrary to other approaches in the literature, the usual positivity condition is not needed. We evaluate the estimator's performance on simulated data and apply the methodology to a real dataset of stock index markets in different countries.

Bye bye, linearity, bye: quantification of the mean for linear CRNs in a random environment

Molecular reactions within a cell are inherently stochastic, and cells often differ in morphological properties or interact with a heterogeneous environment. Consequently, cell populations exhibit heterogeneity both due to these intrinsic and extrinsic causes. Although state-of-the-art studies that focus on dissecting this heterogeneity use single-cell measurements, the bulk data that shows only the mean expression levels is still in routine use. The fingerprint of the heterogeneity is present also in bulk data, despite being hidden from direct measurement. In particular, this heterogeneity can affect the mean expression levels via bimolecular interactions with low-abundant environment species. We make this statement rigorous for the class of linear reaction systems that are embedded in a discrete state Markov environment. The analytic expression that we provide for the stationary mean depends on the reaction rate constants of the linear subsystem, as well as the generator and stationary distribution of the Markov environment. We demonstrate the effect of the environment on the stationary mean. Namely, we show how the heterogeneous case deviates from the quasi-steady state (Q.SS) case when the embedded system is fast compared to the environment.

Optimal EV Fast Charging Station Deployment Based on a Reinforcement Learning Framework

This study aims to determine the optimal deployment plan for EV fast charging stations in a transportation network with a limited budget. The objective of the deployment problem is to maximize the quality of service (QoS) with respect to both waiting time and range anxiety from the perspective of EV customers. With the rapid growth of the electric vehicle (EV) market penetration, state-of-the-art algorithms based on mathematical programming are limited in handling high-dimensional optimization problems adequately. Unlike previous studies, we make the first attempt to formulate the fast charging station deployment problem (FCSDP) as a finite discrete Markov decision process (MDP) in a novel reinforcement learning (RL) framework to alleviate the curse of dimensionality problem. Since creating a supervised training dataset is impractical due to the high computational complexity of the FCSDP, we propose a recurrent neural network (RNN) with an attention mechanism to learn the model parameters and determine the optimal policy in a completely unsupervised manner. Finally, numerical experiments are conducted on multiple problem sizes to evaluate the performance of the RNN-based RL framework. Simulation results show that the proposed approach outperforms the comparing algorithms in terms of solution quality and computation time.

Fetal states identification in cardiotocographic tracings through discrete emissions multivariate hidden Markov models

Background and objectivesComputerized Cardiotocography (cCTG) allows to analyze the Fetal Heart Rate (FHR) objectively and thoroughly, providing valuable insights on fetal condition. A challenging but crucial task in this context is the automatic identification of fetal activity and quiet periods within the tracings. Different neural mechanisms are involved in the regulation of the fetal heart, depending on the behavioral states. Thereby, their correct identification has the potential to increase the interpretability and diagnostic capabilities of FHR quantitative analysis. Moreover, the most common pathologies in pregnancy have been associated with variations in the alternation between quiet and activity states. MethodsWe address the problem of fetal states clustering by means of an unsupervised approach, resorting to the use of a multivariate Hidden Markov Models (HMM) with discrete emissions. A fixed length sliding window is shifted on the CTG traces and a small set of features is extracted at each slide. After an encoding procedure, these features become the emissions of a multivariate HMM in which quiet and activity are the hidden states. After an unsupervised training procedure, the model is used to automatically segment signals. ResultsThe achieved results indicate that our developed model exhibits a high degree of reliability in identifying quiet and activity states within FHR signals. A set of 35 CTG signals belonging to different pregnancies were independently annotated by an expert gynecologist and segmented using the proposed HMM. To avoid any bias, the physician was blinded to the results provided by the algorithm. The overall agreement between the HMM's predictions and the clinician's interpretations was 90%. ConclusionsThe proposed method reliably identified fetal behavioral states, the alternance of which is an important factor in the fetal development. One key strength of our approach lies in the ease of interpreting the obtained results. By utilizing a small set of parameters that are already used in cCTG and possess clear intrinsic meanings, our method provides a high level of explainability. Another significant advantage of our approach is its fully unsupervised learning process. The states identified by our model using the Baum-Welch algorithm are associated with the “Active” and “Quiet” states only after the clustering process, removing the reliance on expert annotations. By autonomously identifying the clusters based solely on the intrinsic characteristics of the signal, our method achieves a more objective evaluation that overcomes the limitations of subjective interpretations. Indeed, we believe it could be integrated in cCTG systems to obtain a more complete signal analysis.

Provably Efficient Causal Model-Based Reinforcement Learning for Systematic Generalization

In the sequential decision making setting, an agent aims to achieve systematic generalization over a large, possibly infinite, set of environments. Such environments are modeled as discrete Markov decision processes with both states and actions represented through a feature vector. The underlying structure of the environments allows the transition dynamics to be factored into two components: one that is environment-specific and another that is shared. Consider a set of environments that share the laws of motion as an example. In this setting, the agent can take a finite amount of reward-free interactions from a subset of these environments. The agent then must be able to approximately solve any planning task defined over any environment in the original set, relying on the above interactions only. Can we design a provably efficient algorithm that achieves this ambitious goal of systematic generalization? In this paper, we give a partially positive answer to this question. First, we provide a tractable formulation of systematic generalization by employing a causal viewpoint. Then, under specific structural assumptions, we provide a simple learning algorithm that guarantees any desired planning error up to an unavoidable sub-optimality term, while showcasing a polynomial sample complexity.

A discrete state Markov chain model and decomposition solution method for the transient behaviour of buffered long serial transfer lines

As demand for greater manufacturing efficiency over shorter production runs in serial manufacturing lines increase, models for transient behaviour are required for insight into short-term production operation and control. This paper proposes a method to model the discrete time transient behaviour of paced long serial transfer lines with finite buffers. The method provides for the solution of lines comprised of machines that represent a variety of complex discrete state behaviours. The solution decouples the line into two-machine, one buffer models and adds external starvation and blockage coupling states and harmonizes the system behaviour by solving for transition probabilities into these external states with optimization. The efficacy of the approach is illustrated with three examples, a twenty-machine serial line with geometric machine behaviour, a twenty-machine serial line with complex machine behaviour, and the design of a transient line for a CONWIP strategy.

Analysing ill-conditioned Markov chains.

Discrete state Markov chains in discrete or continuous time are widely used to model phenomena in the social, physical and life sciences. In many cases, the model can feature a large state space, with extreme differences between the fastest and slowest transition timescales. Analysis of such ill-conditioned models is often intractable with finite precision linear algebra techniques. In this contribution, we propose a solution to this problem, namely partial graph transformation, to iteratively eliminate and renormalize states, producing a low-rank Markov chain from an ill-conditioned initial model. We show that the error induced by this procedure can be minimized by retaining both the renormalized nodes that represent metastable superbasins, and those through which reactive pathways concentrate, i.e. the dividing surface in the discrete state space. This procedure typically returns a much lower rank model, where trajectories can be efficiently generated with kinetic path sampling. We apply this approach to an ill-conditioned Markov chain for a model multi-community system, measuring the accuracy by direct comparison with trajectories and transition statistics. This article is part of a discussion meeting issue 'Supercomputing simulations of advanced materials'.

Mathematical Analysis of the Effectiveness of Screening for Intracranial Aneurysms in First-Degree Relatives of Persons with Subarachnoid Hemorrhage

Population screening for aneurysms in patients with risk factors and preventive surgical treatment are beneficial according to numerous studies. One of the most significant risk factors is heredity, namely, the presence of first-degree relatives (FDR) with aneurysmal subarachnoid hemorrhage (aSAH). Nevertheless, there are still no generally accepted approaches or evidence bases regarding the benefits of the aneurysm screening strategy. Mathematical modeling of the dynamics of aneurysm development in the population was carried out using an algorithm implementing a discrete Markov's chain. To implement the model, all probabilities of events and distributions are taken from available literature sources. Three-dimensional time of flight noncontrast magnetic resonance angiography was chosen as a screening method. Patients underwent preventive surgical treatment if an aneurysm was detected. Screening and preventive treatment in the general population reduces the prevalence of aneurysms by 1.74% (3.44% in the FDR group) and the prevalence of aSAH by 14.36% (37.48% in the FDR group). Mortality due to aSAH was reduced by 14.44%. The number of disabilities also decreases. The occurrence of deep disability was reduced by 20.2% in the FDR group. Economic analysis of the part of the population consisting of FDRs showed annual savings of ies also decr CONCLUSIONS: The mathematical model demonstrated that screening and preventive treatment of cerebral aneurysms can reduce aSAH-associated morbidity and mortality. In the FDR group, there was decrease in the prevalence of aSAH and decrease in associated mortality. Screening for cerebral aneurysms is cost-effective.

Transition Path Theory for Langevin Dynamics on Manifolds: Optimal Control and Data-Driven Solver

We present a data-driven point of view for rare events, which represent conformational transitions in biochemical reactions modeled by overdamped Langevin dynamics on manifolds in high dimensions. We first reinterpret the transition state theory and the transition path theory from the optimal control viewpoint. Given a point cloud probing the manifold, we construct a discrete Markov chain with a -matrix computed from an approximated Voronoi tesselation via the point cloud. We use this -matrix to compute a discrete committor function whose level set automatically orders the point cloud. Then based on the committor function, an optimally controlled random walk on point clouds is constructed and utilized to efficiently sample transition paths, which become an almost sure event in time instead of a rare event in the original reaction dynamics. To compute the mean transition path efficiently, a local averaging algorithm based on the optimally controlled random walk is developed, which adapts the finite temperature string method to the controlled Monte Carlo samples. Numerical examples on sphere/torus including a conformational transition for the alanine dipeptide in vacuum are conducted to illustrate the data-driven solver for the transition path theory on point clouds. The mean transition path obtained via the controlled Monte Carlo simulations highly coincides with the computed dominant transition path in the transition path theory.

THEORETICAL FOUNDATIONS OF FORECASTING THE RESOURCE OF HYDRAULIC STRUCTURES

It is established that in terms of the discrete Markov process, the problem is reduced to the search for unconditional probabilities of the system S at an arbitrary step k in state Si, that is, obtaining a transition probabilities matrix. In this formulation, the model is used for assessing the technical condition of the element; assessing the level of safety of operation of structural elements; ranking elements according to the need for repairs, reconstruction or replacement; in strategic planning of repair or reconstruction costs in conditions of limited funding and forecasting the remaining resource of elements. It is established that the theoretical basis of the study, which aims to predict the resource of hydraulic structures in operation, is the Markov theory of random processes. For a mathematical description of the process of element degradation, the most successful is the mathematical apparatus of the Markov random processes. Determination of the failure intensity parameter is the dominant feature of the Markov phenomenological model of damage accumulation to hydraulic structures' elements. The only parameter of lifecycle management is the failure rate . In the model under consideration, the parameter is determined based on the initial conditions for an individual element obtained from the survey results. Because the parameter λ is determined for an individual element and must be specified each time after the next survey, the accuracy of the model will increase. The proposed model is integral. It does not contain an explicit theoretical apparatus for a material-sensitive element, its static scheme, construction technology, environmental conditions, etc. On the other hand, all these factors and many other secondary ones are taken into account in the model at the moment the state of the element is determined using classification tables containing physical and mechanical signs of degradation. In the theory of structures, the statistical approach to formulating the transition matrix is widespread and is based on historical data from the structure operation system. It is believed that the transition matrix based on the data of the operating system is a more realistic basis for predicting the processes of structures degradation. A large number of foreign studies are devoted to the practical application of the transition matrix based on statistical data, which consider the features of transition matrices related to the bridge operation system in different countries. In this formulation, each element of the transition probability matrix P is the probability that the system in the state will move to state j in one step (i.e., in one year). At the same time, it is considered that there are no operational interventions, so the sub-diagonal elements are zero. As before, the sum of elements of the same line is 1 and the element pjj = 1 because state j is absorbing. For the implementation algorithm of the Markov chain model for forecasting the technical condition of hydraulic structures in general, the initial data are: statistical data of the distribution of structures by the state at the time of the forecast, the rating assessment of the structure is calculated by an expert according to the scale and the forecast time in years. It is established that the degradation properties of structural designs are described by two parameters: the degradation criterion and the failure rate. Any factor of the stress-strain state can be taken as a degradation criterion: reliability, internal forces, or deformations. The degradation criterion can be an arbitrary rating assessment. In our case, the reliability of the element is taken as the degradation criterion, as the most general factor of the stress-strain state.

Random embedded calibrated statistical blind steganalysis using cross validated support vector machine and support vector machine with particle swarm optimization

The evolvement in digital media and information technology over the past decades have purveyed the internet to be an effectual medium for the exchange of data and communication. With the advent of technology, the data has become susceptible to mismanagement and exploitation. This led to the emergence of Internet Security frameworks like Information hiding and detection. Examples of domains of Information hiding and detection are Steganography and steganalysis respectively. This work focus on addressing possible security breaches using Internet security framework like Information hiding and techniques to identify the presence of a breach. The work involves the use of Blind steganalysis technique with the concept of Machine Learning incorporated into it. The work is done using the Joint Photographic Expert Group (JPEG) format because of its wide use for transmission over the Internet. Stego (embedded) images are created for evaluation by randomly embedding a text message into the image. The concept of calibration is used to retrieve an estimate of the cover (clean) image for analysis. The embedding is done with four different steganographic schemes in both spatial and transform domain namely LSB Matching and LSB Replacement, Pixel Value Differencing and F5. After the embedding of data with random percentages, the first order, the second order, the extended Discrete Cosine Transform (DCT) and Markov features are extracted for steganalysis.The above features are a combination of interblock and intra block dependencies. They had been considered in this paper to eliminate the drawback of each one of them, if considered separately. Dimensionality reduction is applied to the features using Principal Component Analysis (PCA). Block based technique had been used in the images for better accuracy of results. The technique of machine learning is added by using classifiers to differentiate the stego image from a cover image. A comparative study had been during with the classifier names Support Vector Machine and its evolutionary counterpart using Particle Swarm Optimization. The idea of cross validation had also been used in this work for better accuracy of results. Further parameters used in the process are the four different types of sampling namely linear, shuffled, stratified and automatic and the six different kernels used in classification specifically dot, multi-quadratic, epanechnikov, radial and ANOVA to identify what combination would yield a better result.

Scaling limits of bisexual Galton–Watson processes

Bisexual Galton–Watson processes are discrete Markov chains where reproduction events are due to mating of males and females. Owing to this interaction, the standard branching property of Galton–Watson processes is lost. We prove tightness for conveniently rescaled bisexual Galton–Watson processes, based on recent techniques developed in [V. Bansaye, M.E. Caballero, and S. Méléard, Scaling limits of population and evolution processes in random environment, Electron. J. Probab. 24(19) (2019), pp. 1–38]. We also identify the possible limits of these rescaled processes as solutions of a stochastic system, coupling two equations through singular coefficients in Poisson terms added to square roots as coefficients of Brownian motions. Under some additional integrability assumptions, pathwise uniqueness of this limiting system of stochastic differential equations and convergence of the rescaled processes are obtained. Two examples corresponding to mutual fidelity are considered.

Geo/D/1/1 모형에서의 실시간 원격 추정값의 오차 분석

In this paper, we study real-time error in the context of monitoring a binary information source through a delay system. To derive the average real-time error, we model the delay system as a discrete time Geo/D/1/1 queueing model. Using a discrete time three-dimensional Markov chain with finite state space, we analyze the queueing model. We also perform some numerical analysis on various system parameters: state transition probabilities of binary information source; transmission times; and transmission frequencies. When the state changes of the information source are positively correlated and negatively correlated, we investigate the relationship between transmission time and transmission frequency.

Development and parameterization of a data likelihood model for geolocation of a bentho-pelagic fish in the North Pacific Ocean

State-space geolocation models feature coupled process (movement) and observation (data likelihood) models to reconstruct fish movement trajectories using electronic tag data. Development of the data likelihood model is therefore a key step in adapting a state-space geolocation model for use with different fish species, geographical regions, or types of electronic data. Here we adapt a discrete hidden Markov model for the geolocation of Pacific spiny dogfish (Squalus suckleyi, n = 154) in the North Pacific Ocean by developing a data likelihood model based on Microwave Telemetry X-tag Pop-up Satellite Archival Tag (PSAT) data. The data likelihood model consists of light-based longitude, light-based latitude, sea surface temperature (SST), temperature-depth profile (TDP), and maximum daily depth. Pacific spiny dogfish tend to occupy coastal waters where small-scale local currents and freshwater inputs make SST and TDP variables difficult to map. To address this issue, we introduce an empirical method for parameterizing SST and TDP likelihoods by calculating root mean square difference between PSAT temperature and depth values recorded at known locations (day of tag deployment and tag pop-up) and mapped values at those locations. For SST observations (n = 85), the difference between measured and mapped values did not vary seasonally or monthly and the overall root mean square error (RMSE) used to parameterize the SST likelihood was 0.9 °C. Likelihood values for SST at known locations were higher for likelihoods parameterized with the empirical value compared to variance specification methods from previous studies. For TDP, measured values differed from mapped values (n = 89) by depth, season, and month. Therefore, RMSE values used to parameterize the TDP likelihood were calculated for each depth bin (n = 27) and month. RMSE values were low (< 1 °C) for all depths during the winter but increased for depths < 100 m during the summer months. Our work provides an example of adapting state-space geolocation models for specific applications. It demonstrates the value of large numbers of tagged animals for parameterizing the data likelihood model in coastal waters as well as flexible data likelihood models with component likelihoods that can be switched on or off depending on geolocation quality.

Supply reliability-driven joint optimization of maintenance and spare parts inventory in a gas pipeline system

This work proposes a method for the joint optimization of the maintenance and spare parts inventory to maximize the reliability of gas supply in a pipeline system. The method trades off maintenance costs and risk of gas shortages. System failure probability calculation and system maximum supply capacity analysis, joint optimization modeling and system maintenance planning, are performed. A gas compressor station is considered as object of the analysis and a discrete Markov model is used to describe the state transition process. A supply capacity model is proposed to analyze the maximum supply capacity of the pipeline system under different failure scenarios. The joint optimization of maintenance and spare parts inventory is performed using a genetic algorithm to optimize the inspection interval and spare ordering time, considering the stochastic behavior of lead time. The effectiveness of the method is validated on a case study of a pipeline system in China. The results demonstrate that the proposed method outperforms others in identifying optimal maintenance strategies. In addition, the sensitivity analysis of critical parameters is analyzed in detail, which can further ensure the robustness of the method.