Rate Matrix Research Articles

A Novel Deep Learning and Ensemble Learning Mechanism for Delta-Type COVID-19 Detection.

Recently, the novel coronavirus disease 2019 (COVID-19) has posed many challenges to the research community by presenting grievous severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) that results in a huge number of mortalities and high morbidities worldwide. Furthermore, the symptoms-based variations in virus type add new challenges for the research and practitioners to combat. COVID-19-infected patients comprise trenchant radiographic visual features, including dry cough, fever, dyspnea, fatigue, etc. Chest X-ray is considered a simple and non-invasive clinical adjutant that performs a key role in the identification of these ocular responses related to COVID-19 infection. Nevertheless, the defined availability of proficient radiologists to understand the X-ray images and the elusive aspects of disease radiographic replies to remnant the biggest bottlenecks in manual diagnosis. To address these issues, the proposed research study presents a hybrid deep learning model for the accurate diagnosing of Delta-type COVID-19 infection using X-ray images. This hybrid model comprises visual geometry group 16 (VGG16) and a support vector machine (SVM), where the VGG16 is accustomed to the identification process, while the SVM is used for the severity-based analysis of the infected people. An overall accuracy rate of 97.37% is recorded for the assumed model. Other performance metrics such as the area under the curve (AUC), precision, F-score, misclassification rate, and confusion matrix are used for validation and analysis purposes. Finally, the applicability of the presumed model is assimilated with other relevant techniques. The high identification rates shine the applicability of the formulated hybrid model in the targeted research domain.

Dynamic event-triggered asynchronous fault detection for Markov jump systems with partially accessible hidden information and subject to aperiodic DoS attacks

This paper is concerned with the event-triggered asynchronous fault detection (FD) problem for Markov jump systems with partially accessible hidden information and subject to aperiodic denial-of-service (DoS) attacks. The hidden Markov model with partially unknown probabilities is introduced to characterize the asynchronous phenomenon between the system and the filter, where the partially unknown probabilities may exist in the transition rate matrix of Markov chain, the conditional probability matrix of detected signal, or in both of them. In order to save the limited network bandwidth while resisting the aperiodic DoS jamming attacks, a new resilient dynamic event-triggered communication strategy is devised. Then, a new switched residual model for asynchronous FD is formulated by comprehensively considering the effects of the event-triggered scheme, asynchronization, and DoS attacks. By means of this model combined with the piecewise stochastic Lyapunov-Krasovskii functional approach, sufficient conditions are derived to guarantee the stochastic stability of the resulting switched residual system with desired dissipativity performance. Based on convex optimization techniques, an explicit expression of the desired asynchronous FD filter is derived. Finally, a single-link robot arm and a mass-spring system model are utilized to demonstrate the effectiveness of the proposed design technique.

Surface urban heat island and its relationship with land cover change in five urban agglomerations in China based on GEE.

The development of urbanization has changed the original land cover and exacerbated the urban heat island effect, seriously affecting the sustainable development of the ecological environment. Research on urban heat island characteristics and land cover changes in five major urban agglomerations in China to provide a reference for preventing thermal environmental risks and urban agglomeration construction planning. This paper estimates the surface urban heat island intensity (SUHII) of the five major urban agglomerations in China from 2003 to 2019 based on Google Earth Engine (GEE) through the urban-rural dichotomy, analyzes their trends through the Sen + M-K trend analysis method, and combines the detrending rate matrix to analyze the impact of land cover type shift on urban heat island change. Research shows that (1) the land cover types of the five major urban agglomerations in China have changed considerably from 2003 to 2019, and all five major urban agglomerations in China experienced varying degrees of urban expansion. (2) The annual average value of SUHII decreases in Beijing-Tianjin-Hebei, Yangtze River Delta, and middle reaches of the urban agglomerations, while the annual average value of SUHII increases in Chengdu-Chongqing and Pearl River Delta urban agglomerations. (3) The spatial composition of land cover types in the five major urban agglomerations in China is highly spatially correlated with urban heat islands, with urban land and bare land urban heat islands being the most pronounced. (4) The land cover type shift has the most significant heat island impact on Beijing-Tianjin-Hebei, Yangtze River Delta, and Chengdu-Chongqing urban agglomerations. (5) The land cover change (LCC) with an increasing trend in SUHII is mainly bare land converted to arable land, and water bodies, grassland, forest land, and arable land converted to urban land.

The stationary and quasi-stationary properties of neutral multi-type branching process diffusions

The stationary asymptotic properties of the diffusion limit of a multi-type branching process with neutral mutations are studied. For the critical and subcritical processes the interesting limits are those of quasi-stationary distributions conditioned on non-extinction. Pedagogical derivations are given for known results that the limiting distributions for supercritical and critical processes are found to collapse onto rays aligned with stationary eigenvectors of the mutation rate matrix, in agreement with discrete multi-type branching processes. For the sub-critical process the previously unsolved quasi-stationary distribution is obtained to first order in the overall mutation rate, which is assumed to be small. The sampling distribution over allele types for a sample of given finite size is found to agree to first order in mutation rates with the analogous sampling distribution for a Wright-Fisher diffusion with constant population size.

Log-Linear Model and Multistate Model to Assess the Rate of Fibrosis in Patients With NAFLD

In this paper, the deleterious effects of obesity, type II diabetes, and insulin resistance, systolic and diastolic hypertension on the rate of progression of fibrosis in patients with non-alcoholic fatty liver disease (NAFLD) are illustrated using a new approach utilizing the Poisson regression to model the transition rate matrix. The observed counts in the transition count matrix are used as the response variables and the covariates are the risk factors for fatty liver. Then, the estimated counts from running the Poisson regression are used to estimate the transition rates using the continuous-time Markov chains (CTMCs) followed by exponentiation of the estimated rate matrix to obtain the transition probability matrix at specific time points. A depicted, hypothetical, observational, prospective longitudinal study of 150 participants followed up every year for a total of 29 years recording their demographic characteristics and their timeline follow-up is demonstrated. The findings revealed that insulin resistance expressed by HOMA2-IR had the most deleterious effects among other factors on increasing the rate of fibrosis progression from state 1 to state 2, from state 2 to state 3, and from state 3 to state 4. The higher the level of HOMA2-IR is, the more rapid the rate of progression is. This analysis helps the health policymakers and medical insurance managers to allocate the financial and human resources for investigating and treating high-risk patients with NAFLD. In addition, this analysis can be used by pharmaceutical companies to conduct longitudinal studies to assess the effectiveness of the newly emerging anti-fibrotic drugs.

MLPNN‐RF: Software fault prediction based on robust weight based optimization and Jacobian adaptive neural network

SummarySoftware fault prediction (SFP) is a vital objective in software engineering. This might permit effective resource allocation and also enhance informed decisions about the release quality. SFP is being a critical issue for software professionals as well as the tech industry. Thus, SFP is necessary. The study intends to perform efficient error rate estimation using the proposed hybrid robust weight based optimization and Jacobian adaptive neural network (RWO‐JANN). It also aims to classify the software faults in an efficient way using multi‐layer perceptron neural network‐random forest (MLPNN‐RF). Various processes are involved to accomplish SFP. At first, the dataset is taken as input. After this, data preprocessing is performed. Subsequently, weights are initialized using the proposed RWO‐JANN. Weight initialization is performed through a series of steps. Then, the position and the weight parameter are updated to perform weight initialization. After this, the error rate is estimated and the weight is updated on the basis of the learning rate and Jacobian matrix calculation. Lastly, the decay rate is analyzed. If the error rate extends beyond the threshold value, the process repeats from weight initialization. If not, the testing process is performed and lastly the classified output for SFP is obtained by the proposed MLPNN‐RF. The proposed system is comparatively analyzed with the existing methods in terms of accuracy, precision, recall, F1 score, sensitivity, specificity, and error rate. The analytical results revealed effective outcomes of proposed system than the existing techniques with accuracy of 99.01%.

TRAF3IP2 regulated by FOXO4 affects fibroblast proliferation, migration, and extracellular matrix deposition in keloid through the TGF-β1/Smad pathway.

Keloids are "tumor-like" scars that grow beyond the boundary of injury. Its pathogenesis is complex. This paper will discuss the pathogenesis of keloid from the transcriptional regulation mechanism of TRAF3IP2. IL-17 was utilized to induce human keloid fibroblasts (KFs) and normal dermal fibroblasts. With the application of RT-qPCR and Western blot, TRAF3IP2 expression was detected. Subsequently, the expression of TRAF3IP2 was interfered by cell transfection and the effects of interfering TRAF3IP2 on cell proliferative rate, migration rate, and extracellular matrix were assessed with CCK-8, Wound Healing, immunofluorescence, and Western blot techniques. Proliferation, migration, and (ECM) deposition were detected by JASPAR software predicted the binding sites of transcription factors FOXO4 and TRAF3IP2 promoters. The relationship between FOXO4 and TRAF3IP2 was verified by Dual luciferase activity assay and ChIP. Finally, the expression of TRAF3IP2 and FOXO4 was interfered simultaneously to further explore the mechanism. TRAF3IP2 was enhanced in IL-17 induced KFs. Interference with TRAF3IP2 imparted suppressive effects on the proliferation, migration, and ECM deposition of KFs. FOXO4 could inhibit TRAF3IP2 transcription, and interference with FOXO4 reversed the effect of TRAF3IP2 down-regulation on KFs via TGF-β1/Smad pathway. TRAF3IP2 was regulated by FOXO4 and affected fibroblast proliferation, migration, and ECM deposition in keloid through the TGF-β1/Smad pathway.

Multi-branch joint adjustment method of mine ventilation based on sensitivity

The sensitivity analysis is an important analysis method to realize the intelligent control of branch air volume for the mine ventilation network. Aiming at the current situation that the ventilation on-demand (VOD) regulation cannot meet the large demand for air volume in specific air consumption branches in mine, this paper proposed a multi-branch wind resistance joint regulation method based on sensitivity analysis of the ventilation network. Based on the sensitivity theory, the concept of sensitivity change rate was proposed to quantitatively analyze the change law of the sensitivity of branch air volume to multi-branch air resistance, and the calculation formula of the sensitivity change rate matrix was obtained. The Lagrangian interpolation method is used to further optimize the relationship between air volume sensitivity and wind resistance. By using the loop air volume method to calculate and analyze a large number of ventilation networks, the adjustable range of wind resistance was determined and the air volume dispatching calculation model for multi-branch was established, which can more efficiently, accurately, and reliably adjust the branch air volume. This theory was applied to the optimal selection of regulating branch set and stability analysis of branch air volume. Finally, the rationality of this theory is verified through case analysis. In addition, this theory can also be applied to the preparation of emergency plans and the rapid prediction of the changing trend of the branch air volume during the disaster period.

AliSim: A Fast and Versatile Phylogenetic Sequence Simulator for the Genomic Era.

Sequence simulators play an important role in phylogenetics. Simulated data has many applications, such as evaluating the performance of different methods, hypothesis testing with parametric bootstraps, and, more recently, generating data for training machine-learning applications. Many sequence simulation programmes exist, but the most feature-rich programmes tend to be rather slow, and the fastest programmes tend to be feature-poor. Here, we introduce AliSim, a new tool that can efficiently simulate biologically realistic alignments under a large range of complex evolutionary models. To achieve high performance across a wide range of simulation conditions, AliSim implements an adaptive approach that combines the commonly used rate matrix and probability matrix approaches. AliSim takes 1.4 h and 1.3 GB RAM to simulate alignments with one million sequences or sites, whereas popular software Seq-Gen, Dawg, and INDELible require 2–5 h and 50–500 GB of RAM. We provide AliSim as an extension of the IQ-TREE software version 2.2, freely available at www.iqtree.org, and a comprehensive user tutorial at http://www.iqtree.org/doc/AliSim.

Hybrid quantum-classical simulations of magic angle spinning dynamic nuclear polarization in very large spin systems.

Solid-state nuclear magnetic resonance can be enhanced using unpaired electron spins with a method known as dynamic nuclear polarization (DNP). Fundamentally, DNP involves ensembles of thousands of spins, a scale that is difficult to match computationally. This scale prevents us from gaining a complete understanding of the spin dynamics and applying simulations to design sample formulations. We recently developed an ab initio model capable of calculating DNP enhancements in systems of up to ∼1000 nuclei; however, this scale is insufficient to accurately simulate the dependence of DNP enhancements on radical concentration or magic angle spinning (MAS) frequency. We build on this work by using ab initio simulations to train a hybrid model that makes use of a rate matrix to treat nuclear spin diffusion. We show that this model can reproduce the MAS rate and concentration dependence of DNP enhancements and build-up time constants. We then apply it to predict the DNP enhancements in core-shell metal-organic-framework nanoparticles and reveal new insights into the composition of the particles' shells.

Co-design of transition rates and sliding mode switched controller for Markovian jumping systems under intermittent DoS attacks

This paper focuses on the stabilization problem for a class of Markovian jumping systems (MJSs) subject to intermittent denial-of-service (IDoS) attacks by synthesizing the sliding mode control (SMC) and the transition rate matrix (TRM). The existing conditions for the transition rates are firstly established to ensure the exponential mean-square stability of the unforced uncertain MJSs. And then, a co-design scheme for both the sliding mode controller and TRM is synthesized to achieve the exponential mean-square stability of the closed-loop system under IDoS, in which a switching estimator is utilized to estimate the unmeasurable system state. By introducing a novel Lyapunov function, both the reachability and the stability of sliding mode dynamics are detailedly analyzed, and an iterative optimization algorithm is given for solving the corresponding sufficient conditions. Finally, the proposed co-design SMC strategy is illustrated via the simulation examples.

The Ediacaran origin of Ecdysozoa: integrating fossil and phylogenomic data

Ecdysozoans (Phyla Arthropoda, Kinorhyncha, Loricifera, Nematoda, Nematomorpha, Onychophora, Priapulida, Tardigrada) are invertebrates bearing a tough, periodically moulted cuticle that predisposes them to exceptional preservation. Ecdysozoans dominate the oldest exceptionally preserved bilaterian animal biotas in the early to mid-Cambrian ( c. 520–508 Ma), with possible trace fossils in the latest Ediacaran (<556 Ma). The fossil record of Ecdysozoa is among the best understood of major animal clades and is believed to document their origins and evolutionary history well. Strikingly, however, molecular clock analyses have implied a considerably deeper Precambrian origin for Ecdysozoa, much older than their earliest fossils. Here, using an improved set of fossil calibrations, we performed Bayesian analyses to estimate an evolutionary time-tree for Ecdysozoa, sampling all eight phyla for the first time. Our results recover Scalidophora as the sister group to Nematoida + Panarthropoda (= Cryptovermes nov.) and suggest that the Ediacaran divergence of Ecdysozoa occurred at least 23 myr before the first potential ecdysozoan trace fossils. This finding is impervious to the use of all plausible phylogenies, fossil prior distributions, evolutionary rate models and matrix partitioning strategies. Arthropods exhibit more precision and less incongruence between fossil- and clock-based estimates of clade ages than other ecdysozoan phyla. Supplementary material: Full methodologies used and an appendix of fossil calibration points are available at https://doi.org/10.6084/m9.figshare.c.5811381 Thematic collection: This article is part of the Advances in the Cambrian Explosion collection available at: https://www.lyellcollection.org/cc/advances-cambrian-explosion

Novel Approach of Multistate Markov Chains to Evaluate Progression in the Expanded Model of Non-alcoholic Fatty Liver Disease

A global increase in the prevalence of obesity and type 2 diabetes is strongly connected to an increased prevalence of non-alcoholic fatty liver disease (NAFLD) worldwide. In this article, the progression of the NAFLD process is modeled by continuous time Markov chains (CTMCs) with nine states. Maximum likelihood is used to estimate the transition intensities among the states. Once the transition intensities are obtained, the mean sojourn time and its variance are estimated, and the state probability distribution and its asymptotic covariance matrix are also estimated. A hypothetical example based on a longitudinal study assessing patients with NAFLD in various stages is discussed. The mean time to absorption is estimated, and the other abovementioned statistical indices are examined. In this article, the maximum likelihood estimation (MLE) function is utilized in a new approach to compensate for the missing values in the follow-up period of patients evaluated in longitudinal studies. A MATLAB code link is provided, at the end of the article, for the estimation of the transition rate matrix and transition probability matrix.

Research on Strawberry Disease Diagnosis Based on Improved Residual Network Recognition Model

Considering the problems of high cost, inefficiency, and time consumption of manual diagnosis of strawberry diseases, G-ResNet50 is proposed based on transfer learning and deep residual network for strawberry disease identification and classification. The G-ResNet50 is based on the ResNet50, and the focal loss function is introduced in G-ResNet50 to make the model devote itself to disease images that are difficult to classify. During the training process of the G-ResNet50 model, its convolutional layer and pooling layer inherit the pre-trained weight parameters from the ResNet50 model on the PlantVillage dataset, while adding dropout regularization and batch regularization methods to optimize the network model. The strawberry disease dataset includes four sample images of healthy plants, powdery mildew, strawberry anthracnose, and leaf spot disease. The dataset is enhanced and expanded by operations including angle rotation, adjusting contrast and brightness, and adding Gaussian noise. Compared with existing models such as VGG16, ResNet50, InceptionV3, and MobileNetV2, the results of model training and testing on 7,525 four-category leaf datasets show that the G-ResNet50 model has faster convergence speed and better classification effect, and its average recognition accuracy rate reached 98.67%, which is significantly higher than other models. Through the three evaluation indicators of precision rate, recall rate, and confusion matrix, it is concluded that the G-ResNet50 has good robustness, high stability, and high recognition accuracy and can provide a feasible solution for strawberry disease detection in practical applications.

A traverse algorithm approach to stochastic stability analysis of Markovian jump systems with unknown and uncertain transition rates

This paper intents to investigate the problem of mean-square stability analysis of Markovian jump systems with generally unknown and uncertain transition rates. Different from pervious works that the transition rates from one mode to others may be partially unknown or uncertain, in this note, the case that the transition rates from one mode to others are totally unknown will be investigated. By means of transition rate estimation, two ways are provided to tackle with the totally unknown case. In general, five cases in the transition rates matrix are studied for the mean-square stability analysis, which almost have covered all types of generally unknown and uncertain transition rates. Simultaneously, corresponding conditions for checking the mean-square stability of the considered Markovian jump systems are developed for the five studied cases. Finally, numerical examples are provided to verify the effectiveness of the proposed results.

Absorption kinetics of vacancies by cavities in aluminum: Numerical characterization of sink strengths and first-passage statistics through Krylov subspace projection and eigenvalue deflation

Modeling the microstructural evolution of metal and alloys, specifically under irradiation, is essential to predict the aging properties of materials. Many models are based on a transition rate matrix describing the jump frequencies of defects and involve a master equation governing the time evolution of a state probability vector. Here, we present non-stochastic numerical techniques to characterize the motion of individual defects migrating over long distances prior to recombining or being absorbed by another defect, resorting to the theory of absorbing Markov chains. These important events are fully determined by their first-passage time distribution to distant locations, no-passage distribution, and walker fluxes to the sinks. We show that these functions can be efficiently computed using a method combining Krylov subspace projection and eigenvalue deflation. For a model system describing the absorption of a vacancy by a cavity in aluminum, the use of a small Krylov subspace deflated by the unique eigenmode corresponding to the quasi-stationary distribution is sufficient to capture the kinetics of the defect absorption faithfully. This method can be used in kinetic Monte Carlo simulations to perform stochastic non-local moves or in cluster dynamics simulations to compute sink strengths.

Performance analysis and system optimization of an energy-saving mechanism in cloud computing with correlated traffic

<p style='text-indent:20px;'>Energy consumption is becoming a significant part of overall operational cost in cloud data centers. For the purpose of satisfying the Service Level Agreement (SLA) of cloud users while enhancing the energy efficiency in cloud computing systems, in this paper we propose an energy-saving mechanism with a sleep mode. Taking into consideration the traffic's correlation and the stochastical behavior of data arrival requests in a random cloud environment with the proposed energy-saving mechanism, we model the system as a MAP/M/<inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula>/<inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula>+<inline-formula><tex-math id="M3">\begin{document}$ K $\end{document}</tex-math></inline-formula> queue with a synchronous multi-vacation. Then, we present a theoretical basis for analyzing and evaluating the system performance by taking a state transition rate matrix in the steady state. Next, we investigate the change trends for the energy saving rate of the system and the average latency of tasks by carrying out numerical experiments. Moreover, we give a</p>

THE POTENTIAL AND EFFECTIVENESS OF REGIONAL RETRIBUTIONS INDRAGIRI HULU DISTRICT

This study aims to determine the potential and effectiveness of regional revenue receipts from regional levies in Indragiri Hulu Regency. This study uses a descriptive approach designed to obtain answers to the research objectives. This research data comes from the collection of documents, interviews and data on regional retribution receipts for 2014 – 2020 from the Transportation Agency of Indragiri Hulu Regency. Data analysis is explained by using Potential Analysis, Contribution, Growth Rate and Potential Matrix, and effectiveness ratio. The results of the study show that the potential for regional levies, on average, has the potential to be developed, but in terms of management and implementation effectiveness, it is still not optimal. Based on quantitative data analysis, parking fees are categorized as potential with positive growth, motor vehicle testing fees have good growth opportunities and have the potential to be developed, while terminal fees and route permit fees are underdeveloped categories. Based on the effectiveness ratio analysis, parking fees are categorized as very effective, motor vehicle testing fees are included in the fairly effective criteria, terminal fees are quite effective, and route permit fees are categorized as quite effective.

Integrated Pharmacogenetics Analysis of the Three Fangjis Decoctions for Treating Arrhythmias Based on Molecular Network Patterns.

Aim: To explore the diverse target distribution and variable mechanisms of different fangjis prescriptions when treating arrhythmias based on the systems pharmacology.Methods: The active ingredients and their corresponding targets were acquired from the three fangjis [Zhigancao Tang (ZT), Guizhigancao Longgumuli Tang (GLT), and Huanglian E'jiao Tang (HET)] and the arrhythmia-related genes were identified based on comprehensive database screening. Networks were constructed between the fangjis and arrhythmia and used to define arrhythmia modules. Common and differential gene targets were identified within the arrhythmia network modules and the cover rate (CR) matrix was applied to compare the contributions of the fangjis to the network and modules. Comparative pharmacogenetics analyses were then conducted to define the arrhythmia-related signaling pathways regulated by the fangjis prescriptions. Finally, the divergence and convergence points of the arrhythmia pathways were deciphered based on databases and the published literature.Results: A total of 187, 105, and 68 active ingredients and 1,139, 1,195, and 811 corresponding gene targets of the three fangjis were obtained and 102 arrhythmia-related genes were acquired. An arrhythmia network was constructed and subdivided into 4 modules. For the target distribution analysis, 65.4% of genes were regulated by the three fangjis within the arrhythmia network. ZT and GLT were more similar to each other, mainly regulated by module two, whereas HET was divided among all the modules. From the perspective of signal transduction, calcium-related pathways [calcium, cyclic guanosine 3′,5′-monophosphate (cGMP)-PKG, and cyclic adenosine 3′,5′-monophosphate (cAMP)] and endocrine system-related pathways (oxytocin signaling pathway and renin secretion pathways) were associated with all the three fangjis prescriptions. Nevertheless, heterogeneity existed between the biological processes and pathway distribution among the three prescriptions. GLT and HET were particularly inclined toward the conditions involving abnormal hormone secretion, whereas ZT tended toward renin-angiotensin-aldosterone system (RAAS) disorders. However, calcium signaling-related pathways prominently feature in the pharmacological activities of the decoctions. Experimental validation indicated that ZT, GLT, and HET significantly shortened the duration of ventricular arrhythmia (VA) and downregulated the expression of CALM2 and interleukin-6 (IL-6) messenger RNAs (mRNAs); GLT and HET downregulated the expression of CALM1 and NOS3 mRNAs; HET downregulated the expression of CRP mRNA.Conclusion: Comparing the various distributions of the three fangjis, pathways provide evidence with respect to precise applications toward individualized arrhythmia treatments.

Reliability modelling for linear and circular [formula omitted]-out-of-[formula omitted]: F systems with shared components

Reliability analysis for k-out-of-n systems has received much attention in the field of reliability due to its practical importance. In this work, models for linear and circular k-out-of-n: F systems with shared components, consisting of m subsystems, are discussed. The finite Markov chain imbedding approach cannot be used directly to derive reliability of the considered systems because of the common area existing between adjacent subsystems. For this reason, we first derive the probability that the number of failed components is a given value in the m common areas. Then, all disjoint cases, each of which is regarded as a series system consisting of m subsystems with different initial state probability distributions, are obtained, with which the nith step transition rate matrix is calculated by using the Markov chain technique, for i=1,2,…,m. Finally, the reliability of the system is obtained by summing the reliabilities of all cases along with the finite Markov chain imbedding approach. Some numerical examples are presented to illustrate the efficiency of the proposed reliability evaluation method.