semi-Markov Process Research Articles

RELIABILITY MODELING AND STOCHASTIC EVALUATION OF A MACHINE OF VARIOUS UNITS WITH IMMEDIATE REPAIR OF FAILED UNIT

This article discusses the design reliability and quality of dairy products. Reliability and stochastic analysis of a dairy plant divided into two units using the regenerative method. At startup time, both units are in active mode. If one of the units fails, the system remains in active mode but goes into a reduced state. If both units fail, the system enters an invalid state. The distribution of the repair time is arbitrary in this system, whereas the malfunctioning time is exponentially distributed. With the data gathered from the milk plant, we assess average time to system failure, availability, busy time, and profitability through a mathematical analysis in this work that makes use of the Laplace transform, semi-Markov process, and regenerative point. Consequently, the conclusion of this paper helps plant managers make timely maintenance decisions and enhance the system’s overall performance.

Analytical model of a computing system with software rejuvenation

The foundational basis of present technological advancements worldwide largely relies on computing systems. The integration of virtualisation amplifies the capabilities of these computing systems. The incorporation of virtualisation requires a virtual machine monitor (VMM) or hypervisor, host or server, and virtual machines (VMs). Over the time, as the system ages, the hypervisor requires rejuvenation, in order to ensure the system continues operating at its full capacity. Security of virtual system also compromises due to ageing. This article aims to examine the impact of software rejuvenation on the reliability, security and the performance of virtualised computing systems by comparing two probabilistic models. In Model-I, VMM quickly recovers from failure, restoring access to the robust state. Whereas, in Model-II, despite prompt repair, access shifts to the rejuvenation state before returning to the robust state. The models are built on the foundation of the theory of semi-Markov processes, employing the regenerative point approach from renewal theory to deduce results for diverse performance indicators. The findings from this research will contribute for making computing systems more resilient and long-lasting in different fields of applications.

Fast Simulation of the Customer Blocking Probability in a Queueing System with Poisson Input Flow Controlled by a Semi-Markov Process

Fast Simulation of the Customer Blocking Probability in a Queueing System with Poisson Input Flow Controlled by a Semi-Markov Process

Sensitivity analysis of an imperfect maintenance policy for Proton-exchange membrane fuel cell based on geometric a semi-Markov model

This paper proposes a maintenance model for a Proton-Exchanged Membrane Fuel Cells system. Two types of failures are considered: non repairable internal failures and possibly repairable external failures. The maintenance involves replacing one or more cells. Therefore the maintenance can be perfect or imperfect. Moreover after each repair, the maintenance operator improves and the down time due to maintenance is reduced. A semi-Markov process is proposed to model the system’s lifetime. In this framework, the reliability measures and the remaining useful lifetime are derived. The long run average maintenance cost is studied. A sensitivity analysis of model parameters and maintenance unit costs is performed and a maintenance optimization strategy is proposed.

Optimizing System Efficiency and Reliability: Integrating Semi-Markov Processes and Regenerative Point Techniques for Maintenance Strategies in Plate Manufacturing

This paper offers a modeling methodology to evaluate similar systems’ performance and optimize system effectiveness with customized maintenance plans. This strategy, which combines regeneration point techniques with semi-Markov processes, is used to assess a plate manufacturing company’s system. Key insights are obtained using a thorough cost-benefit analysis that includes numerical analysis, graphical interpretations, and system efficacy indicators. The study clarifies the dynamics of the system’s reliability under different repair rates by showing an inverse relationship between Mean Time to System Failures (MTSF) and failure rate. Additionally, the analysis clarifies how profit is affected by failure rate, highlighting the necessity of efficient maintenance plans from an economic standpoint. This research highlights the growing significance of availability and dependability in technology-driven sectors and emphasizes the need for dependable systems. This paper makes a substantial contribution to the knowledge already available in the field of reliability engineering by providing a thorough framework that combines regeneration point approaches and semi-Markov processes. The metrics that are obtained offer a comprehensive understanding of the behavior of the system, which in turn helps industry practitioners make well-informed decisions.

ADP-Based Fault-Tolerant Control for Multiagent Systems With Semi-Markovian Jump Parameters.

This article analyzes and validates an approach of integration of adaptive dynamic programming (ADP) and adaptive fault-tolerant control (FTC) technique to address the consensus control problem for semi-Markovian jump multiagent systems having actuator bias faults. A semi-Markovian process, a more versatile stochastic process, is employed to characterize the parameter variations that arise from the intricacies of the environment. The reliance on accurate knowledge of system dynamics is overcome through the utilization of an actor-critic neural network structure within the ADP algorithm. A data-driven FTC scheme is introduced, which enables online adjustment and automatic compensation of actuator bias faults. It has been demonstrated that the signals generated by the controlled system exhibit uniform boundedness. Additionally, the followers' states can achieve and maintain consensus with that of the leader. Ultimately, the simulation results are given to demonstrate the efficacy of the designed theoretical findings.

Adaptive Output Formation Tracking for Nonlinear Multiagent Systems With Double Semi-Markovian Switching Topologies and Time-Varying Actuator Faults.

This article investigates adaptive output formation tracking control of nonlinear multiagent systems with time-varying actuator faults and unknown nonidentical control directions under double semi-Markovian switching topologies. Considering the dynamic changes of communication connections in uncertain environments, a double semi-Markov process is first introduced into the leader-follower structure to describe the random switching of communication topologies. Then, a novel adaptive distributed fault-tolerant output formation tracking control framework is established using the backstepping and Nussbaum gain technique to address matched/mismatched uncertainties and disturbances, time-varying actuator faults, and unknown nonidentical control directions. In this control framework, the independent variable of the Nussbaum function is designed as a non-negative function that monotonically increases with respect to time, thereby overcoming the presence of the absolute value of its derivative in the integration process. Based on the distributed structure, an adaptive fault-tolerant controller is further proposed to achieve the asymptotic output formation tracking in mean-square sense. The stability of the closed-loop nonlinear multiagent systems is analysed through the contradiction argument and Lyapunov theorem. The simulation example verifies the effectiveness of the proposed control strategy.

A Model of Alcohol Consumption with Semi-Markov Variable Coefficients

This paper focuses on the in-depth study of a stochastic approximation methodology involving semi-Markov switches in an averaging scheme with a minor parameter. We present a model where perceived objects are impacted by noise variables dependent on the semi-Markov process. The emphasis is on analyzing the convergence and stability of the stochastic approximation process within this context. Theoretical results are presented alongside numeric simulations, demonstrating the practical applications in managing complex stochastic systems. This work opens promising paths for the use of stochastic approximation approaches in the wider field of semi-Markov processes. Recognizing the mutable nature of alcohol consumption and its dependency on a variety of factors, we propose encoding such dynamism within a semi-Markov parameter structure. This model handles the patterns of individual alcohol consumption as a semi-Markov process; the transition probabilities between different states of alcohol consumption are directed by sociodemographic variables that change over time. Our approach, thus, bridges the gap between the realities of ever-changing alcohol consumption trends and static traditional Markov-chain models. By integrating real-world variables into our innovative model, we offer a cutting-edge analytical tool that lays down new paths for understanding and addressing the challenges with alcohol consumption patterns. Further, our insights have the potential to significantly impact the formulation of effective strategies and public health interventions aimed at alcohol-related harm reduction

The role of health in consumption and portfolio decision-making: Insights from state-dependent models

In this paper, we study an optimal consumption and asset allocation problem accounting for the fact that individuals’ utility differs across various health states. Our study is based on the assumption that an individual’s health status evolves through a semi-Markov process, where the transition probabilities are contingent on both the present state and the duration spent in that state. The optimal form of stock investment and of consumption is determined analytically. The optimal consumption level is significantly shaped by the integration of health-related factors and can be represented as the inverse of the marginal utility function with respect to time and state price density. Additionally, we introduce a Lagrangian multiplier that can be derived by solving a fixed point problem. While our optimal solutions are applicable to a wide range of utility functions, we provide numerical illustrations specifically using the power utility function in a model encompassing three distinct health states. Transition probabilities are estimated from real data collected by the China Insurance Regulatory Commission (CIRC), using a high-order polynomial Perks formula. We find that the investor’s consumption is higher when health is good, but lower when care is needed.

Finite horizon partially observable semi-Markov decision processes under risk probability criteria

This paper deals with a risk probability minimization problem for finite horizon partially observable semi-Markov decision processes, which are the fairly most general models for stochastic dynamic systems. In contrast to the expected discounted and average criteria, the optimality investigated in this paper is to minimize the probability that the accumulated rewards do not reach a prescribed profit level at the finite terminal stage. First, the state space is augmented as the joint conditional distribution of the current unobserved state and the remaining profit goal. We introduce a class of policies depending on observable histories and a class of Markov policies including observable process with the joint conditional distribution. Then under mild assumptions, we prove that the value function is the unique solution to the optimality equation for the probability criterion by using iteration techniques. The existence of (ϵ-)optimal Markov policy for this problem is established. Finally, we use a bandit problem with the probability criterion to demonstrate our main results in which an effective algorithm and the corresponding numerical calculation are given for the semi-Markov model. Moreover, for the case of reduction to the discrete-time Markov model, we derive a concise solution.

Semi-Markov process-driven maintenance scheduling for Tainter gate system considering multiplelimit states

In extended periods of operation, Taiwan’s reservoir electromechanical systems increasingly require substantial maintenance. This research adopts the semi-Markov process, which accommodates non-exponential distribution of state durations, to formulate optimal maintenance strategies for Tainter gate systems that are noted for their prolonged dormancy and significant operational uncertainties. The methodology initiates with the estimation of failure probabilities across four condition states, analyzing deterioration through the Weibull distribution for both general and latent limit states. The general limit state accounts for time-induced deterioration and effects of dormancy using a Bayesian–Weibull first-order reliability method, while the latent limit state addresses activation failures. Employing the semi-Markov process, an annual transition matrix is computed and combined with failure probabilities to assess the Tainter gates’ system reliability. To identify the most efficient maintenance schedule, a genetic algorithm is applied, targeting the minimization of failure probabilities for both limit states below predefined thresholds and cost reduction. Numerical simulations validate the framework’s efficacy, demonstrating its potential to enhance maintenance planning objectivity and decrease dependence on subjective assessments. The findings highlight the predominance of the general limit state in dictating system failure and underscore the risk Tainter gates face during transition from dormancy to activation, emphasizing the need for thorough monitoring.

Stability of a stochastic brucellosis model with semi-Markovian switching and diffusion.

To explore the influence of state changes on brucellosis, a stochastic brucellosis model with semi-Markovian switchings and diffusion is proposed in this paper. When there is no switching, we introduce a critical value and obtain the exponential stability in mean square when by using the stochastic Lyapunov function method. Sudden climate changes can drive changes in transmission rate of brucellosis, which can be modelled by a semi-Markov process. We study the influence of stationary distribution of semi-Markov process on extinction of brucellosis in switching environment including both stable states, during which brucellosis dies out, and unstable states, during which brucellosis persists. The results show that increasing the frequencies and average dwell times in stable states to certain extent can ensure the extinction of brucellosis. Finally, numerical simulations are given to illustrate the analytical results. We also suggest that herdsmen should reduce the densities of animal habitation to decrease the contact rate, increase slaughter rate of animals and apply disinfection measures to kill brucella.

A condition-based maintenance policy for a two-component balanced system with dependent degradation processes

Preventive maintenance of balanced systems has garnered increasing research attention due to its significance in production and servicing processes. This paper investigates the optimization problem of a condition-based maintenance policy for a two-component balanced system with dependent degradation processes. The system’s degradation follows a bivariate gamma process. Once the degradation level of one component exceeds a critical value, the system fails and triggers an immediate corrective replacement. Moreover, if the degradation difference between two components exceeds a threshold, the system is in an unbalanced state, which incurs a penalty cost. Periodic inspections are implemented to reveal the degradation of the system. At each inspection epoch, the decision-maker chooses an action among do-nothing, repair, and preventive replacement. Different from previous research, a repair action does not reduce the degradation levels of either component but increases the degradation level of the lower one to rebalance the system. The objective is to determine the optimal condition-based maintenance policy that minimizes the long-run average cost rate. The optimization problem is analyzed in a semi-Markov decision process framework. A policy-iteration dynamic programming algorithm is developed to derive the optimal policy. A numerical example is presented to illustrate the effectiveness of the proposed approach.

Profit Comparison of Computer System with Hardware Redundancy Subject to Different Repair Activities

In this research paper, main concentrate of the authors on the profit comparison of computer system with hardware redundancy by introducing the concept of priority to software up-gradation, hardware preventive maintenance (PM) and hardware maximum repair time (MRT). The system fails independently from normal mode. All the repair activities such as hardware repair, software up-gradation, hardware preventive maintenance before failure and hardware replacement after maximum repair time are carried out by a single server immediately on need basis. All random variables are statistically independent. The negative exponential distribution is taken for the failure time of the component while the distributions of repair time, up-gradation time, preventive maintenance and replacement time are assumed arbitrary with different probability density functions. Semi-Markov process and regenerative point technique are used. The behaviour of profits of the system models have been examined for different parameters and costs.

A comparative study of numerical methods for reliability assessment based on semi-Markov processes

The semi-Markov process, renowned for its versatile applications, has garnered significant attention in recent years. However, deriving closed-form expressions for computing reliability metrics proves challenging when sojourn times deviate from exponential distributions. This paper investigates three numerical techniques for computing the transition function matrix of the semi-Markov process, including the algebraic method, the truncated method, and the iterative method. It delves into their truncation and discretization errors, as well as their computational complexities. Additionally, the Laplace-based method and the semi-Markov-chain-based method are discussed to contrast their effectiveness with the three devised numerical approaches. Building on the linkage between the Markov renewal equation and system reliability metrics, five computational methods are applied to the evaluation of system reliability and availability. A case study on sequential cyber-attacks is presented to illustrate the applicability of these methods, considering sojourn times that follow exponential, gamma, lognormal, and Weibull distributions respectively. The results reveal that the three proposed numerical methods not only achieve high precision and rapid speed but also address scenarios beyond the capability of the Laplace-based method.

Reliability analysis of multi-component systems subjected to dependent degradation processes and random shocks in dynamic environments

This article presents reliability models for multi-component systems subjected to multiple internally dependent degradation processes and external shocks in dynamic environments. We use the homogeneous semi-Markov process (HSMP) to describe the evolution of the environment state and the non-homogeneous Poisson process (NHPP) to model the arrival of shocks. The degradation processes of the system components are modeled using either general path models (GPM) or stochastic processes. The models highlight the shock-induced degradation acceleration, the degradation-shock dependence, and the influence of dynamic environments on the degradation processes and shock damage. The integrated process evolution (i.e., system degradations, shock arrivals, and environment shifts) follows a non-homogeneous Markov renewal process (NHMRP) in a multidimensional space. The NHMRP semi-Markov kernel and theoretical formulation for system reliability are derived. We provide a simulation algorithm to estimate system reliability. Finally, we demonstrate the theoretical correctness and applicability of the model using a micro-electro-mechanical system (MEMS).

On the Boundary Functional of a Semi-Markov Process

In this paper, we consider the semi-Markov random walk process with negative drift, positive jumps. An integral equation for the Laplace transform of the conditional distribution of the boundary functional is obtained. In this work, we define the residence time of the system by generalized exponential distributions with different parameters via fractional order integral equation. The purpose of this paper is to reduce an integral equation for the Laplace transform of the conditional distribution of a boundary functional of the semi-Markov random walk processes to fractional order differential equation with constant coefficients.

Evaluation of readiness of the technical system using the semi-Markov model with selected sojourn time distributions

Mechanization of forestry work is crucial in forest management, and the specific nature of the tasks performed requires reliable machines with a high level of technical readiness. Therefore, models describing the exploitation process of multi-operational machines used to obtain wood raw materials, and especially assessing the level of their technical readiness, are extremely important. For multi-tasking objects performing random activities in given time intervals, calculating readiness measures is a complex issue. Forecasting subsequent operational states and their durations allows to predict the behavior of technical objects and schedule work. In addition to identifying the sequence of states, it is also important to identify the sojourn time in these states. This article presents a method for identifying the semi-Markov process and then assessing the technical readiness of a Harvester machine. This allowed for conclusions regarding the timely completion of assigned tasks, and also made it possible to adjust the activities carried out to the requirements of forest management.

On the Effect on the Optical Cable Degradation Cycle of the Method of Restoring Communication by Connecting the Cable at the Breakage Point

During operation, various factors affect the fiber optic cable, which lead to failures. It is generally assumed that the weakest link in an optical cable is fiber, and therefore it is fiber failures that are considered in the article. The warranty period of the fiber, which is set by the manufacturer, is 25 years. Currently, on many sections of the routes where the optical cable is laid, its service life already exceeds the warranty period, and the question arises of replacing it with a new one. As a rule, the service life of an optical cable exceeds the warranty period, because the cable is subject to maintenance and repair during operation. This article proposes a method for calculating the degradation cycle of a fiber-optic cable, which determines the time of its replacement with a new one, taking into account the effect of gradual and sudden failures. The reason for the gradual failures is the aging of the optical cable and the increase in signal attenuation, resulting in a degradation failure (a decrease in the level of the received signal below the critical one), which leads to the replacement of the optical cable. The appearance of a sudden failure caused by outside interference is accompanied by repair work on connecting the cable at the breakage point. The appearance of each new connection leads to an additional attenuation of the signal and a reduction in the degradation cycle time. To detect sudden failures, periodic monitoring of the fiber technical condition is carried out. Within the framework of the proposed methodology, it is assumed that the degradation cycle of an optical cable is divided into degradation states of a certain duration, which, in turn, consist of intervals for checking the condition of the optical fiber in the cable. At the same time, mathematical models using semi-Markov processes are considered, reflecting the state of an optical cable during one testing period, in one degradation state, and on a degradation cycle. Expressions and dependencies have also been obtained showing how sudden and gradual failures affect the reduction of the duration of the opyical cable degradation cycle.

Analysis of the technical condition of building structures on the example of a feed preparation workshop

The article aims to develop an instrumental model for a complex assessment of building structures that are subject to changes during their life-cycle process. This model can be used to substantiate the need either to reconstruct the existing building structures with the purpose of creating favorable living conditions or to demolish the building. The research methodology involved the general theory of systems and available techniques for estimating the state of building structures. Brick walls were tested by a nondestructive method using an IPS-MG-4.03 meter. Gypsum block masonry walls were tested by an ONIKS- 2.5 meter. Concrete was tested by an OMSh-1 hardness tester. Bellman dynamic programming was used to optimize the calculations and to find a solution for optimizing reconstruction costs. An example of assessing the condition of a building construction is provided, along with measures for extending its lifecycle. A conclusion is made that evaluation of the reliable operation of a building structure should consider the integral condition of the basic load-bearing structures under the action of environmental factors. This allows the risks of building destruction and the costs of building reconstruction to be reduced. For evaluating the current condition of the building under study, its structure was divided into a number of elements: foundation, wall box, floor slabs, roof. The hydrogeological parameters of the soil were also considered. Basing on the wear of building elements, the weakening coefficients of the building were calculated. The transient states of the system were described by semi-Markov processes. The results obtained can be used when evaluating the building residual life, selecting measures for improving the durability of structures with high wear and tear, and justifying the renovation concepts of territories.