Theory Of semi-Markov Processes Research Articles

Stochastic models of work and rest schedules

Purpose. To develop stochastic models for determining the duration of work and rest schedules that ensure protection of workers from occupational hazards and a high level of labor productivity during a work shift. Methodology. Analysis of scientific literature was applied to determine the purpose and tasks of the research; methods of formalization – to describe the characteristics, dynamics and states of random processes of accumulation and elimination of the consequences of the negative impact of harmful production factors occurring in an worker within the work and rest schedules; methods of semi-Markov processes theory, theories of reliability and recovery – to build stochastic models of work and rest schedules, determine the conditions and probability of a stable mode of their operation. Findings. Stochastic models were developed to determine the duration of work schedules, at the end of which the level of accumulation of the consequences of the negative impact of harmful production factors in the worker will not exceed the set maximum permissible values with a high probability. And such durations of rest schedules, at the end of which this level will be equal to zero (also with a high probability). The condition for ensuring the stable operation of these schedules throughout the entire period of work experience at the workplace was determined by the author. Originality. For the first time an approach to the development of work and rest schedules at workplaces is proposed, which, contrary to others, is based on determining the probabilities of the levels of accumulation of the consequences of the negative impact of harmful production factors in the worker at the end of these schedules, given the actual random and dynamic characteristics of such impact. This, in turn, allows ensuring the protection of the worker both from accidents caused by the fatigue and from professional illness. Practical value. The proposed approach allows increasing the economic efficiency of the enterprise by ensuring a high level of labor productivity, which is achieved by simultaneous development of the maximum possible duration of work and the minimum possible duration of rest schedules, which exclude the worker’s development and occurrence of fatigue and professional illness with a high probability.

Method of determining time for preventive diagnosis in city bus operation systems

The article presents a mathematical model of the process of diagnosing and repairing preventive public transport buses. The mathematical model was developed using the theory of semi-Markov processes. The presented model considers two types of corrective repairs - carried out by the Technical Emergency Service and at Service Stations, and preventive repair. The presented model of the exploitation process system has been described to a small extent in the literature so far. The considered system is analyzed on the basis of two criteria: the availability of public transport buses and the profit per unit time. For the adopted assumptions, the conditions for the existence of an extremum of the criterion function were formulated. In the article, the presented model is illustrated based on sample results.

Stochastic model of the smart power grid section functioning

A feature of large systems, in particular electric power systems, is that the entire system physical state of the is affected by a large number of interacting elements that are randomly in different states. The work relevance is caused by to the fact that construction of mathematical models taking into account the structure and features of the power grid functioning allows us to solve important problems of the energy intellectualization. The goal of the study is due to the desire to show the advantages that arise when using the apparatus of the hidden Markov models theory to semi-Markov models of intelligent electrical systems. To do this, we build a semi-Markov model of a section of a distribution power grid (intelligent electrical grid). Using the theory of semi-Markov processes with a common phase space of states, it is possible to determine the temporal characteristics of the system reliability and obtain a more adequate model. However, territorial dispersion, inaccessibility of some sections of electrical networks and the widespread introduction of automation tools give rise to new problems. For a timely response to various factors, it becomes necessary to as-sess and forecast the states of the system (sections of the system) depending on the signals received in the course of its operation. This can be achieved by applying the hidden Markov models theory. Reliability characteristics are determined and, using the algorithm of stationary phase enlargement, an enlarged semi-Markov model of the section of the intelligent distribution grid is built, which allows passing to the finite state space of the model. Using the merged model, the parameters are determined and a hidden Markov model is developed, for which the most probable states corresponding to a given signal vector are found, the elements subsequent states of the modeled system and signals are predicted. The final part of the article provides an example of finding the reliability characteristics of the system and solving a number of problems in hidden Markov models theory.

Modeling and Performance Analysis of Three Zone-Based Registration Scheme in Wireless Communication Networks

For wireless communication networks, researchers have proposed many schemes to reduce the cost of location registration and paging signals caused by the mobility of user equipment (UE). Among them, a zone-based method that designates one zone (1Z, group of cells) as a registration area (RA) and then performs registration whenever the UE leaves the RA is commonly adopted due to its convenient implementation. However, the performance of 1Z is known to be very poor when the UE frequently crosses the RA’s boundary requesting location updates. Two or three zone-based schemes (2Z or 3Z) have since been recommended to overcome these limitations. In our previous work, we analyzed the performances of 1Z, 2Z, and 3Z systems while assuming a square-shaped zone. However, there is no reason why the shape of the zone is limited to a square. This paper analyzes the performance of 3Z while assuming a hexagonal-shaped rather than a square-shaped zone. Using a semi-Markov process theory, registration and paging costs are evaluated after defining states in 3Z operations and calculating the transition probability between states. Based on various realistic parameters, the numerical results showed that the 3Z outperformed 1Z and 2Z for most call-to-mobility ratio (CMR) values. The performance of 3Z was improved more when the registration cost decreased if the probability of returning to the previously registered zone increased or the time staying in the zone decreased. The 3Z system is easy to implement with simple software modifications. It can be dynamically applied as an efficient mobility management method in the future for various devices that will emerge in the 5G/6G environment.

Модель оценки функциональной устойчивости элементов информационной инфраструктуры для условий воздействия множества компьютерных атак

Information is given about a new approach to the application of methods of the theory of semi-Markov processes to solve the applied problem of assessing the functional stability of elements that make up the information infrastructure, functioning under the influence of multiple computer attacks. The task of assessing functional stability is reduced to the task of finding the survivability function of the element under study and determining its extreme values. The relevance of the study is substantiated. The rationale is based on the assumption that quantitative methods of studying the stability of technical systems, which operate on the theory of reliability, cannot always be used to assess survivability. The concepts of «stability» and «computer attack» are being clarified. Verbal and formal statements of research tasks are formulated. The novelty of the results obtained lies in the application of well-known methods to solve a practically significant problem in a new formulation, taking into account the limitations on the resource allocated to maintain the survivability of the element under study, provided that arbitrary distribution laws are adopted for the random times of the implementation of computer attacks and the recovery times of the functional element. Recommendations on the formation of initial data, the content of the enlarged stages of modeling and a test case to demonstrate the performance of the model are given. The results of the test simulation are presented in the form of graphs of the survivability function. The resulting application can be used in practice to construct a survivability function when implementing up to three computer attacks, as well as a tool for evaluating the reliability of analogous statistical models. The limitation is explained by a progressive increase in the dimension of the analytical model and a decrease in the possibility of its meaningful interpretation.

Defining state-transition diagrams as part of technical state monitoring

State-transition diagrams are given for the situation of monitoring that takes into consideration explicit and hidden failures.Aim. To generate state-transition diagrams that are used in the development of the dependability model and associated calculation methods under conditions of periodic monitoring.Methods. The method is based on the classification of failures by detection, i.e., explicit and hidden failures. Accordingly, facility condition inspection may be continuous and/or periodical. Periodical inspection is conducted at equal intervals. The state-transition diagram is generated taking into consideration the relations between the states and events, i.e., each state can be the cause of a certain event and, at the same time, a state is the consequence of an event. Similarly, each event is the cause of a state change, while at the same time being a consequence of a state. Within the time between two periodic inspections, continuous-time transitions take place due to failures. This process is described by the theory of continuous-time Markov processes. Upon the completion of a periodic inspection operation, the up state ensues if an operable item is subjected to inspection. If an item proves to be down, it is submitted to maintenance. Such transitions in discrete moments in time are described by semi-Markov processes. For a better understanding of the information set forth in the paper, a list is provided of the used terminology referring to state standards.Results. Under the adopted failure modes within a single inspection period, state transitions are described in continuous time, while after the inspection operation they are described in discrete time. Within each inspection period, a system of differential equations is defined and solved based on the assumption of initial up state. Based on the deduced probabilities, the probabilities of a period without failures and with a failure are calculated along with the mean time of up and down states within the period with a failure. Should failures be detected, maintenance state is initiated. The probability of such transition is within the theory of semi-Markov processes. Based on the probabilities of the semi-Markov process, the mean failure-free periods are calculated. Such model adequately reflects the dependability processes of operated equipment. If the method is not fully observed, the dependability indicators may take significantly different values. Three examples of state-transition diagrams are given: taking into account only explicit failures, only hidden failures and both.Discussion and conclusions. The presented approach with a constant (regular) inspection period allows building models that adequately reflect processes occurring in actual systems. All operations involved in the implementation of simulated processes are performed based on the theory of continuous-time Markov processes and semi-Markov processes. The operations are performed in matrix form. Such method enables mathematical operations using computers. The presented approach can be used for improving dependability models of technical systems.

Semi-Markov Model of the System of Repairs and Preventive Replacements by Age of City Buses

The paper presents a mathematical model of the system of repairs and preventive replacements by age of city buses. The mathematical model was developed using the theory of semi-Markov processes. In the model developed, four types of city bus renewal processes are considered and three types of corrective repairs and preventive replacement. Corrective repairs are considered in two types: minimal repairs (repairs carried out by the Technical Service units) and perfect repairs (repairs carried out at the stations of the Service Station). The models of restoration systems that use semi-Markov processes in which minimal repairs, perfect repairs, and preventive replacements by age, have been examined in the literature to a limited extent. The system under consideration is analysed from the point of view of two criteria: profit per time unit and availability of city buses to carry out the assigned transport tasks. Conditions of criterion functions’ extremum (maximum) existence were formulated for the adopted assumptions. The considerations presented in the paper are illustrated by exemplary results of calculations.

Reliability model of an item under unreliable checking conditions with regular period

An analytic model of a long used item functioning with regular period of check-out operations and unreliable checking conditions is represented. The model is based on the theory of semimarkov processes using matrix methods for mathematical operations. Consideration of the model reveals the influence of the first and the second kind check-out errors on the reliability level.

Semi-Markov modeling for assessing reliability of road construction machines in the process of their operation

The article provides a theoretical description of the processes of changing the technical state of road-building machines to assess their reliability at the stage of operation based on semi-Markov models. Reliability is the most important property of machines and characterizes their ability to perform their functions during normal operation. Reliability covers all stages of the life cycle of equipment-its design, manufacture, and operation. The operation stage is the longest in time and the costliest to ensure reliability. This explains the relevance of developing a mathematical model of operational reliability of road-building machines. During the operation of technical objects, their technical condition changes under the influence of objective and subjective factors. As a result, quantitative indicators that characterize the technical condition of machines vary widely and are random in nature. Road-building machines may be in one of the following technical states during operation: perfect, imperfect, upstate, down state. At the same time, a perfect machine is always upstate, and an imperfect one can be either upstate or down state. The transition from one technical state to another is random and can be considered instantaneous with a sufficient degree of accuracy. The probability of any technical state in the future depends only on the state of the process in the present and does not depend on when and how the process was in this state. This makes it possible to use the theory of semi-Markov processes when formalizing the process and developing a mathematical model. Based on semi-Markov modeling, the article shows the sequence of constructing a graph of states and transitions from one state to another for a road construction machine during operation; matrices of transitions and probabilities of transitions from one state to another; systems of differential equations for determining the limiting probabilities of finding a machine in a particular technical state during operation. The developed method of semi-Markov modeling allows estimating the operational reliability of equipment with the reliability allowed by probability theory.

Comparative analysis of the reliability of electric locomotives based on semi-Markov models of restorable systems

Based on the general theory of semi-Markov recovery processes and their application to Markov recovery processes, the applicability of stationary reliability indicators to assessing the reliability of electric locomotives with an asynchronous motor, the functioning of which is considered as a superposition of independent alternating semi-Markov processes, is shown. By calculating the main indicators of system reliability in a steady state, a comparative analysis of the operational reliability of electric locomotives with an asynchronous DC and AC motors, used respectively on the Azerbaijan and Kazakhstan railways, was carried out.

Semi-Markov model of the functioning of redundant measuring instruments, taking into account the frequency of verification

The problem of modeling the process of functioning of redundant measuring instruments is considered. On the basis of the developed model, the determination of the frequency of verification of complex duplicated measuring instruments based on the semi-Markov model of the process of their functioning is justified. The relevance of the application of the theory of semi-Markov processes for modeling the functioning of redundant technical systems, including measuring instruments, is shown. A semi-Markov model of the process of functioning of measuring instruments has been developed, which allows us to take into account the influence of the frequency of verification, the provision of measuring instruments with spare elements, as well as the level of reliability and maintainability on the reliability of measuring instruments. A method is found for defining a semi-Markov process that correctly and adequately approximates the real process. The indicator of the effectiveness of the functioning of the measuring instruments and the corresponding model is the availability factor of the measuring instruments.

Semi-Markov analytical model for assessing the quality of the maintenance system and the state of electromechanical complexes

Analysis of basic research. In practice, the EMC reliability indicators, in particular the failure rate, are time-varying quantities. In view of this, it is necessary to determine such periodicity values and maintenance volumes that would ensure the EMC maximum availability factor. The most adequate mathematical model for analyzing the technical state of EMC can be obtained using the apparatus of the theory of semi-Markov processes (PMF), the basis of which is an oriented graph of the basic states and transitions (GSP) of EMC during operation, which allows to formulate maintenance and maintenance strategies, the content of a separate strategy is determined a specific set of options for all states of EMC.The purpose of the article. Construction and study of a mathematical model of the EMC operation process to determine the optimal intervals and maintenance and repair strategies.The main research material. The solution to the first problem includes: building a model for the operation of the EMC; the derivation of the calculated ratio for the availability factor as a function of the technical condition of the EMC and the service station; obtaining an analytical formula for calculating optimal maintenance intervals.The paper proposes a combination of indicators ranked by importance in the linear programming problem, where the transfer time, which is a function of the SRT parameters and the technical condition of the EMC, is selected as the objective function, and the remaining indicators (readiness level, complex operation costs, and others) are taken into account in the form of restrictions. The result of solving this problem is a set of maintenance and repair options at each step of the operation of the EMC, which comprise the content of the optimal maintenance and repair strategy.Conclusions. An analytical relationship was obtained on the basis of the semi-Markov model of the operation process that allows one to calculate the optimal conditions for ensuring the availability factor of the periodicity of maintenance, which makes it possible to control the unsteady operation of the EMC, to obtain the necessary technical and economic quality indicators to build an adaptive ACS with the technical state of the EMC.

The discrete analysis of relay-races

It is shown that relay-races as a form of concurrency, are widespread in such fields of human activity, as industry, defence, economics, politics, etc. The analytical description of J stages relay-races, in which participate M teams, is considered. The model based on the semi-Markov process theory allows investigating a process evolution in detail but is of little use for computer calculation of forfeit, which the winner team receives from loser teams. To adapt analytical description to computer calculations, the method based on sampling of continual time densities of passing stages is proposed. Due to the method, initial semi-Markov process is converted to semi-Markov process with degenerative distributions, and the task of forfeit calculation is reduced to the task of rigid schedules effectiveness analysis, in which schedules are selected stochastically from the densities samples. Formulae for the stochastic summation of forfeits, is obtained. The results may be used for optimal planning the activity of participant teams when passing the distance.

The discrete analysis of relay-races

It is shown that relay-races as a form of concurrency, are widespread in such fields of human activity, as industry, defence, economics, politics, etc. The analytical description of J stages relay-races, in which participate M teams, is considered. The model based on the semi-Markov process theory allows investigating a process evolution in detail but is of little use for computer calculation of forfeit, which the winner team receives from loser teams. To adapt analytical description to computer calculations, the method based on sampling of continual time densities of passing stages is proposed. Due to the method, initial semi-Markov process is converted to semi-Markov process with degenerative distributions, and the task of forfeit calculation is reduced to the task of rigid schedules effectiveness analysis, in which schedules are selected stochastically from the densities samples. Formulae for the stochastic summation of forfeits, is obtained. The results may be used for optimal planning the activity of participant teams when passing the distance.

Situational planning and operational adjustment of the route of the Autonomous robotic underwater vehicle

Currently, missions (tasks) for the underwater robot formed using imperative programming methods (both text and graphic), describing in detail the sequence of robot actions that need performed to achieve the desired goal. At the same time, only the operator of the underwater robot, which makes up the mission, for example, the delivery of cargo to the target point, has an idea of the goal itself. Such technology is effective if the robot's mission carried out within a priori scenario. In other cases, it can either not be executed at all, or it can be executed with large violations and a threat to the safety of the device.When assessing the effectiveness of an underwater robot, the degree of its information autonomy, i.e. the ability to act independently in an unknown or insufficiently defined environment, is of fundamental importance. Therefore, the "intellectualization" of the Autonomous control system of the underwater robot is extremely important for the mission in unforeseen circumstances. For this propose to use intelligent decision support system. Two ways to implement optimal decision-making strategies based on the mathematical apparatus of the theory of Markov and semi-Markov processes using the Bellman optimality principle propose. The considered ways of implementation of optimal strategies of decision - making process relate to the strategy for a short finite time of cargo delivery, which is the most common in practice, and for a long interval of cargo delivery relative to the entire task. In addition, the article discusses ways to find optimal strategies when the time of making single decisions is fixed or when the time of translation is implement randomly.Hence, the situational approach to decision-making in the planning of the route ARPA is very relevant and allows not only to assess the possible situation on the route, but also to determine the control solutions for the operational adjustment of the route using the intelligent decision support system (ISPR). The development of models of the routing process based on the representation of the situational model in the form of nodes of the graph, the transitions of which correspond to the control solutions.The paper proposes two ways to implement optimal strategies of decision - making based on the mathematical apparatus of the theory of Markov and semi-Markov processes using the Bellman principle of optimality.

Semi-Markov Model and Phase-Merging Scheme of a Multi-Component System with the Group Instantly Replenished Time Reserve

Time redundancy is a method of increasing the reliability and efficiency of the operation of systems for various purposes. A system with time redundancy provides additional time (a time reserve) for restoring characteristics and synchronizing the work of its individual elements. Time reservation is used in production, energy, gas transportation, information, ergonomic systems and some others. In this paper, based on the theory of semi-Markov processes with a common phase space of states, a semi-Markov model of a multicomponent system with a group instantly replenished time reserve is constructed. The reliability characteristics of the system are determined. To approximate the probability and the average time of failure-free operation of the system in conditions of high reliability, asymptotic phase-merging scheme algorithms are used.

Reliability analysis of a rodding anode plant in aluminum industry with multiple units failure and single repairman

The paper presents reliability analysis of a rodding anode plant in aluminum industry with multiple unit failure and single repairman. Manufacturing process of raw aluminum blocks in this plant passes through eight stations. Failure of any of the stations brings the plant to a complete halt, except the butt & thimble removal press stations because of the parallel standby arrangement, and does not affect the system operation completely unless both the units fail. Six years of real maintenance data on component failures, repairs and associated costs are used in this analysis. Measures of system effectiveness is gauged through reliability indices such as mean time to plant failure, availability of the plant, busy period of repairman and expected number of repairs. Effect of repair rate, failure rate and repair cost on system performance w.r.t. revenue is shown graphically. Theory of Semi-Markov and regenerative stochastic processes is used in the analysis.

The reliability of solar energy generating system with inverters in series under common cause failure

• We model the system with the Semi-Markov process theory and the renewal point technique. • A efficient algorithm based on set partition method is presented. • The transient and the steady-state reliability indices are explicitly obtained. • The steady-state availability is unrelated with arbitrary initial condition is demonstrated. With the shortage of nonrenewable energy, renewable energy resources play an important role in practice. One of the most promising renewable energy is viewed as solar energy resources due to its cleanliness and the inexhaustibility. However, the quantitative assessment method on the reliability of a complicated solar energy generating system has been a considerable challenge. In this paper, we propose a Markov model to imitate a solar energy generating system with inverters in series, unreliable by-pass change-over switches, and common cause failure . Employing the renewal point technique and the Semi-Markov process theory, we build the quantitative reliability models of the system. Further, we develop a new algorithm to obtain the explicit form solution of reliability indices. An example illustrates that the proposed models and the algorithm are efficient and concise.

Modeling and performance evaluation of jamming-tolerant wireless systems

Many of the failures suffered by wireless systems are not accidental; they are actually caused by security breaches. Therefore, the conventional QoS measures are not sufficient for quantifying the performance of wireless systems. In fact, there is a need for security-related QoS measures that describe the performance of wireless systems under security-breaches. In this paper we focus on Jamming which is one of the most common security breaches in wireless systems. We propose a mathematical model for jamming-tolerant wireless systems. Our model is based on the theory of semi-Markov processes (SMPs). By analyzing the steady-state behavior of the proposed SMP, we derive security-related performance measures such as the steady-state availability and the “mean time to process a packet” (MTTPP). By studying the transient behavior of the proposed SMP, we derive the “probability of packet failure” (PPF) and the “mean time to packet failure” (MTTPF) performance measures. With the proposed model, we analyze the performance of a real jamming-tolerant wireless system for smart grid applications and the numerical results are presented.

Approach to analysis of technical -organizational system functioning on the base of semi-markov process theory

Relevance of the present task lays in optimization of control over complex systems considering probabilistic and temporal nature of their functioning. Allocated some generic States of the system's normal operation and consider destabilizing situations when the continued functioning of the element in the system becomes difficult, impossible. Including the status when it is necessary to conduct full diagnostic and restoring of the system with explicit damage which makes it impossible for the system to operate in the acceptable mode into the model is described in the work. The proposed solution to this task is based on the mathematical modelling. Considering the general case of nonexponential time of system residence in its own status, the proposed functioning model displays relations between system statuses and probable parameters of its functioning on the base of semi-Markov process theory. Because of this work the explanation of an adequate description of complex system functioning at probabilistic and temporal analysis was presented.