Markov Birth-death Process Research Articles

Accounting for passenger loss in bus bunching reduction: a robust real-time speed control method

ABSTRACT Bus bunching is a common phenomenon resulting from the inherent instability influenced by factors such as traffic and passenger behaviour. However, the issue of passenger loss caused by extended waiting time, is often overlooked despite being a significant concern for passengers. This paper addresses this problem by modelling passenger arrival and loss as a Markov birth–death process. We present a transit route model that incorporates passenger loss while considering additional factors, such as stochasticity, overtaking, passenger alighting, and bus capacity constraints. The primary objective of our model is to minimise passenger loss. To achieve this, we conduct an analysis about the characteristics of passenger loss. Under mild assumptions, we prove that passenger loss increases as the headways between buses exhibit greater variance. We also develop a robust optimisation (RO) method and corresponding algorithms to minimise passenger loss. To validate the effectiveness of our control method, we conduct numerical experiments using both computer-generated and real-world data. The results demonstrate that our method effectively reduces bus bunching, passenger loss, and waiting time compared to the existing literature.

Availability evaluation of solar photovoltaic systems using markov modeling and cuckoo search algorithm

The main objective of present investigation is to evaluate and optimize the operational availability of the solar photovoltaic systems. As the solar energy is a prominent source of renewal energy and contribute a lot in global development having less environmental impacts but the safety and reliability issues of these systems also observed during the operational phase. Availability is an effective tool that is used to discourse the safety and performance issues of renewal energy sources especially solar photovoltaic systems. Here, a stochastic model is developed for solar photovoltaic system having solar photovoltaic plates, solar charger, solar battery, and inverter. The Markov birth-death process is applied for development of the mathematical model of the proposed system. The chapman-Kolmogorov differential difference equations of the proposed solar photovoltaic system used to predict the steady state availability of system. On the basis of literature, the failure and repair rates of all components of solar photovoltaic system are considered as exponentially distributed. In addition, an effort is also made to predict the optimum availability of solar photovoltaic system using well-known optimization technique cuckoo search algorithm. It is revealed that, the predicted availability of the solar photovoltaic system is 0.9988799 at population size 60 after 700 iterations. The estimated parametric values of the failure and repair rates also derived. To highlight the importance of the study the numerical and graphical results are presented and shared with the system designers and maintenance engineers.

Quantifying Reliability Indices of Garbage Data Collection IOT-based Sensor Systems using Markov Birth-death Process

The aim of this paper is to analyze the performance of a Garbage data collecting sensor network system (GDCSNS) through mathematical modelling and a reliability approach. The reliability assessment of such a system is essential to ensuring that it can collect data related to garbage at different locations consistently and accurately. After the determination of the reliability measures of the system, the next aim is to identify the weakest sensors of the system so that a timely maintenance strategy for the weakest sensor can be planned to avoid disruption in the collection of data from the sensor systems. In the considered system, three sensors have been installed at various location in the city that send the information to the center office (hub point) and then from the center office to the person who is responsible for collecting the garbage from the location and dumping it at some predefined places. These sensors collect data related to garbage level, weight, and other information and send it to computers at the city's central office. Markov modelling has been used to model the system. Based on the mathematical model, a state transition diagram and a set of Kolmogorov time-dependent differential equations have been obtained. The various state probabilities (explicit expressions) related to the performance of the system, namely, Reliability, Mean time to failure, have been obtained to understand the different maintenance policies that can be used. A sensitivity analysis has also been performed to determine the weakest sensor among the sensors.

Availability optimization of power generating units used in sewage treatment plants using metaheuristic techniques.

Metaheuristic techniques have been utilized extensively to predict industrial systems' optimum availability. This prediction phenomenon is known as the NP-hard problem. Though, most of the existing methods fail to attain the optimal solution due to several limitations like slow rate of convergence, weak computational speed, stuck in local optima, etc. Consequently, in the present study, an effort has been made to develop a novel mathematical model for power generating units assembled in sewage treatment plants. Markov birth-death process is adopted for model development and generation of Chapman-Kolmogorov differential-difference equations. The global solution is discovered using metaheuristic techniques, namely genetic algorithm and particle swarm optimization. All time-dependent random variables associated with failure rates are considered exponentially distributed, while repair rates follow the arbitrary distribution. The repair and switch devices are perfect and random variables are independent. The numerical results of system availability have been derived for different values of crossover, mutation, several generations, damping ratio, and population size to attain optimum value. The results were also shared with plant personnel. Statistical investigation of availability results justifies that particle swarm optimization outdoes genetic algorithm in predicting the availability of power-generating systems. In present study a Markov model is proposed and optimized for performance evaluation of sewage treatment plant. The developed model is one that can be useful for sewage treatment plant designers in establishing new plants and purposing maintenance policies. The same procedure of performance optimization can be adopted in other process industries too.

Process modeling for decomposition unit of a UFP for reliability indices subject to fail-back mode and degradation

PurposeEnsuring safe operation of a urea fertilizer plant (UFP) is a vital aspect for its functioning and production. Clearly the safe operation of such systems can only be archived with proper and effective maintenance scheduling and through controlling its failures as well as repairs of the components. Also for this, the concern plant management must have the information regarding the failures that affects the system's performance most/least. The objective of this study is to analyze mathematically the factors that are responsible for the failure/degradation of the decomposition unit of UFP.Design/methodology/approachThe considered system has been modeled by the aid of Markov's birth–death process with two types of failures for its components: variable (which are very similar in practical situations) and constant. The mathematical model is solved by the help of Laplace transform and supplementary variable technique.FindingsIn the present paper, the availability, reliability and mean time to failure (MTTF) are computed for the decomposition unit of the UFP. The critical components that affect the reliability and MTTF of the decomposition unit are identified through sensitivity analysis.Originality/valueIn this paper, a mathematical model based on the working of the decomposition unit of a UFP has been developed by considering two types of failure, namely, variable failures rates and constant failure rates (which has not been done in the literature for the decomposition unit). Conclusions in this paper are good references for the improvement of the same.

A fast and efficient analytical method for throughput evaluation of unreliable series-parallel production lines

In this paper, an efficient throughput evaluation method for large series-parallel production lines is developed. The parallel machines follow the exponential reliability model and are not necessarily identical. The proposed method considers different aggregation techniques to transform each group of parallel machines into an equivalent station. Then, the throughput of the obtained buffered serial production line is evaluated using an analytical approach. This evaluation method is based on the analysis of buffer states using a birth-death Markov process. The strength of the approach lies in the analysis of only full and empty buffer states rather than all buffer states. Consequently, through the comparative numerical study against simulation and recent literature, the proposed approach demonstrates its accuracy and efficiency in evaluating throughput of large complex series-parallel systems in only few seconds. In addition, aggregation techniques used are compared and some relevant theoretical properties are derived.

Reliability, Availability, Maintainability, and Dependability Analysis of Cold Standby Series-Parallel System

This paper dealt with the evaluation reliability, availability, maintainability, dependability, mean time between failures, and mean time to failure of series-parallel system. The system under investigation has four subsystems, namely subsystem A containing two units in cold standby, subsystem B and C possess one unit each, and subsystem D has two units in cold standby. Through the transition diagram of each subsystem and the Markov birth–death process, Chapman Kolmogorov forward equations are derived. Both failure and repair rates of units in each subsystem are assumed to follow exponential distribution. The objective is to derive the corresponding reliability models of dependability, availability, maintainability, and reliability and capture the effect of system parameters on reliability, availability, maintainability, and dependability and to determine the critical subsystems. On the basis of numerical results obtained, the system’s performance has been evaluated. Moreover, the outcome of this study shows that the ideal system reliability can be achieved when the overall system failure rate is low and the supporting units are activated.

Formulas for average transition times between states of the Markov birth-death process

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Performance Analysis of Photovoltaic Systems Using (RAMD) Analysis

The primary aim of this present study is to examine how reliability, availability, maintainability, and dependability (RAMD) are used to describe the criticality of each sub-assembly in grid- connected photovoltaic systems. A transition diagram of all subsystems is produced for this analysis, and Chapman-Kolmogorov differential equations for each variable of each subsystem are constructed using the Markov birth-death process. Both random failure and repair time variables have an exponential distribution and are statistically independent. A sufficient repair facility is still available with the device. The numerical results for reliability, maintainability, dependability, and steady-state availability for various photovoltaic device components have been obtained. Other metrics, such as mean time to failure (MTTF), mean time to repair (MTTR), and dependability ratio, which aid in device performance prediction, have also been measured. According to numerical analysis. it is hypothesized that subsystem S4, i.e. the inverter, is the most critical and highly sensitive portion that requires special attention in order to improve the efficiency of the PV device plant. The findings of this research are very useful for photovoltaic system designers and maintenance engineers.

Statistical approach for analyzing the transient reliability of boiler System using CAS Mathematica

The focal point of this manuscript is to address the reliability analysis of Boiler unit in the Thermal power plant driven from coal. It is made up of six subunits laid out in series and parallel manner, alongwith some standby units. The system under consideration is modelled in transient state aided by Markov birth-death process and CAS Mathematica is deployed to solve the resulting complicated system of probabilistic equations arising out of 8 working states.

SIS Epidemic Model Birth-and-Death Markov Chain Approach

We are interested in describing the dynamics of the infected size of the SIS Epidemic model using the Birth-Death Markov process. The Susceptible-Infected-Susceptible (SIS) model is defined within a population of constant size M; the size is kept constant by replacing each death with a newborn healthy individual. The life span of each individual in the population is modelled by an exponential distribution with parameter α; the disease spreads within the population is modelled by a Poisson process with a rate λ_I. λ_I=βI(1-I/M)  is similar to the instantaneous rate in the logistic population growth model. The analysis is focused on the disease outbreak, where the reproduction number (R=β/α) is greater than one. As methodology, we use both numerical and analytical approaches. The numerical approach shows that the infected size dynamics converge to a stationary stochastic process. And the analytical results determine the distribution of the stationary stochastic process as a normal distribution with mean (1-1/R)M and Variance M/R when M becomes larger.

Переходные вероятности марковского процесса на отрезке, лежащем в четверти плоскости

The paper considers a quadratic birth-death Markov process. The points on a line segment located within a quarter-plane represent the states of the random process. We designate the set of vectors that have integer non-negative coordinates as our quarter plane. The process is defined by infinitesimal characteristics, or transition probability densities. These characteristics are determined by a quadratic function of the coordinates at the segment points with integer coordinates. The boundary points of the segment are absorbing; at these points, the random process stops. We investigated a critical case when process jumps are equally probable at the moment of exiting a point. We derived expressions describing transition probabilities of the Markov process as a spectral series. We used a two-dimensional exponential generating function of transition probabilities and a two-dimensional generating function of transition probabilities. The first and second systems of ordinary differential Kolmogorov equations for Markov process transition probabilities are reduced to second-order mixed type partial differential equations for a double generating function. We solve the resulting system of linear equations using separation of variables. The spectrum obtained is discrete. The eigen-functions are expressed in terms of hypergeometric functions. The particular solution constructed is a Fourier series, whose coefficients are derived by means of expo-nential expansion. We employed sums of functional series known in the theory of special functions to construct the exponential expansion required

Simulation modelling for performance optimisation of the skim milk powder production system of a dairy plant

The purpose of this paper is to deal with the simulation modelling and performance optimisation for the skim milk powder production system of a dairy plant using genetic algorithm (GA). This analysis is based on Markov birth-death process, the mathematical formulation of the system is carried out to derive first order Chapman-Kolmogorov differential equations by assuming that the failure and repair rate of each subsystem follows exponential distribution. The proposed method is used to compute long run availability using normalising condition, initial boundary conditions and recursive method. It provides the optimum system availability level for different combinations of failure and repair rates parameters of the subsystems of the skim milk powder production system of a dairy plant concerned. The results are presented and discussed with the plant personnel to adopt and practice suitable maintenance policies/strategies to enhance the productivity and quality of skim milk powder production system of the dairy plant.

Investigation of performance measures of power generating unit of sewage treatment plant

The main objective of present study to investigate the availability and profit of power generation systems established in sewage treatment plants. The sewage treatment plant is an industry in which waste sewage water has been treated and waste is used to generate power. For this purpose, a mathematical model has been proposed by considering constant failure and repair rates. The power generating unit in sewage treatment plant comprises six subsystems as sludge digester, gas holding tank, gas burner, gas scrubber, gas engine and power generation. By using appropriate redundancy technique in model development Chapman-Kolmogorov differential equations has been drawn with the help of the Markov birth death process. Availability and profit have been analysed based on making variation in the failure and repair rates. The numerical and graphical results have been depicted for a particular case. The derived results have been discussed with system designers and found useful.

Modeling and availability analysis of data center: a fuzzy approach

Data Centers are the backbone of any industry that provides a specialized environment to safeguard the company's valuable equipment and intellectual property. The successful operation of company's Data Center is typically shared among multiple departments and personnel. But, the involvement of multiple personal and departments make its configuration complex. In such situations, reliability of data center become a necessary factor. But, the traditional theory of reliability is based on the Bernoulli trials, i.e., either operative or failure. But this situation is unrealistic in case of complex systems like data center. To rectify this problem, here a mathematical model has been developed using the concept of fuzzy reliability. All the failure and repair rates are exponential distributed along with coverage factor. Chapman-Kolmogorov differential equations have been developed for the fuzzy system using Markov birth–death process. A new methodology Runge–Kutta method of order four has been used to solve Chapman-Kolmogorov differential equations using MATLAB (Ode 45 function).

An efficient message broadcasting MAC protocol for VANETs

Future vehicular communication systems must provide inter-vehicle communication to ensure driving safety, and broadcasting safety messages to neighboring vehicles is a critical strategy. However, due to high mobility and variable topology, collisions may occur in message broadcasting. Moreover, channel utilization is also a great challenge for protocol design in a large-scale vehicular network. In this paper, we propose a novel frame structure and then design an efficient messages broadcasting MAC protocol, which allows beacon and emergency messages to share the channel and thus saves bandwidth resources. By distinguishing the priority of these two types messages and setting channel resource preemption mechanism, the protocol greatly reduces message collision probability and meets the latency requirements. In addition, we use Markov model and birth-death process to analyze the performance of channel access and emergency latency, respectively. Finally, simulation results demonstrate the superior performance of the proposed protocol.

Operational availability analysis of generators in steam turbine power plants

The main aim of this study is to explore the availability and profitability of generators, which are key units of steam turbine power plants, considering exponential failure and arbitrary repair time distributions. A Markov birth–death process and supplementary variable technique are used to evaluate system effectiveness measures. Such a generator mainly consists of seven repairable components arranged in a series configuration. A stochastic model of the generator is developed, and the Chapman–Kolmogorov differential difference equation of the system derived. Finally, the results for the steady-state availability and profitability of the system are obtained for a particular case that will be helpful to system designers to enhance the performance of plants to some extent.

Analysis of the Reliability of the Butter-Oil Processing Plant using CAS Mathematica and Maxima

The aim of the present paper is to analysis the reliability of the system by using CAS Mathematica and also, it’s comparative study with CAS Maxima. The butter oil manufacturing plant consists of seven units i.e. separator, pasteurizer, continuous butter making, melting vats, butter oil clarifier, packaging and standby state. Model is developed by using Markov birth-death process. The first order differential equations are derived by using model and then solved for comparative study of reliability to give accuracy in result. By using Mathematica, we calculate probability of each state, which is beneficial to plant owners because better accuracy in result enhance reliability of the plant Graphs are plotted and tables are developed with the help of CAS Mathematica and Maxima, graphs show the variation and tables shows the fluctuations in reliability in comparative manner.

A study of microprocessor systems using RAMD approach

The aim of the present study is to offer results of reliability, availability, maintainability and dependability analysis of a microprocessor system. A microprocessor is designed with the help of seven subsystems including input, terminal, processor, main memory disk, addressing modes, disk controller and disk drives. Reliability, availability, maintainability and dependability analysis has been performed for all the subsystems. For analysis purpose, state transition diagram has been designed for all subsystems and using Markov birth–death process governing equations has been derived. All random variables are exponentially distributed, i.e., follows constant failure rate. Mean time between failures, mean time to repair, availability, reliability, maintainability and dependability ratio parameters for all the subsystems are computed. The reliability and maintainability are estimated at various time instants.

PERFORMABILITY MODELING AND DECISION-MAKING REGARDING MAINTENANCE PRIORTIES FOR POWER GENERATION SYSTEM OF A TYPICAL THERMAL POWER PLANT

The aim of existing paper is to develop performability model and to decide about maintenance priorities for power generation system (PGS) of a typical thermal power plant. PGS comprises of five subsystems, i.e., turbine blades, condenser evacuation and regenerative (CER), turbine governing (TG), turbine lubrication (TL) and generator cooling and seal oil (GCSO). The first order differential equations are derived with the help of transition diagram. After that, these differential equations are further solved recursively and performability model obtained by using normalising condition on the basis of Markov birth-death process. The performability of various subsystems is considered by the various combinations of repair and failure rates. Repair rates are required to decide about criticality of subsystems and propose maintenance priorities. CER is the most critical subsystem with performability level, i.e., 94.97% and require more consideration to improve the performance of this subsystem. Performability enhancement and reduction in maintenance cost can also be achieved by the application of both performability model and decision making regarding maintenance priorities in the concerned plant.