scholarly journals Functioning of a technical system with an instantly replenished time reserve, taking into account preventive maintenance

2021 ◽  
pp. 258-264
Author(s):  
А.Л. Боран-Кешишьян ◽  
М.В. Заморёнов ◽  
П.Н. Флоря ◽  
А.А. Ярошенко ◽  
С.И. Кондратьев
Keyword(s):  
Transition Probabilities ◽  
Distribution Functions ◽  
Technical System ◽  
Embedded Markov Chain ◽  
State Graph ◽  
Total Probability ◽  
System Availability ◽  
Sojourn Times ◽  
Trajectory Method ◽  
Total Probability Theorem

В работе рассматривается функционирование технической системы с мгновенно пополняемым резервом времени с учетом профилактики. Приводится описание функционирования такой системы. При использовании аппарата полумарковских исследований производится построение аналитической модели системы с мгновенно пополняемым резервом времени при учете влияния профилактики на ее производительность. При построении полумарковской модели принимается ограничение на количество профилактик за время восстановления рабочего элемента. Описываются полумарковские состояния исследуемой системы, и приводится граф состояний. Определяются времена пребывания в состояниях системы, вероятности переходов и стационарное распределение вложенной цепи Маркова. Для определения функции распределения времени пребывания системы в подмножестве работоспособных состояний с использованием метода траекторий находятся все траектории переходов системы из этого подмножества в подмножество неработоспособных состояний и вероятности их реализации. Определяются времена пребывания системы в найденных траекториях. На основании теоремы полной вероятности определяются функции распределения времен пребывания системы в подмножествах работоспособных и неработоспособных состояний и коэффициент готовности системы. Приводится пример моделирования исследуемой системы. Проводится сравнение полученных результатов с результатами использования теоремы о среднестационарном времени пребывания системы в подмножестве состояний. The work examines the functioning of a technical system with an instantly replenished reserve of time, taking into account prevention. The description of the functioning of such a system is given. When using the apparatus of semi-Markov studies, an analytical model of the system is constructed with an instantly replenished reserve of time, taking into account the effect of prevention on its performance. When constructing a semi-Markov model, a limitation on the number of preventive measures during the restoration of a working element is adopted. The semi-Markov states of the system under study are described, and the state graph is given. The sojourn times in the states of the system, the transition probabilities, and the stationary distribution of the embedded Markov chain are determined. To determine the distribution function of the time spent by the system in a subset of operable states using the trajectory method, all trajectories of the system's transitions from this subset to the subset of inoperable states and the probability of their realization are found. The residence times of the system in the found trajectories are determined. On the basis of the total probability theorem, the distribution functions of the sojourn times of the system in subsets of the healthy and inoperable states and the system availability factor are determined. The modeling example of th system under study is given. The results obtained are compared with the results of using the theorem on the average stationary sojourn time of the system in a subset of states.

scholarly journals On the Time to Buffer Overflow in a Queueing Model with a General Independent Input Stream and Power-Saving Mechanism Based on Working Vacations

Sensors ◽  
2021 ◽  
Vol 21 (16) ◽  
pp. 5507
Author(s):  
Martyna Kobielnik ◽  
Wojciech Kempa
Keyword(s):  
Distribution Functions ◽  
Cumulative Distribution ◽  
Buffer Overflow ◽  
Embedded Markov Chain ◽  
Total Probability ◽  
Single Server ◽  
Input Stream ◽  
Initial State ◽  
Transient Time ◽  
The Impact

A single server GI/M/1 queue with a limited buffer and an energy-saving mechanism based on a single working vacation policy is analyzed. The general independent input stream and exponential service times are considered. When the queue is empty after a service completion epoch, the server lowers the service speed for a random amount of time following an exponential distribution. Packets that arrive while the buffer is saturated are rejected. The analysis is focused on the duration of the time period with no packet losses. A system of equations for the transient time to the first buffer overflow cumulative distribution functions conditioned by the initial state and working mode of the service unit is stated using the idea of an embedded Markov chain and the continuous version of the law of total probability. The explicit representation for the Laplace transform of considered characteristics is found using a linear algebra-based approach. The results are illustrated using numerical examples, and the impact of the key parameters of the model is investigated.

scholarly journals Estimation of semi-Markov multi-state models: a comparison of the sojourn times and transition intensities approaches

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Azam Asanjarani ◽  
Benoit Liquet ◽  
Yoni Nazarathy
Keyword(s):  
Survival Analysis ◽  
Markov Process ◽  
Transition Probabilities ◽  
Embedded Markov Chain ◽  
Hazard Rates ◽  
Efficient Computation ◽  
Sojourn Times ◽  
Transition Functions ◽  
Semi Markov Process ◽  
State Models

Abstract Semi-Markov models are widely used for survival analysis and reliability analysis. In general, there are two competing parameterizations and each entails its own interpretation and inference properties. On the one hand, a semi-Markov process can be defined based on the distribution of sojourn times, often via hazard rates, together with transition probabilities of an embedded Markov chain. On the other hand, intensity transition functions may be used, often referred to as the hazard rates of the semi-Markov process. We summarize and contrast these two parameterizations both from a probabilistic and an inference perspective, and we highlight relationships between the two approaches. In general, the intensity transition based approach allows the likelihood to be split into likelihoods of two-state models having fewer parameters, allowing efficient computation and usage of many survival analysis tools. Nevertheless, in certain cases the sojourn time based approach is natural and has been exploited extensively in applications. In contrasting the two approaches and contemporary relevant R packages used for inference, we use two real datasets highlighting the probabilistic and inference properties of each approach. This analysis is accompanied by an R vignette.

scholarly journals Degrading systems availability analysis: analytical semi-Markov approach

2021 ◽  
Vol 23 (1) ◽  
pp. 195-208
Author(s):  
Varun Kumar ◽  
Girish Kumar ◽  
Rajesh Kumar Singh ◽  
Umang Soni
Keyword(s):  
Steady State ◽  
Degradation Mechanism ◽  
Continuous Process ◽  
Mechanical Systems ◽  
Embedded Markov Chain ◽  
Steady State Probability ◽  
Opportunistic Maintenance ◽  
System Availability ◽  
Availability Analysis ◽  
Complex Mechanical Systems

This paper deals with modeling and analysis of complex mechanical systems that deteriorate with age. As systems age, the questions on their availability and reliability start to surface. The system is believed to suffer from internal stochastic degradation mechanism that is described as a gradual and continuous process of performance deterioration. Therefore, it becomes difficult for maintenance engineer to model such system. Semi-Markov approach is proposed to analyze the degradation of complex mechanical systems. It involves constructing states corresponding to the system functionality status and constructing kernel matrix between the states. The construction of the transition matrix takes the failure rate and repair rate into account. Once the steady-state probability of the embedded Markov chain is computed, one can compute the steady-state solution and finally, the system availability. System models based on perfect repair without opportunistic and with opportunistic maintenance have been developed and the benefits of opportunistic maintenance are quantified in terms of increased system availability. The proposed methodology is demonstrated for a two-stage reciprocating air compressor with intercooler in between, system in series configuration.

scholarly journals Analysis of the execution time of multi-stage control operations with multiple measurements

2018 ◽  
Vol 224 ◽  
pp. 04021
Author(s):  
Vadim Kopp ◽  
Alexey Balakin ◽  
Natalya Balakina ◽  
Mikhail Zamoryonov
Keyword(s):  
Markov Model ◽  
Execution Time ◽  
Distribution Functions ◽  
Technical System ◽  
Measurement Time ◽  
Production Structure ◽  
Multi Stage ◽  
Output Flow ◽  
Multiple Measurements ◽  
Random Nature

Questions related to the management of the process of multiple measurements are considered, depending on the result, which affects their number. In this case, the measurement time can be significantly increased. The above algorithm for controlling the measurement process justifies the random nature of the duration of its execution, which entails the need to study the time of carrying out multiple measurements. The paper constructs a semi-Markov model that allows to determine the distribution functions of time between events in the output flow of products after measurements, which makes it possible to dock this model with models of elements of higher levels of the hierarchy of the production structure. In the model, it is taken into account that some part of the production after a given number of measurements is defective and leaves the system. The constructed semi-Markov model allows to predict the performance of a technical system associated with multiple measurements.

Time-Dependent Reliability Analysis of Vibratory Systems With Random Parameters

2016 ◽  
Vol 138 (3) ◽  
Author(s):  
Zissimos P. Mourelatos ◽  
Monica Majcher ◽  
Vasileios Geroulas
Keyword(s):  
Large Scale ◽  
Probability Of Failure ◽  
Time Dependent ◽  
Total Probability ◽  
Conditional Probabilities ◽  
Random Parameters ◽  
Random Vibrations ◽  
Input Parameters ◽  
Total Probability Theorem ◽  
Dependent Probability

The field of random vibrations of large-scale systems with millions of degrees-of-freedom (DOF) is of significant importance in many engineering disciplines. In this paper, we propose a method to calculate the time-dependent reliability of linear vibratory systems with random parameters excited by nonstationary Gaussian processes. The approach combines principles of random vibrations, the total probability theorem, and recent advances in time-dependent reliability using an integral equation involving the upcrossing and joint upcrossing rates. A space-filling design, such as optimal symmetric Latin hypercube (OSLH) sampling, is first used to sample the input parameter space. For each design point, the corresponding conditional time-dependent probability of failure is calculated efficiently using random vibrations principles to obtain the statistics of the output process and an efficient numerical estimation of the upcrossing and joint upcrossing rates. A time-dependent metamodel is then created between the input parameters and the output conditional probabilities allowing us to estimate the conditional probabilities for any set of input parameters. The total probability theorem is finally applied to calculate the time-dependent probability of failure. The proposed method is demonstrated using a vibratory beam example.

Reliability Model of Gear With Correlated Failure Modes Based on Joint Distribution

2014 ◽  
Author(s):  
Lai Yue-Hua ◽  
Dong Hai-Ping ◽  
Yi Xiao-Jian ◽  
Ding Juan ◽  
Lei Hua-Jin
Keyword(s):  
Joint Distribution ◽  
Failure Modes ◽  
Physical Models ◽  
Total Probability ◽  
Traditional Model ◽  
Reliability Model ◽  
Interference Theory ◽  
Calculation Results ◽  
Set Up ◽  
Total Probability Theorem

In this paper, a precise reliability model for gear with correlated failure modes is presented. Firstly, physical models of stress and strength are set up against two main failure modes of a gear respectively. Then, the corresponding performance functions to the two failure modes are obtained according to stress-strength interference theory regarding randomness of variables in physical models of stress and strength. Furthermore, joint distribution of the two performance functions is deduced by Total Probability Theorem considering the correlation of random variables. So, reliability of a gear with correlated failure modes can be computed based on the joint distribution. Finally, an example is given and in this example the precise reliability model based on joint distribution and the traditional reliability model without considering the correlation of failure modes, are respectively adopted to calculate reliability of a gear. The calculation results are compared with that by Monte Carlo simulation and the compared results show that the gear’s reliability obtained by considering correlation of failure modes based on joint distribution is more accurate than that by the traditional model without considering the correlation of failure modes.

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