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

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.

On the stability of queues with the dropping function

In this paper, the stability of the queueing system with the dropping function is studied. In such system, every incoming job may be dropped randomly, with the probability being a function of the queue length. The main objective of the work is to find an easy to use condition, sufficient for the instability of the system, under assumption of Poisson arrivals and general service time distribution. Such condition is found and proven using a boundary for the dropping function and analysis of the embedded Markov chain. Applicability of the proven condition is demonstrated on several examples of dropping functions. Additionally, its correctness is confirmed using a discrete-event simulator.

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

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.

Optimal Design of Photovoltaic Connected Energy Storage System Using Markov Chain Models

This study improves an approach for Markov chain-based photovoltaic-coupled energy storage model in order to serve a more reliable and sustainable power supply system. In this paper, two Markov chain models are proposed: Embedded Markov and Absorbing Markov chain. The equilibrium probabilities of the Embedded Markov chain completely characterize the system behavior at a certain point in time. Thus, the model can be used to calculate important measurements to evaluate the system such as the average availability or the probability when the battery is fully discharged. Also, Absorbing Markov chain is employed to calculate the expected duration until the system fails to serve the load demand, as well as the failure probability once a new battery is installed in the system. The results show that the optimal condition for satisfying the availability of 3 nines (0.999), with an average load usage of 1209.94 kWh, is the energy storage system capacity of 25 MW, and the number of photovoltaic modules is 67,510, which is considered for installation and operation cost. Also, when the initial state of charge is set to 80% or higher, the available time is stable for more than 20,000 h.

Stochastic monotonicity approach for a non-Markovian priority retrial queue

This paper considers a non-Markovian priority retrial queue which serves two types of customers. Customers in the regular queue have priority over the customers in the orbit. This means that the customer in orbit can only start retrying when the regular queue becomes empty. If another customer arrives during a retrial time, this customer is served and the retrial has to start over when the regular queue becomes empty again. In this study, a particular interest is devoted to the stochastic monotonicity approach based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary joint distribution of the embedded Markov chain of the considered system.

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

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.

Degrading systems availability analysis: analytical semi-Markov approach

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.

TWO–COMPONENT RESTORABLE TOOL SYSTEM MULTI–PURPOSE MACHINE WITH AN INSTANTLY REFILLED TIME RESERVE

The object of the research is the technical system of a multipurpose machine tool, the tools of which can fail and be restored. A failed tool remains functional for some time due to a temporary reserve until a parametric failure occurs, the magnitude of which is random. All random variables describing the system have general distributions. The apparatus for constructing a mathematical model of the described system is a semi–Markov process with a discrete–continuous phase space of states. The stationary distribution of the embedded Markov chain is found explicitly. For systems with parallel connection, series connection with disconnection and without disconnection of elements, the stationary time between failures of the system, the stationary time spent in the state of failure and the stationary system availability factor are found. A numerical example shows the dependence of the stationary characteristics of the system on the size of the time reserve.

Stochastic analysis of a single server unreliable queue with balking and general retrial time

In this investigation, we consider an M/G/1 queue with general retrial times allowing balking and server subject to breakdowns and repairs. In addition, the customer whose service is interrupted can stay at the server waiting for repair or leave and return while the server is being repaired. The server is not allowed to begin service on other customers until the current customer has completed service, even if current customer is temporarily absent. This model has a potential application in various fields, such as in the cognitive radio network and the manufacturing systems, etc. The methodology is strongly based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary distribution of the embedded Markov chain of the considered system.

Alternative approach based on roots for computing the stationary queue-length distributions in GIX/M(1, b)/1 single working vacation queue

The purpose of this paper is to present an alternative algorithm for computing the stationary queue-length and system-length distributions of a single working vacation queue with renewal input batch arrival and exponential holding times. Here we assume that a group of customers arrives into the system, and they are served in batches not exceeding a specific number b. Because of batch arrival, the transition probability matrix of the corresponding embedded Markov chain for the working vacation queue has no skip-free-to-the-right property. Without considering whether the transition probability matrix has a special block structure, through the calculation of roots of the associated characteristic equation of the generating function of queue-length distribution immediately before batch arrival, we suggest a procedure to obtain the steady-state distributions of the number of customers in the queue at different epochs. Furthermore, we present the analytic results for the sojourn time of an arbitrary customer in a batch by utilizing the queue-length distribution at the pre-arrival epoch. Finally, various examples are provided to show the applicability of the numerical algorithm.