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.

Determining Risk Factors That Affect Progression in Patients with Nonproliferative Diabetic Retinopathy

Purpose. To determine risk factors that affect nonproliferative diabetic retinopathy (NPDR) progression and establish a predictive model to estimate the probability of and time to progression in NPDR. Patients and Methods. Charts of diabetic patients who received an initial eye exam between 2010 and 2017 at our county hospital were included. Patients with proliferative diabetic retinopathy (PDR), fewer than 2 years of follow-up, or fewer than 3 clinic visits were excluded. Demographics and baseline systemic and ocular characteristics were recorded. Follow-up mean annual HbA1c and blood pressure, best-corrected visual acuity, and the number of antivascular endothelial growth factor treatments were recorded. Stage and date of progression were recorded. A 5-state nonhomogeneous continuous-time Markov chain with a backward elimination model was used to identify risk factors and estimate their effects on progression. Results. Two hundred thirty patients were included. Initially, 65 eyes (28.3%) had no retinopathy; 73 (31.7%) mild NPDR; 60 (26.1%) moderate NPDR; and 32 (13.9%) severe NPDR. Patients were followed for a mean of 5.8 years (±2.0 years; range 2.1–9.4 years). 164 (71.3%) eyes progressed during the follow-up. Time-independent risk factors affecting progression rate were age (hazard ratio (HR) = 0.99, P = 0.047 ), duration of diabetes (HR = 1.02, P = 0.018 ), and Hispanic ethnicity (HR = 1.31, P = 0.068 ). Mean sojourn times at mean age, duration of diabetes, and annual HbA1c for a non-Hispanic patient were estimated to be 3.03 (±0.97), 4.63 (±1.21), 6.18 (±1.45), and 4.85 (±1.25) years for no retinopathy, mild NPDR, moderate NPDR, and severe NPDR, respectively. Each 1% increase in HbA1c annually diminished sojourn times by 15%, 10%, 7%, and 10% for no retinopathy, mild NPDR, moderate NPDR, and severe NPDR, respectively. Conclusion. HbA1c level is a significant modifiable risk factor in controlling the progression of DR. The proposed model could be used to predict the time and rate of progression based on an individual’s risk factors. A prospective multicenter study should be conducted to further validate our model.

Reliability and Inference for Multi State Systems: The Generalized Kumaraswamy Case

Semi-Markov processes are typical tools for modeling multi state systems by allowing several distributions for sojourn times. In this work, we focus on a general class of distributions based on an arbitrary parent continuous distribution function G with Kumaraswamy as the baseline distribution and discuss some of its properties, including the advantageous property of being closed under minima. In addition, an estimate is provided for the so-called stress–strength reliability parameter, which measures the performance of a system in mechanical engineering. In this work, the sojourn times of the multi-state system are considered to follow a distribution with two shape parameters, which belongs to the proposed general class of distributions. Furthermore and for a multi-state system, we provide parameter estimates for the above general class, which are assumed to vary over the states of the system. The theoretical part of the work also includes the asymptotic theory for the proposed estimators with and without censoring as well as expressions for classical reliability characteristics. The performance and effectiveness of the proposed methodology is investigated via simulations, which show remarkable results with the help of statistical (for the parameter estimates) and graphical tools (for the reliability parameter estimate).

Simulation of epidemic models with generally distributed sojourn times

Epidemic models are used to analyze the progression or outcome of an epidemic under different control policies like vaccinations, quarantines, lockdowns, use of face-masks, pharmaceutical interventions, etc. When these models accurately represent real-life situations, they may become an important tool in the decision-making process. Among these models, compartmental models are very popular and assume individuals move along a series of compartments that describe their current health status. Nevertheless, these models are mostly Markovian, that is, the time in each compartment follows an exponential distribution. Here, we introduce a novel approach to simulate general stochastic epidemic models that accepts any distribution for the sojourn times.

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.

Monte Carlo Simulation Approach to Reliability Analysis of Complex Systems

The article presents new results concerned with general procedures and algorithms to assess the reliability of complex systems with various reliability structures. The analytical method and based on it the simulation method were used to estimate the reliability characteristics of the port grain transportation system. Finally, the general simulation algorithm was developed to evaluate the reliability characteristics of ageing complex systems. In this case, the systems operating processes were described by any distributions of sojourn times in operation states and the reliability functions of their components were modified in such a way that these components are not characterized by a “lack of memory”. The application of this algorithm has been illustrated by the results for exemplary complex two-state systems.

Quantum time delay for unitary operators: General theory

We present a suitable framework for the definition of quantum time delay in terms of sojourn times for unitary operators in a two-Hilbert spaces setting. We prove that this time delay defined in terms of sojourn times (time-dependent definition) exists and coincides with the expectation value of a unitary analogue of the Eisenbud–Wigner time delay operator (time-independent definition). Our proofs rely on a new summation formula relating localization operators to time operators and on various tools from functional analysis such as Mackey’s imprimitivity theorem, Trotter–Kato formula and commutator methods for unitary operators. Our approach is general and model-independent.