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Generic open quantum systems are notoriously difficult to simulate unless one looks at specific regimes. In contrast, classical dissipative systems can often be effectively described by stochastic processes, which are generally less computationally expensive. Here, we use the paradigmatic case of a dissipative quantum oscillator to give a pedagogic introduction to the modeling of open quantum systems using quasiclassical methods, i.e., classical stochastic methods that use a "quantum" noise spectrum to capture the influence of the environment on the system. Such quasiclassical methods have the potential to offer insights into the impact of the quantum nature of the environment on the dynamics of the system of interest while still being computationally tractable. [ABSTRACT FROM AUTHOR]

Spontaneous gas-bubble nucleation in weak gas–liquid solutions has been a challenging topic in theory, experimentation, and computer simulations. In analogy with recent advances in crystallization and droplet formation studies, the diffusive-shielding stabilization and thermodynamic irreversibility of bulk nanobubble (bNB) mechanisms are revisited and deployed to characterize nucleation processes in a stochastic framework of computer experiments using the large-scale atomic/molecular massively parallel simulator code. Theoretical bases, assumptions, and limitations underlying the irreversibility hypothesis of bNBs, and their computational counterparts, are extensively described and illustrated. In essence, it is established that the irreversibility hypothesis can be numerically investigated by converging the system volume (due to the finiteness of interatomic forces) and the initial dissolved-gas concentration in the solution (due to the single-bNB limitation). Helium nucleation in liquid Pb17Li alloy is selected as a representative case study, where it exhibits typical characteristics of noble-gas/liquid-metal systems. The proposed framework lays down the bases on which the stability of gas-bNBs in weak and supersaturated gas–liquid solutions can be inferred and explained from a novel perspective. In essence, it stochastically marches toward a unique irreversible state along out-of-equilibrium nucleation/growth trajectories. Moreover, it does not attempt to characterize the interface or any interface-related properties, neither theoretically nor computationally. It was concluded that bNBs of a few tens of He-atoms are irreversible when dissolved-He concentrations in the weak gas–liquid solution are at least ∼50 and ∼105 mol m−3 at 600 and 1000 K (and ∼ 80 MPa), respectively, whereas classical molecular dynamics -estimated solubilities are at least two orders of magnitude smaller. [ABSTRACT FROM AUTHOR]

We consider a stochastic capacitated lot sizing problem with in-line rework. Both defective and defect-free products can be the result of an imperfect production process in real-world production systems. However, the proportion of defective items in a production lot is subject to uncertainty. For economic and environmental reasons, the possibility to rework defective products appears reasonable since these products usually have considerable value. In this paper, a nonlinear model formulation for integrated lot sizing and rework planning is proposed. We use a sample average approach to approximate the generic nonlinear model formulation. To cope with the randomness of the proportion of defective products, we apply a flexible planning approach that allows the adjustment of production and rework quantities. We conduct extensive numerical investigations to evaluate the performance of the proposed planning approach. [ABSTRACT FROM AUTHOR]

Age and longevity are key parameters for demography and life-history evolution of organisms. In clonal species, a widespread life history among animals, plants, macroalgae and fungi, the sexually produced offspring (genet) grows indeterminately by producing iterative modules, or ramets, and so obscure their age. Here we present a novel molecular clock based on the accumulation of fixed somatic genetic variation that segregates among ramets. Using a stochastic model, we demonstrate that the accumulation of fixed somatic genetic variation will approach linearity after a lag phase, and is determined by the mitotic mutation rate, without direct dependence on asexual generation time. The lag phase decreased with lower stem cell population size, number of founder cells for the formation of new modules, and the ratio of symmetric versus asymmetric cell divisions. We calibrated the somatic genetic clock on cultivated eelgrass Zostera marina genets (4 and 17 years respectively). In a global data set of 20 eelgrass populations, genet ages were up to 1,403 years. The somatic genetic clock is applicable to any multicellular clonal species where the number of founder cells is small, opening novel research avenues to study longevity and, hence, demography and population dynamics of clonal species.

We consider a single-machine scheduling problem with due-date assignment and stochastic processing times, where only the mean and support (i.e. an interval bounded with lower and upper values) of processing times are known to the decision maker. The objective is to jointly determine a scheduling policy and a set of due dates for all jobs, so as to minimise the total expected individually weighted costs of earliness, tardiness and due-date assignment. By identifying an upper bound with the robust optimisation approach and a lower bound, and using a linear function of them to approximate the studied objective function, we establish an approximated problem. Then, a branch-and-bound algorithm is proposed to find an optimal solution for the approximated problem. Finally, a series of computational experiments are conducted to examine the performance of problem approximation and two developed heuristic algorithms. HIGHLIGHTS Study due-date assignment scheduling problem with stochastic processing times. Apply the mean and support (interval data) to model stochastic processing times. Use a linear function of identified lower and upper bounds to make an approximation. Derive optimal due-date assignment and develop branch-and-bound algorithms. Evaluate the efficiency of branch-and-bound algorithm and heuristic algorithms. [ABSTRACT FROM AUTHOR]

Cellular signaling, crucial for biological processes like immune response and homeostasis, relies on specificity and fidelity in signal transduction to accurately respond to stimuli amidst biological noise. Kinetic proofreading (KPR) is a key mechanism enhancing signaling specificity through time-delayed steps, although its effectiveness is debated due to intrinsic noise potentially reducing signal fidelity. In this study, we reformulate the theory of kinetic proofreading (KPR) by convolving multiple intermediate states into a single state and then define an overall processing time required to traverse these states. This simplification allows us to succinctly describe kinetic proofreading in terms of a single waiting time parameter, facilitating a more direct evaluation and comparison of KPR performance across different biological contexts such as DNA replication and T cell receptor (TCR) signaling. We find that loss of fidelity for longer proofreading steps relies on the specific strategy of information extraction and show that in the first-passage time (FPT) discrimination strategy, longer proofreading steps can exponentially improve the accuracy of KPR at the cost of speed. Thus, KPR can still be an effective discrimination mechanism in the high noise regime. However, in a product concentration-based discrimination strategy, longer proofreading steps do not necessarily lead to an increase in performance. However, by introducing activation thresholds on product concentrations, can we decompose the product-based strategy into a series of FPT-based strategies to better resolve the subtleties of KPR-mediated product discrimination. Our findings underscore the importance of understanding KPR in the context of how information is extracted and processed in the cell.

The aim of this study is to suggest a novel approach for estimating the intramolecular mobility of a charge carrier that migrates within a polymer chain and is involved in a pair reaction with a particle located on the same chain. The approach is based on the effect of an external electric field on the migration rate and, consequently, the kinetics of the reaction. As a first step, this problem is considered a stochastic one-step process with absorbing and reflecting boundaries, and an analytical solution is obtained in the case that the second reactant is immobile. With the use of computer simulations of stochastic migration, the effect of the mobility of both reactants and the influence of the Coulomb interaction between them are considered. It is found that the ratio of the pair reaction rates with and without an external field is relatively little dependent on these factors and that the analytical expressions derived can be applied to estimate the relative mobility of recombining particles with accuracy better than a factor of two in many realistic situations. [ABSTRACT FROM AUTHOR]

Generative Adversarial Networks (GANs) have shown immense potential in fields such as text and image generation. Only very recently attempts to exploit GANs to statistical-mechanics models have been reported. Here we quantitatively test this approach by applying it to a prototypical stochastic process on a lattice. By suitably adding noise to the original data we succeed in bringing both the Generator and the Discriminator loss functions close to their ideal value. Importantly, the discreteness of the model is retained despite the noise. As typical for adversarial approaches, oscillations around the convergence limit persist also at large epochs. This undermines model selection and the quality of the generated trajectories. We demonstrate that a simple multi-model procedure where stochastic trajectories are advanced at each step upon randomly selecting a Generator leads to a remarkable increase in accuracy. This is illustrated by quantitative analysis of both the predicted equilibrium probability distribution and of the escape-time distribution. Based on the reported findings, we believe that GANs are a promising tool to tackle complex statistical dynamics by machine learning techniques. [ABSTRACT FROM AUTHOR]

The brain produces diverse functions, from perceiving sounds to producing arm reaches, through the collective activity of populations of many neurons. Determining if and how the features of these exogenous variables (e.g., sound frequency, reach angle) are reflected in population neural activity is important for understanding how the brain operates. Often, high-dimensional neural population activity is confined to low-dimensional latent spaces. However, many current methods fail to extract latent spaces that are clearly structured by exogenous variables. This has contributed to a debate about whether or not brains should be thought of as dynamical systems or representational systems. Here, we developed a new latent process Bayesian regression framework, the orthogonal stochastic linear mixing model (OSLMM) which introduces an orthogonality constraint amongst time-varying mixture coefficients, and provide Markov chain Monte Carlo inference procedures. We demonstrate superior performance of OSLMM on latent trajectory recovery in synthetic experiments and show superior computational efficiency and prediction performance on several real-world benchmark data sets. We primarily focus on demonstrating the utility of OSLMM in two neural data sets: μECoG recordings from rat auditory cortex during presentation of pure tones and multi-single unit recordings form monkey motor cortex during complex arm reaching. We show that OSLMM achieves superior or comparable predictive accuracy of neural data and decoding of external variables (e.g., reach velocity). Most importantly, in both experimental contexts, we demonstrate that OSLMM latent trajectories directly reflect features of the sounds and reaches, demonstrating that neural dynamics are structured by neural representations. Together, these results demonstrate that OSLMM will be useful for the analysis of diverse, large-scale biological time-series datasets.

"Gating" is a widely observed phenomenon in biochemistry that describes the transition between the activated (or open) and deactivated (or closed) states of an ion-channel, which makes transport through that channel highly selective. In general, gating is a mechanism that imposes an additional restriction on a transport, as the process ends only when the "gate" is open and continues otherwise. When diffusion occurs in the presence of a constant bias to a gated target, i.e., to a target that switches between an open and a closed state, the dynamics essentially slow down compared to ungated drift-diffusion, resulting in an increase in the mean completion time, ⟨TG⟩ > ⟨T⟩, where T denotes the random time of transport and G indicates gating. In this work, we utilize stochastic resetting as an external protocol to counterbalance the delay due to gating. We consider a particle in the positive semi-infinite space that undergoes drift-diffusion in the presence of a stochastically gated target at the origin and is moreover subjected to rate-limiting resetting dynamics. Calculating the minimal mean completion time ⟨ T r ⋆ G ⟩ rendered by an optimal resetting rate r⋆ for this exactly solvable system, we construct a phase diagram that owns three distinct phases: (i) where resetting can make gated drift-diffusion faster even compared to the original ungated process, ⟨ T r ⋆ G ⟩ < ⟨ T ⟩ < ⟨ T G ⟩ , (ii) where resetting still expedites gated drift-diffusion but not beyond the original ungated process, ⟨ T ⟩ ≤ ⟨ T r ⋆ G ⟩ < ⟨ T G ⟩ , and (iii) where resetting fails to expedite gated drift-diffusion, ⟨ T ⟩ < ⟨ T G ⟩ ≤ ⟨ T r ⋆ G ⟩. We also highlight various non-trivial behaviors of the completion time as the resetting rate, gating parameters, and geometry of the set-up are carefully ramified. Gated drift-diffusion aptly models various stochastic processes such as chemical reactions that exclusively take place in certain activated states of the reactants. Our work predicts the conditions under which stochastic resetting can act as a useful strategy to enhance the rate of such processes without compromising their selectivity. [ABSTRACT FROM AUTHOR]

In this paper, we derive a condition-based maintenance strategy for a degradation-shock dependence system subject to two causes of failure, degradation and random shocks. We develop a reliability model for a system under warranty for failure in which we consider the dependence between the degradation process and random shocks and do a warranty cost analysis based on the suggested model. We consider a degradation-shock dependence system that is under warranty with three separate services: repair, replacement, and preventive maintenance. These are triggered when a product degradation crosses one of three predetermined degradation thresholds. We also investigate the dependence between random shocks and degradation modelled by a time-scaled covariate factor. We study the random shock considering fatal shocks that require replacement and nonfatal shocks that require only repair and the degradation caused by nonfatal shocks. The condition-based maintenance strategy for a degradation-shock dependence system is a novel approach to analysing system degradation, with a goal of optimising warranty costs; we also minimise the expected total system cost to find an optimal warranty length. We present several numeric examples and real-world applications to illustrate our proposed approach. [ABSTRACT FROM AUTHOR]

Revealing the mechanisms underlying community organization has long been a central interest for ecologists and biogeographers. Recent findings have suggested that different dimensions of biodiversity may be shaped by contrasting ecological processes, offering complementary insights about community assembly. However, studies integrating multiple diversity facets across the Qinghai‐Tibetan Plateau (QTP) remain insufficient. We combined multiple analytical frameworks to unravel the patterns (taxonomic, functional and phylogenetic) and correlates (local environment, land use and dispersal pathways) of aquatic arthropod diversity in lotic (stream) and lentic (wetland) ecosystems of the QTP. We observed strong phylogenetic signals in most functional traits, pointing to the influence of evolutionary processes on these faunas. Null models indicated that community structure in most streams and wetlands was mostly the result of random draws from the regional functional and phylogenetic species pool. The prevalence of stochasticity was most likely associated with the interplay of the paleogeographical history, the extremely harsh environmental conditions at high elevations and the subsequent impoverishment of the regional species pool. However, some streams and wetlands also exhibited phylogenetic overdispersion and functional clustering, emphasizing the potential importance of competitive exclusion and habitat filtering, respectively. Variation partitioning further revealed that both environmental filtering and dispersal limitation accounted for the spatial variation of diversity measures, with local environment overriding the effects of space and catchment land use. Overall, this study improves our understanding of community organization and diversity patterns in environmentally extreme alpine catchments, with broad implications for the conservation and management of one of the world’s most important high‐altitude treasures. [ABSTRACT FROM AUTHOR]

This article presents a robust security‐based model predictive control (MPC) framework for a class of discrete‐time nonlinear cyber‐physical systems (CPSs) suffering random deception attacks and persistent actuator saturation. In CPSs, the attacks can significantly degrade the control performance. Naturally, a larger degree of control is required, prompting the consideration of actuator saturation in CPS's structure. In our work, the cyberattacks on the controller‐actuator (C‐A) channel are described as a stochastic process with specific probability and bounded energy. The security‐related constraint is introduced to improve the resistance and robustness against attacks. Mean‐square quadratic boundedness is derived to estimate the robustly invariant set. By placing the saturated input into a convex hull, a robust security‐based MPC optimization problem is addressed less conservatively with linear matrix inequality technique. Furthermore, the feasibility of the proposed method is proved recursively, and the stability and security of the closed‐loop system are established in mean‐square sense. In the end, the laboratory tank and reactor‐separator process are applied to validate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

AbstractContinuous processes in most applications are measured discretely with error. This complicates the task of detecting intersections and the number of intersections between two continuous processes (i.e., when the processes have the same value). Intersections of continuous processes are scientifically important, but challenging to estimate from data. For example, in the field of animal ecology, intersections of the paths of moving animals tracked with satellite technologies can be used to understand disease transmission. We illustrate how to quantify contact between animals using telemetry data (i.e., the recorded locations of an animal over time). We introduce our method to quantify contact time with accessible concepts from introductory stochastic process literature, such as Brownian motion. Then, we provide two data examples using white-tailed deer (Odocoileus virginianus) and mule deer (Odocoileus hemionus) telemetry data in a region with high prevalence of chronic wasting disease. Our work provides a needed connection between existing model-based literature for animal movement and rule-based literature for animal interaction. Further, our work illustrates a unique statistical problem receiving minimal attention with broad applicability in human and livestock tracking. [ABSTRACT FROM AUTHOR]

A set of ATMs in commercial areas were considered for business transactions. Customer demand for money requires prompt cash supply; to harmonize optimality options for ATM cash loading to sustain customers. We formulated this tis decision as a multi-stage stochastic process with Markovian demand of ATM cash among customers. The loading and operations of the ATMs were governed by cash demand and supply; with unit (loading, operational, and shortage) costs. The study considered the ATM cash demand of ten ATMs in regions within commercial centres of Kampala city. The relevant data was collected every hour throughout four months. The task focused on optimizing cash-loading decisions that minimize the operational costs of managing ATMs. The optimality of cash loading decisions were determined using the Markov decision processes. The tested model revealed optimal state-dependent options and operational costs of managing ATMs cost-effectively. As a cost optimization strategy for managers in the banking industry, the productivity of ATMs towards improved efficiency and utilization of resources for cash provision to customers emerged; supporting a larger customer base for business transactions. [ABSTRACT FROM AUTHOR]

Microorganisms in lakes are sensitive to salinity fluctuations. Despite extensive prior research on bacterial communities, our understanding of their characteristics and assembly mechanisms in lakes, especially in desert lakes with different salinities. To address this issue, we collected three samples from freshwater lakes, six from brackish lakes, and five from salt lakes in the Badanjilin Desert. The 16S rRNA gene sequencing was applied to investigate the bacterial interactions with rising salinity, community coexistence patterns, and assembly mechanisms. Our findings suggested that the increased lake salinity significantly reduces the bacterial community diversity and enhanced the community differentiation. Significant variations were observed in the contribution of biomarkers from Cyanobacteria, Chloroflexi, and Halobacterota to the composition of the lake bacterial communities. The bacterial communities in the salt lakes exhibited a higher susceptibility to salinity limitations than those in the freshwater and brackish lakes. In addition, the null modeling analyses confirmed the quantitative biases in the stochastic assembly processes of bacterial communities across freshwater, brackish, and saline lakes. With the increasing lake salinity, the significance of undominated and diffusion limitation decreased slightly, and the influence of homogenizing dispersal on community assembly increased. However, the stochasticity remained the dominant process across all lakes in the Badanjilin Desert. The analysis of co-occurring networks revealed that the rising salinity reduced the complexity of bacterial network structures and altered the interspecific interactions, resulting in the increased interspecies collaboration with increasing salinity levels. Under the influence of salinity stress, the key taxon Cyanobacteria in freshwater lakes (Schizothrix_LEGE_07164) was replaced by Proteobacteria (Thalassobaculum and Polycyclovorans) in brackish lakes, and Thermotogota (SC103) in salt lakes. The results indicated the symbiotic patterns of bacterial communities across varying salinity gradients in lakes and offer insights into potential mechanisms of community aggregation, thereby enhancing our understanding of bacterial distribution in response to salinity changes. [ABSTRACT FROM AUTHOR]

Introduction: Accurately predicting the remaining mechanical equipment is of great significance for ensuring the safe operation of the equipment and improving economic efficiency. Methods: To accurately assess the mechanical equipment degradation, predict its remaining useful life, and ensure efficient, stable, and safe operation, a degradation and life prediction model for mechanical equipment based on multivariate stochastic processes is studied. The study innovatively predicts the remaining life of mechanical equipment using multivariate stochastic processes, and facilitates the correlation analysis between performance indicators based on the characteristics of Copula functions. Results and discussion: The results showed that the Root Mean Squared Error value of the prediction results based on the trivariate Wiener process was 2.58, which decreased by 46.91% and 35.82% compared with the univariate and bivariate Wiener processes, respectively. The prediction value based on the trivariate gamma process was 3.49, which decreased by 44.95% and 40.54% compared with the univariate and bivariate gamma processes, respectively. In conclusion, the degradation and life prediction model with multivariate stochastic processes can effectively assess the mechanical equipment degradation and predict its remaining useful life. This provides an important reference for the maintenance and management of mechanical equipment, improving equipment efficiency and extend its service life. [ABSTRACT FROM AUTHOR]

Without heritable variation natural selection cannot effect evolutionary change. In the case of group selection, there must be variation in the population of groups. Where does this variation come from? One source of variation is from the stochastic birth–death processes that occur within groups. This is where variation between groups comes from in most mathematical models of group selection. Here, we argue that another important source of variation between groups is fission, the (generally random) group-level reproduction where parent groups split into two or more offspring groups. We construct a simple model of the fissioning process with a parameter that controls how much variation is produced among the offspring groups. We then illustrate the effect of that parameter with some examples. In most models of group selection in the literature, no variation is produced during group reproduction events; that is, groups "clone" themselves when they reproduce. Fission is often a more biologically realistic method of group reproduction, and it can significantly increase the efficacy of group selection. [ABSTRACT FROM AUTHOR]

In this paper, random and stochastic processes are defined on fractal curves. Fractal calculus is used to define the cumulative distribution function, probability density function, moments, variance, and correlation function of stochastic processes on fractal curves. A new framework, which is a generalization of mean square calculus, is formulated. The sequence of random variables on the fractal curve, fractal mean square continuity, mean square F α -derivative, and fractal mean square integral are discussed. The mean square solution of a fractal stochastic equation is derived and plotted to illustrate the details. [ABSTRACT FROM AUTHOR]

Flow lines are often used to perform assembly operations in multi-stage processes. During these assembly operations, components that are relatively small, compared to the work pieces travelling down the flow line, are mounted to the work pieces at a given stage. Those components, or more generally, any kind of auxiliary material, are provisioned to the corresponding production stage in a repetitive but not necessarily deterministic manner using a certain delivery frequency, each time filling the local storage up to a predetermined order-up-to level. Just like random processing times, machine failures, and repairs, the randomness of the provisioning process can impact the long-term throughput of such a flow line. In this paper, we develop a fast and accurate analytical performance evaluation method to estimate the long-term throughput of a Markovian flow line of this type for the practically important case of limited buffer capacities between the production stages. We first give an exact characterization of a two-machine line of that type and show how to determine system state probabilities and aggregate performance measures. Furthermore, we show how to use this two-machine model as the building block of an approximate decomposition approach for longer flow lines. As opposed to previous decomposition approaches, even the state space of the two-machine lines can become so large that an exact solution of the Markov chains can become impractical. We hence show how to set up, train, and use an artificial neural network to replace the Markov chain solver embedded in the decomposition approach, which then leads to an accurate and extremely fast flow line evaluation tool. The proposed methodology is evaluated by a comparison with simulation results and used to characterize the structural patterns describing the behaviour of flow lines of this type. The method can be used to systematically consider the combined impact of the delivery frequency and the local order-up-to levels for the auxiliary material when designing a flow line of this type. [ABSTRACT FROM AUTHOR]

Aerobic anoxygenic phototrophic bacteria (AAPB) are significant bacterial groups in aquatic ecosystems, known for their rapid growth and photoheterotrophic characteristics. However, the distribution and ecological assembly process of AAPB in low irradiation freshwater basins remain unclear, warranting further investigation. In this study, we present the diversity, abundance, spatial variations, ecological process, and community interaction of AAPB in sediment of Three Gorges Reservoir (TGR) under low irradiation. Our findings demonstrate the dominant genera of AAPB community that exist in the TGR area also are appeared in different waters, with some regional preference. Moreover, the concentration of pufM gene, an indicator for AAPB, maintains a consistently high numerical level ranging from (2.21 ± 0.44) × 104 to (9.98 ± 0.30) × 107 gene copies/g. Although solar irradiation is suggested as the major factor affecting AAPB, it remains unclear whether and how AAPB differ between regions due to varying solar irradiation levels. Our results show spatial differences between total bacteria and AAPB communities, with significant differences observed only in AAPB. Geographical and environmental factor contributed less than 10% to the spatial difference of community, with sediment type and environmental factors being the key factors influencing microbial community structure. The stochastic process plays a dominant role in the aggregation and replacement of AAPB communities, among which the most contribution is dispersal limitation. For AAPB network, Yoonia and Gemmobacter are the hubs for modules. Those results valuable insights into the AAPB communities in TGR with low irradiation. [Display omitted] [ABSTRACT FROM AUTHOR]

The present paper develops a new procedure for finding the most unfavorable long-period excitation for isolated building structures based on the probabilistic critical excitation method. The upper bound of earthquake input energy per unit mass is considered the problem's constraint, and a modified power spectral density (PSD) function is proposed to conform to the new constraint. It is shown that the new constraint can be used as a criterion to estimate the possible range of input energy of similar earthquakes. In addition, the new PSD function can control the excitation's intensity. Finally, three base-isolated shear buildings modeled as multi degree of freedom systems with nonproportional damping are considered, and the critical excitations for each model are found based on the proposed method. To examine the reliability of the results, three actual accelerograms with a considerably high level of total input energy are selected as benchmarks and linear dynamic analyses are conducted using these accelerograms along with the generated critical excitations. The results show that the generated long-period excitations can better estimate the structural behavior (i.e., maximum displacement, drift, and absolute acceleration of stories) than the actual benchmark accelerograms. [ABSTRACT FROM AUTHOR]

We propose a new threshold selection method for nonparametric estimation of the extremal index of stochastic processes. The discrepancy method was proposed as a data-driven smoothing tool for estimation of a probability density function. Now it is modified to select a threshold parameter of an extremal index estimator. A modification of the discrepancy statistic based on the Cramér–von Mises–Smirnov statistic $ \omega ^2 $ ω 2 is calculated by k largest order statistics instead of an entire sample. Its asymptotic distribution as $ k\to \infty $ k → ∞ is proved to coincide with the $ \omega ^2 $ ω 2 -distribution. Its quantiles are used as discrepancy values. The convergence rate of an extremal index estimate coupled with the discrepancy method is derived. The discrepancy method is used as an automatic threshold selection for the intervals and K-gaps estimators. It may be applied to other estimators of the extremal index. The performance of our method is evaluated by simulated and real data examples. [ABSTRACT FROM AUTHOR]

In this study we solve the following problem: Simulate random 2D Poisson point processes with a desired correlation function. To solve this problem we suggest the following algorithm: (1) simulate a positive valued random process with the desired correlation function, (2) use this process as an intensity of the doubly stochastic Poisson random point process. We apply this algorithm to simulate random distribution of nanocrystals on a plane. Then we apply the developed methods to calculate excitonic fluxes to the family of generated nanocrystals. [ABSTRACT FROM AUTHOR]

Tasks of different nature and difficulty levels are a part of people's lives. In this context, there is a scientific interest in the relationship between the difficulty of the task and the persistence need to accomplish it. Despite the generality of this problem, some tasks can be simulated in the form of games. In this way, we employ data from a large online platform, called Steam, to analyze games and the performance of their players. More specifically, we investigated persistence in completing tasks based on the proportion of players who accomplished game achievements. Overall, we present five major findings. First, the probability distribution for the number of achievements is log-normal distribution. Second, the distribution of game players also follows a log-normal. Third, most games require neither a very high degree of persistence nor a very low one. Fourth, players also prefer games that demand a certain intermediate persistence. Fifth, the proportion of players as a function of the number of achievements declines approximately exponentially. As both the log-normal and the exponential functions are memoryless, they are mathematical forms that describe random effects arising from the nature of the system. Therefore our first two findings describe random processes of fragmenting achievements and players while the last three provide a quantitative measure of the human preference in the pursuit of challenging, achievable and justifiable tasks. [ABSTRACT FROM AUTHOR]

We derive consistency and asymptotic normality results for quasi‐maximum likelihood methods for drift parameters of ergodic stochastic processes observed in discrete time in an underlying continuous‐time setting. The special feature of our analysis is that the stochastic integral part is unobserved and nonparametric. Additionally, the drift may depend on the (unknown and unobserved) stochastic integrand. Our results hold for ergodic semi‐parametric diffusions and backward SDEs. Simulation studies confirm that the methods proposed yield good convergence results. [ABSTRACT FROM AUTHOR]

We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan [Probab. Theory Related Fields 93 (1992), pp. 377–392]. Our starting point is a version of Strassen's functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero–one law for functionals of convex hulls of walks with drift, of some independent interest. As another application of our approach, we obtain iterated-logarithm laws for intrinsic volumes of the convex hull of the centre of mass (running average) process associated to the random walk. [ABSTRACT FROM AUTHOR]

Geometric properties of exceedance regions above a given quantile level provide meaningful theoretical and statistical characterizations for stochastic processes defined on Euclidean domains. Many theoretical results have been obtained for excursions of Gaussian processes and include expected values of the so-called Lipschitz–Killing curvatures (LKCs), such as the area, perimeter and Euler characteristic in two-dimensional Euclidean space. In this paper, we derive novel results for the expected LKCs of excursion sets of more general processes whose construction is based on location or scale mixtures of a Gaussian process, which means that the mean or the standard deviation, respectively, of a stationary Gaussian process is a random variable. We first present exact formulas for peaks-over-threshold-stable limit processes (so-called Pareto processes) arising from the use of Gaussian or log-Gaussian spectral functions in the spectral construction of max-stable processes. These peaks-over-threshold limits are known to arise for Gaussian location or scale mixtures if the mixing distributions satisfies certain regular-variation properties. As a second important result, we show that expected LKCs of excursion sets of such general mixture processes converge to the corresponding expressions of their Pareto process limits. We further provide exact subasymptotic formulas of expected LKCs for various specific choices of the distribution of the mixing variable. Finally, we discuss consistent empirical estimation of LKCs of exceedance regions and implement numerical experiments to illustrate the rate of convergence towards asymptotic expressions. An application to daily temperature data simulated by climate models for the period 1951–2005 over a regular pixel grid covering continental France showcases the practical utility of the new results. [ABSTRACT FROM AUTHOR]

This study proposes a novel velocity divergence‐generalized adaptive probability density evolution method (VD‐GAPDEM) for calculating the probability density function of the stochastic response process of stochastic structures under stochastic dynamic loads. First, based on the principle of probability conservation, the velocity divergence‐generalized adaptive probability density evolution equation (VD‐GAPDEE) is derived for a stochastic system that can effectively consider the shape and location changes of the joint transitional probability density of representative points (RPs) in the stochastic response process. Second, a novel VD‐GAPDEM is proposed to solve the VD‐GAPDEE directly using the point selection technique based on the generalized F discrepancy and the second‐order Runge–Kutta method with a smoothing kernel method (Runge–Kutta‐SKFAM). Furthermore, the differences and connections between VD‐GAPDEM and the existing probability density evolution method are analyzed. Additionally, the high computational efficiency and accuracy of the proposed VD‐GAPDEM are demonstrated through three typical examples of stochastic response analysis, involving stochastic systems subjected to stochastic dynamic loads. [ABSTRACT FROM AUTHOR]

In [Trans. Am. Math. Soc. 375 (2022), no. 1, 627–668], Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let r⩾3$$ r\geqslant 3 $$ and let n$$ n $$ be divisible by r$$ r $$. Then, in the random r$$ r $$‐uniform hypergraph process on n$$ n $$ vertices, as soon as the last isolated vertex disappears, a perfect matching emerges. In the present work, we prove the analogue of this result for clique factors in the random graph process: at the time that the last vertex joins a copy of the complete graph Kr$$ {K}_r $$, the random graph process contains a Kr$$ {K}_r $$‐factor. Our proof draws on a novel sequence of couplings which embeds the random hypergraph process into the cliques of the random graph process. An analogous result is proved for clique factors in the s$$ s $$‐uniform hypergraph process (s⩾3$$ s\geqslant 3 $$). [ABSTRACT FROM AUTHOR]

This paper gives a new property for stochastic processes, called square-mean μ - pseudo-S-asymptotically Bloch-type periodicity. We show how this property is preserved under some operations, such as composition and convolution, for stochastic processes. Our main results extend the classical results on S-asymptotically Bloch-type periodic functions. We also apply our results to some problems involving semilinear stochastic integrodifferential equations in abstract spaces [ABSTRACT FROM AUTHOR]

We revisit a deterministic model for studying the dynamics of allelopathy. The model is formulated in terms of a non-homogeneous linear system of differential equations whose forcing or source term is a piecewise constant function (square wave). To account for the inherent uncertainties present in this natural phenomenon, we reformulate the model as a system of random differential equations where all model parameters and the initial condition are assumed to be random variables, while the forcing term is a stochastic process. Taking extensive advantage of the so-called Random Variable Transformation (RVT) method, we obtain the solution of the randomized model by providing explicit expressions of the first probability density function of the solution under very general assumptions on the model data. We also determine the joint probability density function of the non-trivial equilibrium point, which is a random vector. If the source term is a time-dependent stochastic process, the RVT method might not be applicable since no explicit solution of the model is available. We then show an alternative approach to overcome this drawback by applying the Liouville–Gibbs partial differential equation. All the theoretical findings are illustrated through several examples, including the application of the randomized model to real-world data on alkaloid contents from leaching thornapple seed. [ABSTRACT FROM AUTHOR]

For a 24/7 database system, backups should be implemented right after a large volume of data has been updated, putting their backup windows in non-busy states with user's convenience.From this viewpoint, this paper studies periodic and random incremental backup policies, in which, incremental backup is implemented right after data update and full backup is performed at periodic times KT , or at a number N of data updates, respectively. We firstly describe the stochastic processes of data update and database failure, and then model the expected cost rates for data backup and data restoration.Respective K ∗ , N ∗ , K f ∗ , and N f ∗ are obtained to minimize their expected cost rates in analytical ways, respectively. Finally, numerical examples are given to illustrate the optimum policies. [ABSTRACT FROM AUTHOR]

The study of myxomycete biogeography has a long-standing history and has consistently drawn scholarly interest. Nevertheless, studies focusing specifically on the spatial and temporal distribution patterns of myxomycete diversity are relatively limited, with even fewer investigating the mechanisms driving the generation and maintenance of myxomycete diversity. Therefore, this study selected two geographically distant sampling sites within northern Chinese forests to investigate myxomycete species composition, community structure, environmental drivers, and assembly patterns under geographic barriers. We established plots in the Altai Mountains (ALE) and the Greater Khingan Mountains (GKM), gathered bark and litter, and conducted 80-day moist chamber cultures of myxomycetes. Additionally, myxomycete specimens were collected in the field simultaneously to supplement the data set. This study collected 541 myxomycete specimens belonging to 73 species from 28 genera, spanning 12 families and eight orders. The ALE and the GKM had 20 identical species, accounting for 27% of the total species. Myxomycetes from both regions exhibited abundant occurrence 18 days after cultivation, with the quantity on bark substrates notably higher than on litter. Arcyria pomiformis and Comatricha elegans were the most common species in moist chamber cultures. Mantel test outcomes revealed that environmental factors had no significant impact on myxomycete community similarity between the two areas, aligning with findings from the neutral community model analysis, indicating a predominant influence of stochastic processes on myxomycete community structure in moist chamber cultures. This study represents the first application of a quantitative framework to analyze myxomycete community assembly cultivated in moist chambers. [ABSTRACT FROM AUTHOR]

Plant communities are complex systems shaped by a combination of deterministic and stochastic ecological processes. Bryophytes are an essential component of plant diversity in natural ecosystems, yet our understanding of their community ecology needs to catch up to that of other organisms. The unique geological history, alpine climatic conditions, and high habitat heterogeneity of Tibet provide suitable areas for bryophytes to survive in the alpine regions. Therefore, field surveys were conducted across 184 plots in forest, thicket, and herbaceous vegetation of Tibet to investigate the role of deterministic processes such as biological interactions and abiotic effects, along with stochastic processes, in shaping the distribution of bryophyte assemblages. We employed various analytical methods, including mixed effects models, partial least squares path modeling, null model analysis, and neutral community models. The study showed that bryophyte richness was highest in forests. Bryophyte assemblages showed greater segregation in forest and thicket environments compared to herbaceous vegetation. As the influence of stochastic processes increased, that of deterministic processes decreased from forests through thickets to herbaceous vegetation. Deterministic processes were the main driving forces for the bryophyte assemblage pattern. Soil properties and climatic factors, particularly pH played a key role in determining bryophyte patterns in Tibet. This study has deepened our comprehension of how deterministic and stochastic ecological processes interplay and shape bryophyte distribution patterns in Tibet. [ABSTRACT FROM AUTHOR]

AbstractFueled in part by recent applications in neuroscience, the multivariate Hawkes process has become a popular tool for modeling the network of interactions among high-dimensional point process data. While evaluating the uncertainty of the network estimates is critical in scientific applications, existing methodological and theoretical work has primarily addressed estimation. To bridge this gap, we develop a new statistical inference procedure for high-dimensional Hawkes processes. The key ingredient for the inference procedure is a new concentration inequality on the first- and second-order statistics for integrated stochastic processes, which summarize the entire history of the process. Combining recent martingale central limit theorem with the new concentration inequality, we then characterize the convergence rate of the test statistics in a continuous time domain. Finally, to account for potential non-stationarity of the process in practice, we extend our statistical inference procedure to a flexible class of Hawkes processes with time-varying background intensities and unknown transition functions. The finite sample validity of the inferential tools is illustrated via extensive simulations and further applied to a neuron spike train data set. [ABSTRACT FROM AUTHOR]

This research explores the reliability of deep learning, specifically Long Short‐Term Memory (LSTM) networks, for estimating the Hurst parameter in fractional stochastic processes. The study focuses on three types of processes: fractional Brownian motion (fBm), fractional Ornstein–Uhlenbeck (fOU) process, and linear fractional stable motions (lfsm). The work involves a fast generation of extensive datasets for fBm and fOU to train the LSTM network on a large volume of data in a feasible time. The study analyses the accuracy of the LSTM network's Hurst parameter estimation regarding various performance measures like root mean squared error (RMSE), mean absolute error (MAE), mean relative error (MRE), and quantiles of the absolute and relative errors. It finds that LSTM outperforms the traditional statistical methods in the case of fBm and fOU processes; however, it has limited accuracy on lfsm processes. The research also delves into the implications of training length and valuation sequence length on the LSTM's performance. The methodology is applied to estimating the Hurst parameter in li‐ion battery degradation data and obtaining confidence bounds for the estimation. The study concludes that while deep learning methods show promise in parameter estimation of fractional processes, their effectiveness is contingent on the process type and the quality of training data. [ABSTRACT FROM AUTHOR]

Time‐varying group formation tracking control problems for multi‐agent systems are investigated based on distributed multi‐sensor multi‐target filtering with intermittent observations. First, in order to estimate the states of multiple targets under the phenomenon of intermittent observations accurately, a distributed multi‐sensor multi‐target filtering algorithm is proposed based on cubature Kalman filtering. Second, a time‐varying group formation tracking protocol is designed for multi‐agent systems by using the state estimations obtained from the filtering algorithm and the neighboring interaction. The protocol enables multi‐agent systems to form time‐varying subformations and track multiple targets in the same subgroups, respectively. Third, the boundedness of the error covariance matrices is proved under the condition that the observation probability is higher than the minimum threshold. Then the estimation errors of the filtering algorithm are proved to be stochastically bounded by introducing a stochastic process. Furthermore, the boundedness of the group formation tracking errors is proved. Finally, a numerical example is used to verify the performance of the proposed algorithm and protocol. [ABSTRACT FROM AUTHOR]

AbstractLet us consider a sequence of binary (zero-one) trials. A counter registers ones but once an one is registered the mechanism is locked for a fixed number of trials following each registration. On such a sequence of a fixed length we define a random variable enumerating runs of ones of length exceeding a threshold. Exact recursive expressions are obtained for the probability mass function, generating functions and moments, as well as an exact closed formula for the mean of this random variable. A simulation algorithm is also provided for approximating values of this random variable on such a modified binary sequence. Statistical inference problems associated with the probability of ones are examined by numerical techniques and simulation. An application connected to the chance that a stochastic process remains or not in statistical control is discussed. Illustrative numerical examples exemplify further the theoretical results. [ABSTRACT FROM AUTHOR]

Subsampling is a practical strategy for analyzing vast survival data, which are progressively encountered across diverse research domains. While the optimal subsampling method has been applied to inferences for Cox models and parametric accelerated failure time (AFT) models, its application to semi‐parametric AFT models with rank‐based estimation have received limited attention. The challenges arise from the non‐smooth estimating function for regression coefficients and the seemingly zero contribution from censored observations in estimating functions in the commonly seen form. To address these challenges, we develop optimal subsampling probabilities for both event and censored observations by expressing the estimating functions through a well‐defined stochastic process. Meanwhile, we apply an induced smoothing procedure to the non‐smooth estimating functions. As the optimal subsampling probabilities depend on the unknown regression coefficients, we employ a two‐step procedure to obtain a feasible estimation method. An additional benefit of the method is its ability to resolve the issue of underestimation of the variance when the subsample size approaches the full sample size. We validate the performance of our estimators through a simulation study and apply the methods to analyze the survival time of lymphoma patients in the surveillance, epidemiology, and end results program. [ABSTRACT FROM AUTHOR]

Continuous time Bayesian networks (CTBNs) represent a class of stochastic processes, which can be used to model complex phenomena, for instance, they can describe interactions occurring in living processes, social science models or medicine. The literature on this topic is usually focused on a case when a dependence structure of a system is known and we are to determine conditional transition intensities (parameters of a network). In the paper, we study a structure learning problem, which is a more challenging task and the existing research on this topic is limited. The approach, which we propose, is based on a penalized likelihood method. We prove that our algorithm, under mild regularity conditions, recognizes a dependence structure of a graph with high probability. We also investigate properties of the procedure in numerical studies. [ABSTRACT FROM AUTHOR]

In the analysis of clustered failure time data, Cox frailty models have been extensively studied by incorporating frailty with a prespecified distribution to address potential correlation of data within clusters. In this paper, we propose a frailty proportional mean residual life regression model to analyze such data. A novel conditional quasi‐likelihood inference procedure is developed, utilizing a stochastic process and the inverse probability of censoring weighting (IPCW) to form estimating equations for regression parameters. Our proposal employs conditional inference based on a penalized quasi‐likelihood to address within‐cluster correlation without need to specify the frailty distribution, bringing the method closer to what suffices for real‐world applications. By adopting the Buckley–James estimator in the IPCW, the method further allows for dependent censoring. We establish asymptotic properties of the proposed estimator and evaluate its finite sample performance via simulation studies. An application to the data from a multi‐institutional breast cancer study is presented for illustration. [ABSTRACT FROM AUTHOR]

Microorganisms are important members of seagrass bed ecosystems and play a crucial role in maintaining the health of seagrasses and the ecological functions of the ecosystem. In this study, we systematically quantified the assembly processes of microbial communities in fragmented seagrass beds and examined their correlation with environmental factors. Concurrently, we explored the relative contributions of species replacement and richness differences to the taxonomic and functional β-diversity of microbial communities, investigated the potential interrelation between these components, and assessed the explanatory power of environmental factors. The results suggest that stochastic processes dominate community assembly. Taxonomic β-diversity differences are governed by species replacement, while for functional β-diversity, the contribution of richness differences slightly outweighs that of replacement processes. A weak but significant correlation (p < 0.05) exists between the two components of bdiversity in taxonomy and functionality, with almost no observed significant correlation with environmental factors. This implies significant differences in taxonomy, but functional convergence and redundancy within microbial communities. Environmental factors are insufficient to explain the β-diversity differences. In conclusion, the assembly of microbial communities in fragmented seagrass beds is governed by stochastic processes. The patterns of taxonomic and functional β-diversity provide new insights and evidence for a better understanding of these stochastic assembly rules. This has important implications for the conservation and management of fragmented seagrass beds. [ABSTRACT FROM AUTHOR]

A network evolution with predicted tail and extremal indices of PageRank and the Max-Linear Model used as node influence indices in random graphs is considered. The tail index shows a heaviness of the distribution tail. The extremal index is a measure of clustering (or local dependence) of the stochastic process. The cluster implies a set of consecutive exceedances of the process over a sufficiently high threshold. Our recent results concerning sums and maxima of non-stationary random length sequences of regularly varying random variables are extended to random graphs. Starting with a set of connected stationary seed communities as a hot spot and ranking them with regard to their tail indices, the tail and extremal indices of new nodes that are appended to the network may be determined. This procedure allows us to predict a temporal network evolution in terms of tail and extremal indices. The extremal index determines limiting distributions of a maximum of the PageRank and the Max-Linear Model of newly attached nodes. The exposition is provided by algorithms and examples. To validate our theoretical results, our simulation and real data study concerning a linear preferential attachment as a tool for network growth are provided. [ABSTRACT FROM AUTHOR]

AbstractFunctional Time Series (FTS) are sequences of dependent random elements taking values on some functional space. Most of the research on this domain focuses on producing a predictor able to forecast the next function, having observed a part of the sequence. For this, the Autoregressive Hilbertian process is a suitable framework. Here, we address the problem of constructing simultaneous predictive confidence bands for a stationary FTS. The method is based on an entropy measure for stochastic processes. To construct predictive bands, we use a Reproducing Kernel Hilbert Space (RKHS) to represent the functions and a functional bootstrap procedure that allows us to estimate the prediction law and a Reproducing Kernel Hilbert Spaces (RKHS) to represent the functions, considering then the basis associated to the reproducing kernel. We then classify the points on the projected space according to those that belong to the minimum entropy set (MES) and those that do not. We map the minimum entropy set back to the functional space and construct a band using the regularity property of the RKHS. The proposed methodology is illustrated through artificial and real data sets. [ABSTRACT FROM AUTHOR]

A cruise was conducted in the summer of 2023 from the Pearl River Estuary (PRE) to the adjacent waters of the Xisha Islands in the northern South China Sea (NSCS) to investigate the distribution, community structure, and assembly patterns of eukaryotic and prokaryotic phytoplankton using high-throughput sequencing (HTS) and microscopic observation. Dinophyta were the most abundant phylum in the eukaryotic phytoplankton community based on HTS, accounting for 92.17% of the total amplicon sequence variants (ASVs). Syndiniales was the most abundant order among eukaryotic phytoplankton, whereas Prochlorococcus was the most abundant genus within cyanobacteria. The alpha diversity showed the lowest values in the PRE area and decreased gradually with depth, while cyanobacteria exhibited higher alpha diversity indices in the PRE and at depths ranging from 75 m to 750 m. The morphological results were different from the data based on HTS. Diatoms (37 species) dominated the phytoplankton community, with an average abundance of 3.01 × 104 cells L-1, but only six species of dinoflagellate were observed. Spearman correlation analysis and redundancy analysis (RDA) showed that the distribution and community structure of phytoplankton were largely influenced by geographical location and environmental parameters in the NSCS. The neutral community model (NCM) and null model indicated that deterministic processes played a significant role in the assembly of eukaryotic phytoplankton, with heterogeneous selection and homogeneous selection accounting for 47.27 and 29.95%, respectively. However, stochastic processes (over 60%) dominated the assembly of cyanobacteria and undominated processes accounted for 63.44%. In summary, the formation of eukaryotic phytoplankton was mainly influenced by environmental factors and geographic location, but the assembly of cyanobacteria was shaped by both stochastic processes, which accounted for over 60%, and environmental selection in the NSCS. [ABSTRACT FROM AUTHOR]

The step-stress accelerated degradation test (ADT) provides a feasible method for assessing the storage life of high-reliability, long-life products. However, this method results in a slower rate of performance degradation at the beginning of the test, significantly reducing the test efficiency. Therefore, this article proposes an assessment method for the step-down stress ADT that considers random effects and detection errors (SDRD). Firstly, a new Inverse Gaussian (IG) model is proposed. The model introduces the Gamma distribution to characterize the randomness of the product degradation path and uses the normal distribution to describe the detection errors of performance parameters. In addition, to solve the problem that the likelihood function of the IG model is complex and has no explicit expression, the Monte Carlo (MC) method is used to estimate unknown parameters of the model. This approach enhances computational accuracy and efficiency. Finally, to verify the effectiveness of the SDRD method, it is applied to the step-down stress ADT data from a specific missile tank to assess its storage life. Comparing the life assessment results of different methods, the conclusion shows that the SDRD method is more effective for assessing the storage life of high-reliability, long-life products. [ABSTRACT FROM AUTHOR]

For the sensorless control in a low-speed range of synchronous reluctance motors (SynRMs), injecting random high-frequency (HF) square-wave-type voltages has become a widely used and technologically mature method. It can solve the noise problem of traditional injection signal methods. However, all injection signal methods will cause problems such as torque ripple, which causes speed fluctuations. This article proposes a self-regulating random model algorithm for the random injection signal method, which includes a quantity adaptive module for adding additional random processes, an evaluation module for evaluating torque deviation degree, and an updated model module that is used to receive signals from the other two modules and complete model changes and output random model elements. The main function of this algorithm is to create a model that updates to suppress the evaluation value deviation based on the evaluation situation and outputs an optimal sequence of random numbers, thereby limiting speed bias always in a small range; this can reduce unnecessary changes in the output value of the speed regulator. The feasibility and effectiveness of the proposed algorithm and control method have been demonstrated in experiments based on a 5-kW synchronous reluctance motor. [ABSTRACT FROM AUTHOR]

To ensure the efficient functioning of solar energy generation systems, it is crucial to have dependable designs and regular maintenance. However, when these systems or their components operate at multiple working levels, optimizing reliability becomes a complex task for models and analyses. In the context of reliability modeling in solar energy generation systems, researchers often assume that random variables follow an exponential distribution (binary-state representation) as a simplification, although this may not always hold true for real-world engineering systems. In the present paper, a multi-state solar energy generating system with inverters in series configuration is investigated, in which unreliable by-pass changeover switches, common cause failures (CCFs), and multiple repairman vacations are also considered. Furthermore, the arrivals of CCFs and the repair processes of the failed system due to CCFs are governed by different Markovian arrival processes (MAPs), and the lifetimes and repair times of inverters and by-pass changeover switches and the repairman vacation time in the system have different phase-type (PH) distributions. Therefore, the behavior of the system is represented using a Markov process methodology, and reliability measures for the proposed system are derived utilizing aggregated stochastic process theory. Finally, a numerical example and a comparison analysis are presented to demonstrate the findings. [ABSTRACT FROM AUTHOR]

This paper establishes a new fractional frSIS model utilizing a continuous time random walk method. There are two main innovations in this paper. On the one hand, the model is analyzed from a mathematical perspective. First, unlike the classic SIS infectious disease model, this model presents the infection rate and cure rate in a fractional order. Then, we proved the basic regeneration number R0 of the model and studied the influence of orders a and b on R0. Second, we found that frSIS has a disease-free equilibrium point E0 and an endemic equilibrium point E∗. Moreover, we proved frSIS global stability of the model using R0. If R0<1, the model of E0 is globally asymptotically stable. If R0>1, the model of E∗ is globally asymptotically stable. On the other hand, from the perspective of infectious diseases, we discovered that appropriately increasing a and decreasing b are beneficial for controlling the spread of diseases and ultimately leading to their disappearance. This can help us provide some dynamic adjustments in prevention and control measures based on changes in the disease. [ABSTRACT FROM AUTHOR]