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EN
We investigate a class of exponentially weakly ergodic inhomogeneous birth and death processes. We consider special transformations of the reduced intensity matrix of the process and obtain uniform (in time) error bounds of truncations. Our approach also guarantees that we can find limiting characteristics approximately with an arbitrarily fixed error. As an example, we obtain the respective bounds of the truncation error for an Mt/Mt/S queue for any number of servers S. Arbitrary intensity functions instead of periodic ones can be considered in the same manner.
EN
This paper describes an analytical study of open two-node (tandem) network models with blocking and truncation. The study is based on semi-Markov process theory, and network models assume that multiple servers serve each queue. Tasks arrive at the tandem in a Poisson fashion at the rate [...], and the service times at the first and the second node are nonexponentially distributed with means sA and sB, respectively. Both nodes have buffers with finite capacities. In this type of network, if the second buffer is full, the accumulation of new tasks by the second node is temporarily suspended (a blocking factor) and tasks must wait on the first node until the transmission process is resumed. All new tasks that find the first buffer full are turned away and are lost (a truncation factor). First, a Markov model of the tandem is investigated. Here, a twodimensional state graph is constructed and a set of steady-state equations is created. These equations allow calculating state probabilities for each graph state. A special algorithm for transforming the Markov model into a semi-Markov process is presented. This approach allows calculating steady-state probabilities in the semi-Markov model. Next, the algorithms for calculating the main measures of effectiveness in the semi-Markov model are presented. In the numerical part of this paper, the author investigates examples of several semi-Markov models. Finally, the results of calculating both the main measures of effectiveness and quality of service (QoS) parameters are presented.
EN
The design of complex concurrent systems often involves intricate performance and dependability considerations. Continuous-time Markov chains (CTMCs) are a widely used modeling formalism that captures such performance and dependability properties, and makes them analyzable by model checking. In this paper, we focus on time-bounded probabilistic properties of infinite-state CTMCs, expressible in a subset of continuous stochastic logic (CSL). This comprises important dependability measures, such as time-bounded probabilistic reachability, performability, survivability, and various availability measures like instantaneous, conditional instantaneous and interval availabilities. Conventional model checkers explore the given model exhaustively, which is often costly, due to state explosion, and sometimes impossible because the model is infinite. This paper presents a method that only explores the model up to a finite depth. The required depth is determined on the fly by an algorithm that is configurable in order to adapt to the characteristics of different classes of models. We provide experimental evidence showing that our method is effective.
EN
Extremes of stream flow and precipitation are commonly modeled by heavy-tailed distributions. While scrutinizing annual flow maxima or the peaks over threshold, the largest sample elements are quite often suspected to be low quality data, outliers or values corresponding to much longer return periods than the obser-vation period. Since the interest is primarily in the estimation of the right tail (in the case of floods or heavy rainfalls), sensitivity of upper quantiles to largest elements of a series constitutes a problem of special concern. This study investigated the sen-sitivity problem using the log-Gumbel distribution by generating samples of different sizes (n) and different values of the coefficient of variation by Monte Carlo ex-periments. Parameters of the log-Gumbel distribution were estimated by the prob-ability weighted moments (PWMs) method, method of moments (MOMs) and maximum likelihood method (MLM), both for complete samples and the samples deprived of their largest elements. In the latter case, the distribution censored by the non-exceedance probability threshold, FT , was considered. Using FT instead of the censored threshold T creates possibility of controlling estimator property. The effect of the FT value on the performance of the quantile estimates was then examined. It is shown that right censoring of data need not reduce an accuracy of large quantile estimates if the method of PWMs or MOMs is employed. Moreover allowing bias of estimates one can get the gain in variance and in mean square error of large quantiles even if ML method is used.
5
Content available A non-singular description of parametrical resonance
EN
Employing the Mathieu equations we present a method for construction of Ince-Strutt diagrams, in which no cut-off of the infinite chain of equations for Fourier coefficients is necessary.
PL
Niesingularny opis rezonansu parametrycznego. Na przykładzie równania Mathieu przedstawiamy niesingularną metodę konstrukcji diagramów Incea-Strutta, które obrazują na płaszczyźnie F parametrów równania obszary stabilności i niestabilności. Niesingularność metody została osiągnięta dzięki nie obrywaniu nieskończonych łańcuchów równań dla współczynników Fouriera. Pozwala to na zachowanie pierwotnego charakteru rozważanych równań (zagadnienie początkowe). Przedstawione rozważania sugerują poszerzenie oszarów niestabilności.
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