Stationary representation of queues. I

1986 ◽  
Vol 18 (3) ◽  
pp. 815-848 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Weak Convergence ◽  
Asymptotic Behaviour ◽  
Waiting Time ◽  
Initial Conditions ◽  
Probability Distributions ◽  
Stationary Sequence ◽  
Limiting Distribution ◽  
Stationary Representation

The paper deals with the asymptotic behaviour of queues for which the generic sequence is not necessarily stationary but is asymptotically stationary in some sense. The latter property is defined by an appropriate type of convergence of probability distributions of the sequences to the distribution of a stationary sequence We consider six types of convergence of to The main result is as follows: if the sequence of the distributions converges in one of six ways then the sequence of distributions of the sequences converges in the same way, independently of initial conditions. Furthermore the limiting distribution is the same as the limiting distribution obtained by the weak convergence of the distributions Here wk and w∗k denote the waiting time of the kth unit in the queue generated by (v, u) and (v0, u0) respectively.

Stationary representation of queues. I

1986 ◽  
Vol 18 (03) ◽  
pp. 815-848 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Weak Convergence ◽  
Asymptotic Behaviour ◽  
Waiting Time ◽  
Initial Conditions ◽  
Probability Distributions ◽  
Stationary Sequence ◽  
Limiting Distribution ◽  
Stationary Representation

The paper deals with the asymptotic behaviour of queues for which the generic sequence is not necessarily stationary but is asymptotically stationary in some sense. The latter property is defined by an appropriate type of convergence of probability distributions of the sequences to the distribution of a stationary sequence We consider six types of convergence of to The main result is as follows: if the sequence of the distributions converges in one of six ways then the sequence of distributions of the sequences converges in the same way, independently of initial conditions. Furthermore the limiting distribution is the same as the limiting distribution obtained by the weak convergence of the distributions Here wk and w∗ k denote the waiting time of the kth unit in the queue generated by ( v, u ) and ( v 0, u 0) respectively.

Stationary representation of queues. II

1986 ◽  
Vol 18 (3) ◽  
pp. 849-859 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Asymptotic Behaviour ◽  
Waiting Time ◽  
Heavy Traffic ◽  
The Other ◽  
Traffic Situation ◽  
Queue Size ◽  
Stationary Representation

The paper is a continuation of [7]. One of the main results is as follows: if the sequence (w, v, u) is asymptotically stationary in some sense then (l, w, v, u) is asymptotically stationary in the same sense. The other main result deals with an asymptotic behaviour of the vector of the queue size and the waiting time in the heavy-traffic situation. This result resembles a formula of the Little type.

Stationary representation of queues. II

1986 ◽  
Vol 18 (03) ◽  
pp. 849-859 ◽  
Author(s):  
Władysław Szczotka
Keyword(s):  
Asymptotic Behaviour ◽  
Waiting Time ◽  
Heavy Traffic ◽  
The Other ◽  
Traffic Situation ◽  
Queue Size ◽  
Stationary Representation

The paper is a continuation of [7]. One of the main results is as follows: if the sequence ( w, v, u ) is asymptotically stationary in some sense then ( l , w, v, u ) is asymptotically stationary in the same sense. The other main result deals with an asymptotic behaviour of the vector of the queue size and the waiting time in the heavy-traffic situation. This result resembles a formula of the Little type.

Numerical simulation of soliton gas within the Korteweg-de Vries type equations

2019 ◽  
pp. 52-66
Author(s):  
Екатерина Геннадьевна Диденкулова ◽  
Анна Витальевна Кокорина ◽  
Алексей Викторович Слюняев
Keyword(s):  
Numerical Scheme ◽  
Initial Conditions ◽  
Probability Distributions ◽  
Scattering Problem ◽  
Spectral Composition ◽  
Computation Time ◽  
Qualitative Description ◽  
Statistical Characteristics ◽  
De Vries ◽  
Pseudo Spectral Method

Приведены детали численной схемы и способа задания начальных условий для моделирования нерегулярной динамики ансамблей солитонов в рамках уравнений типа Кортевега-де Вриза на примере модифицированного уравнения Кортевега-де Вриза с фокусирующим типом нелинейности. Дано качественное описание эволюции статистических характеристик для ансамблей солитонов одной и разных полярностей. Обсуждаются результаты тестовых экспериментов по столкновению большого числа солитонов. The details of the numerical scheme and the method of specifying the initial conditions for the simulation of the irregular dynamics of soliton ensembles within the framework of equations of the Korteweg - de Vries type are given using the example of the modified Korteweg - de Vries equation with a focusing type of nonlinearity. The numerical algorithm is based on a pseudo-spectral method with implicit integration over time and uses the Crank-Nicholson scheme for improving the stability property. The aims of the research are to determine the relationship between the spectral composition of the waves (the Fourier spectrum or the spectrum of the associated scattering problem) and their probabilistic properties, to describe transient processes and the equilibrium states. The paper gives a qualitative description of the evolution of statistical characteristics for ensembles of solitons of the same and different polarities, obtained as a result of numerical simulations; the probability distributions for wave amplitudes are also provided. The results of test experiments on the collision of a large number of solitons are discussed: the choice of optimal conditions and the manifestation of numerical artifacts caused by insufficient accuracy of the discretization. The numerical scheme used turned out to be extremely suitable for the class of the problems studied, since it ensures good accuracy in describing collisions of solitons with a short computation time.

On weak convergence within the hnbue family of life distributions

1984 ◽  
Vol 21 (03) ◽  
pp. 654-660 ◽  
Author(s):  
Sujit K. Basu ◽  
Manish C. Bhattacharjee
Keyword(s):  
Weak Convergence ◽  
Exponential Distribution ◽  
Limiting Distribution ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Moment Sequence ◽  
Positive Order ◽  
Life Distributions ◽  
Necessary And Sufficient

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions.

Extreme values, range and weak convergence of integrals of Markov chains

1982 ◽  
Vol 19 (02) ◽  
pp. 272-288 ◽  
Author(s):  
P. J. Brockwell ◽  
S. I. Resnick ◽  
N. Pacheco-Santiago
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Extreme Values ◽  
Finite State ◽  
Integral Process ◽  
Maximum Minimum ◽  
Finite State Space

A study is made of the maximum, minimum and range on [0,t] of the integral processwhereSis a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

scholarly journals The mean waiting time to a repetition

1983 ◽  
Vol 15 (01) ◽  
pp. 216-218
Author(s):  
Gunnar Blom
Keyword(s):  
Waiting Time ◽  
Mild Condition ◽  
Random Variables ◽  
Stationary Sequence ◽  
Mean Waiting Time ◽  
The Mean

Let X 1, X2, · ·· be a stationary sequence of random variables and E 1 , E 2 , · ··, EN mutually exclusive events defined on k consecutive X's such that the probabilities of the events have the sum unity. In the sequence E j1 , E j2 , · ·· generated by the X's, the mean waiting time from an event, say E j1 , to a repetition of that event is equal to N (under a mild condition of ergodicity). Applications are given.

scholarly journals Rapid variation with remainder and rates of convergence

1990 ◽  
Vol 22 (04) ◽  
pp. 787-801 ◽  
Author(s):  
J. Beirlant ◽  
E. Willekens
Keyword(s):  
Weak Convergence ◽  
Asymptotic Behaviour ◽  
Rates Of Convergence ◽  
Convergence Result ◽  
Hill Estimator ◽  
Rapid Variation ◽  
Double Exponential Distribution ◽  
Double Exponential ◽  
The Hill

In this paper, we refine the concept of Γ-variation up to second order, and we give a characterization of this type of asymptotic behaviour. We apply our results to obtain uniform rates of convergence in the weak convergence of renormalised sample maxima to the double exponential distribution. In a second application we derive a rate of convergence result for the Hill estimator.

scholarly journals Aspects of Geodesical Motion with Fisher-Rao Metric: Classical and Quantum

2018 ◽  
Vol 25 (01) ◽  
pp. 1850005 ◽  
Author(s):  
Florio M. Ciaglia ◽  
Fabio Di Cosmo ◽  
Domenico Felice ◽  
Stefano Mancini ◽  
Giuseppe Marmo ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Geometric Structure ◽  
Initial Conditions ◽  
Probability Distributions ◽  
End Points ◽  
Quantum Properties ◽  
Statistical Manifolds ◽  
Classical Setting ◽  
Qualitative Effect

The purpose of this paper is to exploit the geometric structure of quantum mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that the end points of geodesics in the classical setting coincide with the probability distributions that minimise Shannon’s entropy, i.e. with distributions of zero dispersion. In the quantum setting this happens only for particular initial conditions, which in turn correspond to classical submanifolds. This result can be interpreted as a geometric manifestation of the uncertainty principle.

Investigations on the power-law burst lifetime distribution characteristics of the magnetospheric system

2020 ◽  
Author(s):  
Prince Prasad ◽  
Santhosh Kumar G ◽  
Sumesh Gopinath
Keyword(s):  
Power Law ◽  
Waiting Time ◽  
Probability Distributions ◽  
Sunspot Cycle ◽  
Lifetime Distribution ◽  
Characteristic Time Scale ◽  
General Strategy ◽  
Scaling Exponent ◽  
Time Duration ◽  
Scaling Properties

<p>The waiting time distributions and associated statistical relationships can be considered as a general strategy for analyzing space weather and inner magnetospheric processes to a large extent. It measures the distribution of delay times between subsequent hopping events in such processes. In a physical system the time duration between two events is called a waiting-time, like the time between avalanches. The burst lifetime can be considered as the time duration when magnitude of fluctuations are above a given threshold intensity.  If a characteristic time scale is absent then the probability densities vary with power-law relations having a scaling exponent. The burst lifetime distribution of the substorm index called as the Wp index (Wave and planetary), which reflects Pi2 wave power at low-latitude is considered for the present analysis. Our analysis shows that the lifetime probability distributions of Wp index yield power-law exponents. Even though power-law exponents are observed in magnetospheric proxies for different solar activity periods, not many studies were made to analyze whether these features will repeat or differ depending on sunspot cycle. We compare the variations of power-law exponents of Wp index and other magnetospheric proxies, such as AE index, during solar maxima and solar minima. Thus the study classifies the activity bursts in Wp and other magnetospheric proxies that may have different dynamical critical scaling features. We also expect that the study sheds light into certain stochastic aspects of scaling properties of the magnetosphere which are not developed as global phenomena, but in turn generated due to inherent localized properties of the magnetosphere.</p>

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