scholarly journals Stable Process Approach to Analysis of Systems Under Heavy-Tailed Noise: Modeling and Stochastic Linearization

2019 ◽  
Vol 64 (4) ◽  
pp. 1344-1357 ◽  
Author(s):  
Kenji Kashima ◽  
Hiroki Aoyama ◽  
Yoshito Ohta
Keyword(s):  
Stable Process ◽  
Process Approach ◽  
Noise Modeling ◽  
Stochastic Linearization ◽  
Heavy Tailed

Finite-pool queueing with heavy-tailed services

2017 ◽  
Vol 54 (3) ◽  
pp. 921-942
Author(s):  
Gianmarco Bet ◽  
Remco van der Hofstad ◽  
Johan S. H. van Leeuwaarden
Keyword(s):  
Head Start ◽  
Stable Process ◽  
Scaling Limit ◽  
Asymptotic Result ◽  
Single Server ◽  
Single Server Queue ◽  
Long Period ◽  
Server Queue ◽  
Service Times ◽  
Heavy Tailed

AbstractWe consider the Δ(i)/G/1 queue, in which a total ofncustomers join a single-server queue for service. Customers join the queue independently after exponential times. We considerheavy-tailedservice-time distributions with tails decaying asx-α, α ∈ (1, 2). We consider the asymptotic regime in which the population size grows to ∞ and establish that the scaled queue-length process converges to an α-stable process with a negative quadratic drift. We leverage this asymptotic result to characterize the head start that is needed to create a long period of uninterrupted activity (a busy period). The heavy-tailed service times should be contrasted with the case of light-tailed service times, for which a similar scaling limit arises (Betet al.(2015)), but then with a Brownian motion instead of an α-stable process.

scholarly journals Transient Asymptotics of Lévy-Driven Queues

2010 ◽  
Vol 47 (1) ◽  
pp. 109-129 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Abdelghafour Es-Saghouani ◽  
Michel Mandjes
Keyword(s):  
Steady State ◽  
Stable Process ◽  
Sample Path ◽  
Rare Event ◽  
Compound Poisson Process ◽  
Input Process ◽  
Deterministic Function ◽  
Heavy Tailed ◽  
Stationary Workload ◽  
Sample Path Large Deviations

With (Qt)t denoting the stationary workload process in a queue fed by a Lévy input process (Xt)t, this paper focuses on the asymptotics of rare event probabilities of the type P(Q0 > pB, QTB > qB) for given positive numbers p and q, and a positive deterministic function TB. We first identify conditions under which the probability of interest is dominated by the ‘most demanding event’, in the sense that it is asymptotically equivalent to P(Q > max{p, q}B) for large B, where Q denotes the steady-state workload. These conditions essentially reduce to TB being sublinear (i.e. TB/B → 0 as B → ∞). A second condition is derived under which the probability of interest essentially ‘decouples’, in that it is asymptotically equivalent to P(Q > pB)P(Q > qB) for large B. For various models considered in the literature, this ‘decoupling condition’ reduces to requiring that TB is superlinear (i.e. TB / B → ∞ as B → ∞). This is not true for certain ‘heavy-tailed’ cases, for instance, the situations in which the Lévy input process corresponds to an α-stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to TB / B2 → ∞. For these input processes, we also establish the asymptotics of the probability under consideration for TB increasing superlinearly but subquadratically. We pay special attention to the case TB = RB for some R > 0; for light-tailed input, we derive intuitively appealing asymptotics, intensively relying on sample path large deviations results. The regimes obtained can be interpreted in terms of the most likely paths to overflow.

scholarly journals Application of Lévy processes in modelling (geodetic) time series with mixed spectra

2021 ◽  
Vol 28 (1) ◽  
pp. 121-134
Author(s):  
Jean-Philippe Montillet ◽  
Xiaoxing He ◽  
Kegen Yu ◽  
Changliang Xiong
Keyword(s):  
Time Series ◽  
Lévy Processes ◽  
Stable Process ◽  
Functional Model ◽  
Noise Spectrum ◽  
Levy Processes ◽  
Infinite Variance ◽  
Residual Time ◽  
First Case ◽  
Heavy Tailed

Abstract. Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series together with the estimated geophysical signals. The noise spectrum of these time series is generally modelled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series after deterministically subtracting geophysical signals from the observations. This residual time series is then assumed to be a sum of three stochastic processes, including the family of Lévy processes. The introduction of a third stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series, we identify three classes of Lévy processes, namely Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. The fractional Lévy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity properties. The stable process is here restrained to the special case of infinite variance, which can be only satisfied in the case of heavy-tailed distributions in the application to geodetic time series. Therefore, the model implies potential anxiety in the functional model selection, where missing geophysical information can generate such residual time series.

scholarly journals Application of Levy Processes in Modelling (Geodetic) Time Series With Mixed Spectra

2019 ◽  
Author(s):  
Jean-Philippe Montillet ◽  
Xiaoxing He ◽  
Kegen Yu
Keyword(s):  
Time Series ◽  
Lévy Processes ◽  
Stable Process ◽  
Functional Model ◽  
Noise Spectrum ◽  
Levy Processes ◽  
Residual Time ◽  
Heavy Tailed Distributions ◽  
First Case ◽  
Heavy Tailed

Abstract. Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series, together with the extraction of geophysical signals. The noise spectrum of these time series is generally modeled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series, after deterministically subtracting geophysical signals from the observations. This residual time series is then assumed to be a sum of three random variables (r.v.), with the last r.v. belonging to the family of Levy processes. This stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series, we identify three classes of Levy processes: Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. Fractional Levy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity property. Stable process is characterized by a large variance, which can be satisfied in the case of heavy-tailed distributions. The application to geodetic time series implies potential anxiety in the functional model selection where missing geophysical information can generate such residual time series.

scholarly journals LIMITS FOR CUMULATIVE INPUT PROCESSES TO QUEUES

2000 ◽  
Vol 14 (2) ◽  
pp. 123-150 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Probability Distributions ◽  
Heavy Traffic ◽  
Stable Process ◽  
Continuous Functions ◽  
Fluid Queue ◽  
Busy Time ◽  
Sample Paths ◽  
Heavy Tailed ◽  
Functional Central Limit Theorems ◽  
Stochastic Fluid

We establish functional central limit theorems (FCLTs) for a cumulative input process to a fluid queue from the superposition of independent on–off sources, where the on periods and off periods may have heavy-tailed probability distributions. Variants of these FCLTs hold for cumulative busy-time and idle-time processes associated with standard queueing models. The heavy-tailed on-period and off-period distributions can cause the limit process to have discontinuous sample paths (e.g., to be a non-Brownian stable process or more general Lévy process) even though the converging processes have continuous sample paths. Consequently, we exploit the Skorohod M1 topology on the function space D of right-continuous functions with left limits. The limits here combined with the previously established continuity of the reflection map in the M1 topology imply both heavy-traffic and non-heavy-traffic FCLTs for buffer-content processes in stochastic fluid networks.

scholarly journals Transient Asymptotics of Lévy-Driven Queues

2010 ◽  
Vol 47 (01) ◽  
pp. 109-129 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Abdelghafour Es-Saghouani ◽  
Michel Mandjes
Keyword(s):  
Steady State ◽  
Stable Process ◽  
Sample Path ◽  
Rare Event ◽  
Compound Poisson Process ◽  
Input Process ◽  
Deterministic Function ◽  
Heavy Tailed ◽  
Stationary Workload ◽  
Sample Path Large Deviations

With (Q t ) t denoting the stationary workload process in a queue fed by a Lévy input process (X t ) t , this paper focuses on the asymptotics of rare event probabilities of the type P(Q 0 > pB, Q T B > qB) for given positive numbers p and q, and a positive deterministic function T B . We first identify conditions under which the probability of interest is dominated by the ‘most demanding event’, in the sense that it is asymptotically equivalent to P(Q > max{p, q}B) for large B, where Q denotes the steady-state workload. These conditions essentially reduce to T B being sublinear (i.e. T B /B → 0 as B → ∞). A second condition is derived under which the probability of interest essentially ‘decouples’, in that it is asymptotically equivalent to P(Q > pB)P(Q > qB) for large B. For various models considered in the literature, this ‘decoupling condition’ reduces to requiring that T B is superlinear (i.e. T B / B → ∞ as B → ∞). This is not true for certain ‘heavy-tailed’ cases, for instance, the situations in which the Lévy input process corresponds to an α-stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to T B / B 2 → ∞. For these input processes, we also establish the asymptotics of the probability under consideration for T B increasing superlinearly but subquadratically. We pay special attention to the case T B = RB for some R > 0; for light-tailed input, we derive intuitively appealing asymptotics, intensively relying on sample path large deviations results. The regimes obtained can be interpreted in terms of the most likely paths to overflow.

scholarly journals Application of Lévy Processes in Modelling (Geodetic) Time Series With Mixed Spectra

2020 ◽  
Author(s):  
Jean-Philippe Montillet ◽  
Xiaoxing He ◽  
Kegen Yu ◽  
Changliang Xiong
Keyword(s):  
Time Series ◽  
Lévy Processes ◽  
Stable Process ◽  
Functional Model ◽  
Noise Spectrum ◽  
Levy Processes ◽  
Infinite Variance ◽  
Residual Time ◽  
First Case ◽  
Heavy Tailed

Abstract. Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series, together with the estimated geophysical signals. The noise spectrum of these time series is generally modelled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series, after deterministically subtracting geophysical signals from the observations. This residual time series is then assumed to be a sum of three stochastic processes, including the family of Lévy processes. The introduction of a third stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series,we identify three classes of Lévy processes: Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. Fractional Lévy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity property. Stable process is here restrained to the special case of infinite variance, which can be only satisfied in the case of heavy-tailed distributions in the application to geodetic time series. Therefore, it implies potential anxiety in the functional model selection where missing geophysical information can generate such residual time series.

Perspectives: A Process Approach to Social Communication for Adolescents With Asperger's Syndrome Using Radio Interviewing

2011 ◽  
Vol 12 (4) ◽  
pp. 110-120
Author(s):  
Amy Thrasher ◽  
Jennifer Wilger ◽  
Matthew Goldman ◽  
Catharine Whitlatch
Keyword(s):  
Social Communication ◽  
Social Skill ◽  
Asperger's Syndrome ◽  
Asperger’S Syndrome ◽  
Process Approach ◽  
Communication Intervention ◽  
Social Thinking ◽  
Social Communication Intervention

Abstract The Perspectives program is a unique collaborative social communication intervention for adolescents with Asperger's syndrome and similar learning profiles. Clinicians use radio interviews as the vehicle to explicitly teach the process of social communication. Social skill objectives are addressed through this process approach, which was adapted from the framework of Social Thinking (Winner, 2002)

An Alternative to Cohen's κ

2006 ◽  
Vol 11 (1) ◽  
pp. 12-24 ◽  
Author(s):  
Alexander von Eye
Keyword(s):  
Simulation Study ◽  
Null Hypothesis ◽  
Categorical Variables ◽  
Alternative Measure ◽  
Rater Agreement ◽  
Verbal Processing ◽  
Heavy Tailed ◽  
Applicant Selection

At the level of manifest categorical variables, a large number of coefficients and models for the examination of rater agreement has been proposed and used. The most popular of these is Cohen's κ. In this article, a new coefficient, κ s , is proposed as an alternative measure of rater agreement. Both κ and κ s allow researchers to determine whether agreement in groups of two or more raters is significantly beyond chance. Stouffer's z is used to test the null hypothesis that κ s = 0. The coefficient κ s allows one, in addition to evaluating rater agreement in a fashion parallel to κ, to (1) examine subsets of cells in agreement tables, (2) examine cells that indicate disagreement, (3) consider alternative chance models, (4) take covariates into account, and (5) compare independent samples. Results from a simulation study are reported, which suggest that (a) the four measures of rater agreement, Cohen's κ, Brennan and Prediger's κ n , raw agreement, and κ s are sensitive to the same data characteristics when evaluating rater agreement and (b) both the z-statistic for Cohen's κ and Stouffer's z for κ s are unimodally and symmetrically distributed, but slightly heavy-tailed. Examples use data from verbal processing and applicant selection.

Sign in / Sign up

Export Citation Format

Share Document