scholarly journals On the excursions of reflected local-time processes and stochastic fluid queues

2011 ◽  
Vol 48 (A) ◽  
pp. 79-98 ◽  
Author(s):  
Takis Konstantopoulos ◽  
Andreas E. Kyprianou ◽  
Paavo Salminen
Keyword(s):  
Local Time ◽  
Service System ◽  
Macroscopic Model ◽  
Time Process ◽  
Fluid Queues ◽  
Maximum Value ◽  
Stationary Version ◽  
Negative Drift ◽  
Stochastic Fluid ◽  
Strong Markov

In this paper we extend our previous work. We consider the local-time process L of a strong Markov process X, add negative drift to L, and reflect it à la Skorokhod to obtain a process Q. The reflection of X, together with Q, is, in some sense, a macroscopic model for a service system with two priorities. We derive an expression for the joint law of the duration of an excursion, the maximum value of the process on it, and the time between successive excursions. We work with a properly constructed stationary version of the process. Examples are also given in the paper.

scholarly journals On the excursions of reflected local-time processes and stochastic fluid queues

2011 ◽  
Vol 48 (A) ◽  
pp. 79-98
Author(s):  
Takis Konstantopoulos ◽  
Andreas E. Kyprianou ◽  
Paavo Salminen
Keyword(s):  
Local Time ◽  
Service System ◽  
Macroscopic Model ◽  
Time Process ◽  
Fluid Queues ◽  
Maximum Value ◽  
Stationary Version ◽  
Negative Drift ◽  
Stochastic Fluid ◽  
Strong Markov

In this paper we extend our previous work. We consider the local-time process L of a strong Markov process X, add negative drift to L, and reflect it à la Skorokhod to obtain a process Q. The reflection of X, together with Q, is, in some sense, a macroscopic model for a service system with two priorities. We derive an expression for the joint law of the duration of an excursion, the maximum value of the process on it, and the time between successive excursions. We work with a properly constructed stationary version of the process. Examples are also given in the paper.

scholarly journals Analysis of stochastic fluid queues driven by local-time processes

2008 ◽  
Vol 40 (04) ◽  
pp. 1072-1103 ◽  
Author(s):  
Takis Konstantopoulos ◽  
Andreas E. Kyprianou ◽  
Paavo Salminen ◽  
Marina Sirviö
Keyword(s):  
Local Time ◽  
Lévy Process ◽  
Point Processes ◽  
Levy Process ◽  
Fluid Queue ◽  
Idle Period ◽  
Priority Service ◽  
Stationary Version ◽  
Palm Calculus ◽  
Stochastic Fluid

We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically (but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator), hence making the theory of Lévy processes applicable. Another important ingredient in our approach is the use of Palm calculus for stationary random point processes and measures.

scholarly journals Analysis of stochastic fluid queues driven by local-time processes

2008 ◽  
Vol 40 (4) ◽  
pp. 1072-1103 ◽  
Author(s):  
Takis Konstantopoulos ◽  
Andreas E. Kyprianou ◽  
Paavo Salminen ◽  
Marina Sirviö
Keyword(s):  
Local Time ◽  
Lévy Process ◽  
Point Processes ◽  
Levy Process ◽  
Fluid Queue ◽  
Idle Period ◽  
Priority Service ◽  
Stationary Version ◽  
Palm Calculus ◽  
Stochastic Fluid

We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically (but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator), hence making the theory of Lévy processes applicable. Another important ingredient in our approach is the use of Palm calculus for stationary random point processes and measures.

Limiting conditional and conditional invariant distributions for the Poisson process with negative drift

1999 ◽  
Vol 36 (04) ◽  
pp. 1194-1209 ◽  
Author(s):  
Raúl Fierro ◽  
Servet Martínez ◽  
Jaime San Martín
Keyword(s):  
Initial Point ◽  
Time Process ◽  
Conditional Distributions ◽  
Exponential Law ◽  
Invariant Distributions ◽  
Virtual Waiting Time ◽  
The Family ◽  
Waiting Time Process ◽  
Negative Drift ◽  
Limiting Behaviour

In this paper we study the conditional limiting behaviour for the virtual waiting time process for the queue M/D/1. We describe the family of conditional invariant distributions which are continuous and parametrized by the eigenvalues λ ∊ (0, λ c ], as it happens for diffusions. In this case, there is a periodic dependence of the limiting conditional distributions on the initial point and the minimal conditional invariant distribution is a mixture, according to an exponential law, of the limiting conditional distributions.

scholarly journals On the local time process of a skew Brownian motion

2019 ◽  
Vol 372 (5) ◽  
pp. 3597-3618 ◽  
Author(s):  
Andrei Borodin ◽  
Paavo Salminen
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
Time Process ◽  
Skew Brownian Motion

Drift Parameter Estimation for a Reflected Fractional Brownian Motion Based on its Local Time

2013 ◽  
Vol 50 (02) ◽  
pp. 592-597 ◽  
Author(s):  
Yaozhong Hu ◽  
Chihoon Lee
Keyword(s):  
Parameter Estimation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Local Time ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Service Rate ◽  
Time Process ◽  
State Process ◽  
Drift Parameter

We consider a drift parameter estimation problem when the state process is a reflected fractional Brownian motion (RFBM) with a nonzero drift parameter and the observation is the associated local time process. The RFBM process arises as the key approximating process for queueing systems with long-range dependent and self-similar input processes, where the drift parameter carries the physical meaning of the surplus service rate and plays a central role in the heavy-traffic approximation theory for queueing systems. We study a statistical estimator based on the cumulative local time process and establish its strong consistency and asymptotic normality.

A discussion on orbital analysis - Variations in air density at 280 km height

1967 ◽  
Vol 262 (1124) ◽  
pp. 195-199
Keyword(s):  
Diurnal Variation ◽  
Local Time ◽  
Atmospheric Density ◽  
Air Drag ◽  
Average Value ◽  
Air Density ◽  
Maximum Value ◽  
Orbital Analysis

The effect of air drag on the orbits of six Cosmos satellites having low perigee heights has been investigated. The diurnal variation of neutral atmospheric density at about 280 km is shown to have an amplitude of about 25% of its average value and to have a maximum value at about 14 h local time.

scholarly journals A Lévy-Driven Stochastic Queueing System with Server Breakdowns and Vacations

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1239
Author(s):  
Yi Peng ◽  
Jinbiao Wu
Keyword(s):  
Communication Networks ◽  
Queueing System ◽  
Stopping Times ◽  
Stochastic Decomposition ◽  
Martingale Method ◽  
Computer Communication ◽  
Fluid Queues ◽  
Random Jumps ◽  
Stochastic Fluid ◽  
Decomposition Properties

Motivated by modelling the data transmission in computer communication networks, we study a Lévy-driven stochastic fluid queueing system where the server may subject to breakdowns and repairs. In addition, the server will leave for a vacation each time when the system is empty. We cast the workload process as a Lévy process modified to have random jumps at two classes of stopping times. By using the properties of Lévy processes and Kella–Whitt martingale method, we derive the limiting distribution of the workload process. Moreover, we investigate the busy period and the correlation structure. Finally, we prove that the stochastic decomposition properties also hold for fluid queues with Lévy input.

Limiting conditional and conditional invariant distributions for the Poisson process with negative drift

1999 ◽  
Vol 36 (4) ◽  
pp. 1194-1209 ◽  
Author(s):  
Raúl Fierro ◽  
Servet Martínez ◽  
Jaime San Martín
Keyword(s):  
Initial Point ◽  
Time Process ◽  
Conditional Distributions ◽  
Exponential Law ◽  
Invariant Distributions ◽  
Virtual Waiting Time ◽  
The Family ◽  
Waiting Time Process ◽  
Negative Drift ◽  
Limiting Behaviour

In this paper we study the conditional limiting behaviour for the virtual waiting time process for the queue M/D/1. We describe the family of conditional invariant distributions which are continuous and parametrized by the eigenvalues λ ∊ (0, λc], as it happens for diffusions. In this case, there is a periodic dependence of the limiting conditional distributions on the initial point and the minimal conditional invariant distribution is a mixture, according to an exponential law, of the limiting conditional distributions.

scholarly journals A generalized occupation time formula for continuous semimartingales

2010 ◽  
Vol 47 (1) ◽  
pp. 54-58
Author(s):  
Raouf Ghomrasni
Keyword(s):  
Local Time ◽  
Wide Class ◽  
Quadratic Variation ◽  
Occupation Time ◽  
Time Process ◽  
Continuous Semimartingale ◽  
Class Of Functions ◽  
Variation Process

We show that for a wide class of functions F we have lim \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim }\limits_{\varepsilon \downarrow 0} \frac{1}{\varepsilon }\int\limits_0^t {\{ F(s,X_s ) - F(s,X_s - \varepsilon )\} d\left\langle {X,X} \right\rangle _s = - } \int\limits_0^t {\int\limits_\mathbb{R} {F(s,x)dL_s^x } }$$ \end{document} where Xt is a continuous semimartingale, ( Ltx , x ∈ ℝ, t ≧ 0) its local time process and (〈 X, X 〉 t , t ≧ 0) its quadratic variation process.

Sign in / Sign up

Export Citation Format

Share Document