Transient behavior of the M/M/1 queue via Laplace transforms

1988 ◽  
Vol 20 (1) ◽  
pp. 145-178 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
First Passage Time ◽  
Operational Calculus ◽  
Transient Behavior ◽  
Building Blocks ◽  
Passage Time ◽  
Reflected Brownian Motion ◽  
Dependent Behavior ◽  
The Laplace Transform ◽  
First Moment ◽  
Special Case

This paper shows how the Laplace transform analysis of Bailey (1954), (1957) can be continued to yield additional insights about the time-dependent behavior of the queue-length process in the M/M/1 model. A transform factorization is established that leads to a decomposition of the first moment as a function of time into two monotone components. This factorization facilitates developing approximations for the moments and determining their asymptotic behavior as . All descriptions of the transient behavior are expressed in terms of basic building blocks such as the first-passage-time distributions. The analysis is facilitated by appropriate scaling of space and time so that regulated or reflected Brownian motion (RBM) appears as the special case in which the traffic intensity ρ equals the critical value 1. An operational calculus is developed for obtaining M/M/1 results directly from corresponding RBM results as well as vice versa. The analysis thus provides useful insight about RBM approximations for queues.

Transient behavior of the M/M/1 queue via Laplace transforms

1988 ◽  
Vol 20 (01) ◽  
pp. 145-178 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
First Passage Time ◽  
Operational Calculus ◽  
Transient Behavior ◽  
Building Blocks ◽  
Passage Time ◽  
Reflected Brownian Motion ◽  
Dependent Behavior ◽  
The Laplace Transform ◽  
First Moment ◽  
Special Case

This paper shows how the Laplace transform analysis of Bailey (1954), (1957) can be continued to yield additional insights about the time-dependent behavior of the queue-length process in theM/M/1 model. A transform factorization is established that leads to a decomposition of the first moment as a function of time into two monotone components. This factorization facilitates developing approximations for the moments and determining their asymptotic behavior as. All descriptions of the transient behavior are expressed in terms of basic building blocks such as the first-passage-time distributions. The analysis is facilitated by appropriate scaling of space and time so that regulated or reflected Brownian motion (RBM) appears as the special case in which the traffic intensity ρ equals the critical value 1. An operational calculus is developed for obtainingM/M/1 results directly from corresponding RBM results as well as vice versa. The analysis thus provides useful insight about RBM approximations for queues.

Spectral Theory for Skip-Free Markov Chains

1989 ◽  
Vol 3 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Markov Chains ◽  
Spectral Theory ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage Times ◽  
Simple Relationship ◽  
First Passage ◽  
Birth And Death Processes ◽  
The Laplace Transform ◽  
Passage Times

The distribution of upward first passage times in skip-free Markov chains can be expressed solely in terms of the eigenvalues in the spectral representation, without performing a separate calculation to determine the eigenvectors. We provide insight into this result and skip-free Markov chains more generally by showing that part of the spectral theory developed for birth-and-death processes extends to skip-free chains. We show that the eigenvalues and eigenvectors of skip-free chains can be characterized in terms of recursively defined polynomials. Moreover, the Laplace transform of the upward first passage time from 0 to n is the reciprocal of the nth polynomial. This simple relationship holds because the Laplace transforms of the first passage times satisfy the same recursion as the polynomials except for a normalization.

Bounds and Approximations for the Transient Behavior of Continuous-Time Markov Chains

1989 ◽  
Vol 3 (2) ◽  
pp. 175-198 ◽  
Author(s):  
Bok Sik Yoon ◽  
J. George Shanthikumar
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
First Passage Time ◽  
Transient Behavior ◽  
Passage Time ◽  
Lower And Upper Bounds ◽  
Simple Method ◽  
Continuous Time Markov Chains ◽  
Recent Addition ◽  
Dependent Probability

Discretization is a simple, yet powerful tool in obtaining time-dependent probability distribution of continuous-time Markov chains. One of the most commonly used approaches is uniformization. A recent addition to such approaches is an external uniformization technique. In this paper, we briefly review these different approaches, propose some new approaches, and discuss their performances based on theoretical bounds and empirical computational results. A simple method to get lower and upper bounds for first passage time distribution is also proposed.

Hitting-Time Densities of a Two-Dimensional Markov Process

1992 ◽  
Vol 6 (4) ◽  
pp. 561-580
Author(s):  
C. H. Hesse
Keyword(s):  
Global Analysis ◽  
First Passage Time ◽  
Type Equation ◽  
Passage Time ◽  
Two Dimensional ◽  
First Passage ◽  
Wide Range ◽  
The Laplace Transform ◽  
Mean And Variance

This paper deals with the two-dimensional stochastic process (X(t), V(t)) where dX(t) = V(t)dt, V(t) = W(t) + ν for some constant ν and W(t) is a one-dimensional Wiener process with zero mean and variance parameter σ2= 1. We are interested in the first-passage time of (X(t), V(t)) to the plane X = 0 for a process starting from (X(0) = −x, V(0) = ν) with x > 0. The partial differential equation for the Laplace transform of the first-passage time density is transformed into a Schrödinger-type equation and, using methods of global analysis, such as the method of dominant balance, an approximation to the first-passage density is obtained. In a series of simulations, the quality of this approximation is checked. Over a wide range of x and ν it is found to perform well, globally in t. Some applications are mentioned.

scholarly journals Three Essays on Stopping

Risks ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 105
Author(s):  
Eberhard Mayerhofer
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Alternative Proof ◽  
Sufficient Condition ◽  
Diffusion Characteristic ◽  
Reflected Brownian Motion ◽  
First Passage ◽  
Brownian Motion With Drift ◽  
Spectrally Negative Lévy Process ◽  
Closed Form Formula

First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This corrects a formula by Perry et al. (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown, if and only if the diffusion characteristic μ / σ 2 is constant. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give an alternative proof for the fact that the maximum before a fixed drawdown is exponentially distributed for any spectrally negative Lévy process, a result due to Mijatović and Pistorius (2012). Our proof is similar, but simpler than Lehoczky (1977) or Landriault et al. (2017).

scholarly journals The First Passage Time and the Dividend Value Function for One-Dimensional Diffusion Processes between Two Reflecting Barriers

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Chuancun Yin ◽  
Huiqing Wang
Keyword(s):  
Boundary Conditions ◽  
Value Function ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
One Dimensional ◽  
The Laplace Transform ◽  
The Value Function ◽  
Reflecting Barriers

We consider the general one-dimensional time-homogeneous regular diffusion process between two reflecting barriers. An approach based on the Itô formula with corresponding boundary conditions allows us to derive the differential equations with boundary conditions for the Laplace transform of the first passage time and the value function. As examples, the explicit solutions of them for several popular diffusions are obtained. In addition, some applications to risk theory are considered.

scholarly journals AN INTEGRAL EQUATION FOR THE DISTRIBUTION OF THE FIRST EXIT TIME OF A REFLECTED BROWNIAN MOTION

2009 ◽  
Vol 50 (4) ◽  
pp. 445-454 ◽  
Author(s):  
VICTOR DE-LA-PEÑA ◽  
GERARDO HERNÁNDEZ-DEL-VALLE ◽  
CARLOS G. PACHECO-GONZÁLEZ
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
First Passage Time ◽  
Exit Time ◽  
Numerical Procedure ◽  
Passage Time ◽  
Reflected Brownian Motion ◽  
First Exit Time ◽  
Time Problem ◽  
First Passage Time Problem

AbstractReflected Brownian motion is used in areas such as physiology, electrochemistry and nuclear magnetic resonance. We study the first-passage-time problem of this process which is relevant in applications; specifically, we find a Volterra integral equation for the distribution of the first time that a reflected Brownian motion reaches a nondecreasing barrier. Additionally, we note how a numerical procedure can be used to solve the integral equation.

The first passage time to the equilibrium state of a first order reaction

1987 ◽  
Vol 52 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Milan Šolc
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Order Reaction ◽  
First Order Reaction ◽  
First Passage ◽  
First Order ◽  
Second Moments ◽  
Large Systems ◽  
First Moment ◽  
Number Of Particles

The first passage time to the state of exact equilibrium (the most probable stationary state) for a first order reaction is described in terms of the probability density and the first and second moments. For large systems, the first moment is proportional to the logarithm of the number of particles in the system, while the dispersion is independent of the size of the system.

scholarly journals A Damped Telegraph Random Process with Logistic Stationary Distribution

2010 ◽  
Vol 47 (01) ◽  
pp. 84-96 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Barbara Martinucci
Keyword(s):  
Stochastic Process ◽  
Real Line ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Finite Velocity ◽  
Telegraph Process ◽  
Velocity Changes ◽  
Random Times ◽  
Special Case

We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

scholarly journals First passage upwards for state-dependent-killed spectrally negative Lévy processes

2019 ◽  
Vol 56 (2) ◽  
pp. 472-495 ◽  
Author(s):  
Matija Vidmar
Keyword(s):  
Branching Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Additive Functional ◽  
First Passage ◽  
State Dependent ◽  
Continuous State ◽  
The Laplace Transform ◽  
Exit Probability ◽  
The One

AbstractFor a spectrally negative Lévy process X, killed according to a rate that is a function ω of its position, we complement the recent findings of [12] by analysing (in greater generality) the exit probability of the one-sided upwards passage problem. When ω is strictly positive, this problem is related to the determination of the Laplace transform of the first passage time upwards for X that has been time-changed by the inverse of the additive functional $$\int_0^ \cdot \omega ({X_u}){\kern 1pt} {\rm{d}}u$$. In particular, our findings thus shed extra light on related results concerning first passage times downwards (resp. upwards) of continuous-state branching processes (resp. spectrally negative positive self-similar Markov processes).

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