Burke's theorem on passage times in Gordon–Newell networks

In a closed cycle of exponential queues where the first and the last nodes are multiserver queues while the other nodes are single-server queues, the cycle-time distribution has a simple product form. The same result holds for passage-time distributions on overtake-free paths in Gordon–Newell networks. In brief, we prove Burke's theorem on passage times in closed networks.

Download Full-text

Laplace transform inversion and passage-time distributions in Markov processes

Journal of Applied Probability ◽

10.1017/s0021900200038432 ◽

1990 ◽

Vol 27 (01) ◽

pp. 74-87 ◽

Cited By ~ 4

Author(s):

Peter G. Harrison

Keyword(s):

Differential Equation ◽

Queueing Networks ◽

Time Distribution ◽

Passage Time ◽

Weighted Sums ◽

Order Partial Differential Equation ◽

Exponential Distributions ◽

First Order ◽

Laplace Transform Inversion ◽

Passage Times

Products of the Laplace transforms of exponential distributions with different parameters are inverted to give a mixture of Erlang densities, i.e. an expression for the convolution of exponentials. The formula for these inversions is expressed both as an explicit sum and in terms of a recurrence relation which is better suited to numerical computation. The recurrence for the inversion of certain weighted sums of these transforms is then solved by converting it into a linear first-order partial differential equation. The result may be used to find the density function of passage times between states in a Markov process and it is applied to derive an expression for cycle time distribution in tree-structured Markovian queueing networks.

Download Full-text

Laplace transform inversion and passage-time distributions in Markov processes

Journal of Applied Probability ◽

10.2307/3214596 ◽

1990 ◽

Vol 27 (1) ◽

pp. 74-87 ◽

Cited By ~ 24

Author(s):

Peter G. Harrison

Keyword(s):

Differential Equation ◽

Queueing Networks ◽

Time Distribution ◽

Passage Time ◽

Weighted Sums ◽

Order Partial Differential Equation ◽

Exponential Distributions ◽

First Order ◽

Laplace Transform Inversion ◽

Passage Times

Products of the Laplace transforms of exponential distributions with different parameters are inverted to give a mixture of Erlang densities, i.e. an expression for the convolution of exponentials. The formula for these inversions is expressed both as an explicit sum and in terms of a recurrence relation which is better suited to numerical computation. The recurrence for the inversion of certain weighted sums of these transforms is then solved by converting it into a linear first-order partial differential equation. The result may be used to find the density function of passage times between states in a Markov process and it is applied to derive an expression for cycle time distribution in tree-structured Markovian queueing networks.

Download Full-text

First passage time for the state of exact chemical equilibrium

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19800777 ◽

1980 ◽

Vol 45 (3) ◽

pp. 777-782 ◽

Cited By ~ 1

Author(s):

Milan Šolc

Keyword(s):

Chemical Equilibrium ◽

First Passage Time ◽

Passage Time ◽

The State ◽

Recurrence Time ◽

First Passage ◽

First Order ◽

The Mean ◽

Passage Times ◽

Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

Download Full-text

Spectral Theory for Skip-Free Markov Chains

Probability in the Engineering and Informational Sciences ◽

10.1017/s026996480000098x ◽

1989 ◽

Vol 3 (1) ◽

pp. 77-88 ◽

Cited By ~ 9

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.

Download Full-text

Dynamical behavior of simplified FitzHugh–Nagumo neural system with time delay driven by Lévy noise

International Journal of Modern Physics B ◽

10.1142/s0217979221502404 ◽

2021 ◽

pp. 2150240

Author(s):

Weida Qiu ◽

Yongfeng Guo ◽

Xiuxian Yu

Keyword(s):

Time Delay ◽

First Passage Time ◽

Dynamical Behavior ◽

Passage Time ◽

Neural System ◽

The Other ◽

Lévy Noise ◽

First Passage ◽

Levy Noise ◽

System With Time Delay

In this paper, the dynamical behavior of the FitzHugh–Nagumo (FHN) neural system with time delay driven by Lévy noise is studied from two aspects: the mean first-passage time (MFPT) and the probability density function (PDF) of the first-passage time (FPT). Using the Janicki–Weron algorithm to generate the Lévy noise, and through the order-4 Runge–Kutta algorithm to simulate the FHN system response, the time that the system needs from one stable state to the other one is tracked in the process. Using the MATLAB software to simulate the process above 20,000 times and recording the PFTs, the PDF of the FPT and the MFPT is obtained. Finally, the effects of the Lévy noise and time-delay on the FPT are discussed. It is found that the increase of both time-delay feedback intensity and Lévy noise intensity can promote the transition of the particle from the resting state to the excited state. However, the two parameters produce the opposite effects in the other direction.

Download Full-text

Iterative algorithm for the first passage time distribution in a jump–diffusion model with regime-switching, and its applications

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2015.08.015 ◽

2016 ◽

Vol 294 ◽

pp. 177-195 ◽

Cited By ~ 3

Author(s):

Jerim Kim ◽

Bara Kim ◽

In-Suk Wee

Keyword(s):

Iterative Algorithm ◽

Diffusion Model ◽

Regime Switching ◽

Time Distribution ◽

First Passage Time ◽

Jump Diffusion ◽

Passage Time ◽

First Passage ◽

Jump Diffusion Model

Download Full-text

Probing Protein Shelf Lives from Inverse Mean First Passage Times

10.26434/chemrxiv.9529163 ◽

2019 ◽

Author(s):

Vishal Singh ◽

Parbati Biswas

Keyword(s):

Protein Aggregation ◽

Rate Constants ◽

Survival Probability ◽

First Passage Time ◽

Passage Time ◽

First Passage ◽

First Order Kinetics ◽

The Mean ◽

Passage Times ◽

Good Agreement

Protein aggregation is investigated theoretically via protein turnover, misfolding, aggregation and degradation. The Mean First Passage Time (MFPT) of aggregation is evaluated within the framework of Chemical Master Equation (CME) and pseudo first order kinetics with appropriate boundary conditions. The rate constants of aggregation of different proteins are calculated from the inverse MFPT, which show an excellent match with the experimentally reported rate constants and those extracted from the ThT/ThS fluorescence data. Protein aggregation is found to be practically independent of the number of contacts and the critical number of misfolded contacts. The age of appearance of aggregation-related diseases is obtained from the survival probability and the MFPT results, which matches with those reported in the literature. The calculated survival probability is in good agreement with the only available clinical data for Parkinson’s disease.<br>

Download Full-text

Identifying Coefficients in the Spectral Representation for First Passage Time Distributions

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800000309 ◽

1987 ◽

Vol 1 (1) ◽

pp. 69-74 ◽

Cited By ~ 26

Author(s):

Mark Brown ◽

Yi-Shi Shao

Keyword(s):

Markov Processes ◽

Time Distribution ◽

Infinitesimal Generator ◽

Spectral Representation ◽

First Passage Time ◽

Passage Time ◽

Spectral Approach ◽

Generator Matrix ◽

Eigenvalues And Eigenvectors ◽

First Passage

The spectral approach to first passage time distributions for Markov processes requires knowledge of the eigenvalues and eigenvectors of the infinitesimal generator matrix. We demonstrate that in many cases knowledge of the eigenvalues alone is sufficient to compute the first passage time distribution.

Download Full-text

Single server queues with modified service mechanisms

Journal of the Australian Mathematical Society ◽

10.1017/s144678870002749x ◽

1962 ◽

Vol 2 (4) ◽

pp. 499-507 ◽

Cited By ~ 32

Author(s):

G. F. Yeo

Keyword(s):

Time Distribution ◽

Busy Period ◽

Queueing System ◽

Probability Generating Function ◽

Single Server ◽

Waiting Time Distribution ◽

Period Distribution ◽

Time Dependent Problem ◽

Stationary Waiting Time ◽

Special Case

SummaryThis paper considers a generalisation of the queueing system M/G/I, where customers arriving at empty and non-empty queues have different service time distributions. The characteristic function (c.f.) of the stationary waiting time distribution and the probability generating function (p.g.f.) of the queue size are obtained. The busy period distribution is found; the results are generalised to an Erlangian inter-arrival distribution; the time-dependent problem is considered, and finally a special case of server absenteeism is discussed.

Download Full-text