On the Strong Ratio Limit Property for Discrete-Time Birth-Death Processes

The Strong Ratio Limit Property for Some General Markov Processes

The Annals of Mathematical Statistics ◽

10.1214/aoms/1177697603 ◽

1969 ◽

Vol 40 (3) ◽

pp. 986-992 ◽

Cited By ~ 1

Author(s):

Naresh C. Jain

Keyword(s):

Markov Processes ◽

Limit Property ◽

Ratio Limit

Spectral Representation of Discrete-Time Birth–Death Chains

10.1017/9781009030540.002 ◽

2021 ◽

pp. 1-56

Keyword(s):

Discrete Time ◽

Spectral Representation ◽

Birth Death

On the parity of individuals in a branching process

Journal of Applied Probability ◽

10.1017/s0021900200094286 ◽

1976 ◽

Vol 13 (02) ◽

pp. 219-230 ◽

Cited By ~ 9

Author(s):

J. Gani ◽

I. W. Saunders

Keyword(s):

Discrete Time ◽

Continuous Time ◽

Branching Process ◽

Yeast Cells ◽

Death Process ◽

Second Moments ◽

The Mean ◽

Watson Process ◽

Number Of Cells ◽

Birth Death

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

A birth and death model of neuron firing

Journal of Applied Probability ◽

10.1017/s0021900200036822 ◽

1974 ◽

Vol 11 (02) ◽

pp. 369-373

Author(s):

B. F. Logan ◽

L. A. Shepp

Keyword(s):

Discrete Time ◽

Random Variables ◽

Nerve Cells ◽

Neuron Firing ◽

The Times ◽

Number Of Particles ◽

Birth Death

A simple birth-death model of particle fluctuations is studied where at each discrete time a birth and/or death may occur. We show that if the probability of a birth does not depend on the number of particles present and if births and deaths are independent, then the times between successive deaths are independent geometrically distributed random variables, which is false in the general case. Since the above properties of the times between successive neuron firings have been observed in nerve cells, the model proposed in [2] obtains added credence.

Newton-Shamanskii Method for a Quadratic Matrix Equation Arising in Quasi-Birth-Death Problems

East Asian Journal on Applied Mathematics ◽

10.4208/eajam.040914.301014a ◽

2014 ◽

Vol 4 (4) ◽

pp. 386-395

Author(s):

Pei-Chang Guo

Keyword(s):

Markov Chains ◽

Stationary Distribution ◽

Discrete Time ◽

Matrix Equation ◽

Nonnegative Solution ◽

Minimal Nonnegative Solution ◽

The Minimal Nonnegative Solution ◽

Quadratic Matrix Equation ◽

Quadratic Matrix ◽

Birth Death

AbstractIn order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. The Newton-Shamanskii method is applied to solve this equation, and the sequence of matrices produced is monotonically increasing and converges to its minimal nonnegative solution. Numerical results illustrate the effectiveness of this procedure.

Performance Modeling of Rule-Based Architectures as Discrete-Time Quasi Birth-Death Processes

2005 14th IEEE Workshop on Local & Metropolitan Area Networks ◽

10.1109/lanman.2005.1541542 ◽

2006 ◽

Cited By ~ 1

Author(s):

S. Palugyai ◽

M.J. Csorba

Keyword(s):

Discrete Time ◽

Performance Modeling ◽

Rule Based ◽

Birth Death

Strong ratio limit property

Bulletin of the American Mathematical Society ◽

10.1090/s0002-9904-1961-10694-0 ◽

1961 ◽

Vol 67 (6) ◽

pp. 571-575 ◽

Cited By ~ 26

Author(s):

Steven Orey

Keyword(s):

Limit Property ◽

Ratio Limit

Strong ratio limit property and R-recurrence of reversible Markov chains

Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete ◽

10.1007/bf00532621 ◽

1974 ◽

Vol 30 (4) ◽

pp. 343-356 ◽

Cited By ~ 7

Author(s):

G�tz Kersting

Keyword(s):

Markov Chains ◽

Limit Property ◽

Reversible Markov Chains ◽

Ratio Limit

On the convergence to stationarity of birth-death processes

Journal of Applied Probability ◽

10.1017/s0021900200018854 ◽

2001 ◽

Vol 38 (03) ◽

pp. 696-706 ◽

Cited By ~ 2

Author(s):

Pauline Coolen-Schrijner ◽

Erik A. Van Doorn

Keyword(s):

Discrete Time ◽

Death Process ◽

Speed Of Convergence ◽

Recent Proposal ◽

The Many ◽

Birth Death

Taking up a recent proposal by Stadje and Parthasarathy in the setting of the many-server Poisson queue, we consider the integral ∫0∞[limu→∞E(X(u))-E(X(t))]dtas a measure of the speed of convergence towards stationarity of the process {X(t),t≥0}, and evaluate the integral explicitly in terms of the parameters of the process in the case that {X(t),t≥0} is an ergodic birth-death process on {0,1,….} starting in 0. We also discuss the discrete-time counterpart of this result, and examine some specific examples.

Weak convergence of conditioned birth-death processes in discrete time

Journal of Applied Probability ◽

10.1017/s0021900200100683 ◽

1997 ◽

Vol 34 (01) ◽

pp. 46-53

Author(s):

Pauline Schrijner ◽

Erik A. Van Doorn

Keyword(s):

Weak Convergence ◽

Discrete Time ◽

Transition Probabilities ◽

Death Process ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Absorbing State ◽

Limiting Behaviour ◽

Necessary And Sufficient ◽

Birth Death

We consider a discrete-time birth-death process on the non-negative integers with −1 as an absorbing state and study the limiting behaviour asn →∞ of the process conditioned on non-absorption until timen.By proving that a condition recently proposed by Martinez and Vares is vacuously true, we establish that the conditioned process is always weakly convergent when all self-transition probabilities are zero. In the aperiodic case we obtain a necessary and sufficient condition for weak convergence.