# On the convergence to stationarity of birth-death processes

P. Coolen-Schrijner, Erik A. van Doorn

Research output: Book/ReportReportOther research output

### Abstract

Taking up a recent proposal by Stadje and Parthasarathy in the \linebreak[4] setting of the many-server Poisson queue, we consider the integral \linebreak[4] $\int_0^{\infty}[\lim_{u\to\infty} E(X(u))-E(X(t))]dt$ as a measure of the speed of convergence towards stationarity of the process $\{X(t), t \geq 0\}$, and evaluate the integral explicitly in terms of the parameters of the process in the case that $\{X(t), t \geq 0\}$ is an ergodic birth-death process on $\{0,1,\ldots\}$ starting in 0. We also discuss the discrete-time counterpart of this result, and examine some specific examples.
Original language Undefined Enschede University of Twente, Department of Applied Mathematics Published - 2000

### Publication series

Name Memorandum / Department of Applied Mathematics Department of Applied Mathematics, University of Twente 1554 0169-2690

• MSC-60J80
• IR-65741
• EWI-3374

### Cite this

Coolen-Schrijner, P., & van Doorn, E. A. (2000). On the convergence to stationarity of birth-death processes. (Memorandum / Department of Applied Mathematics; No. 1554). Enschede: University of Twente, Department of Applied Mathematics.
Coolen-Schrijner, P. ; van Doorn, Erik A. / On the convergence to stationarity of birth-death processes. Enschede : University of Twente, Department of Applied Mathematics, 2000. (Memorandum / Department of Applied Mathematics; 1554).
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abstract = "Taking up a recent proposal by Stadje and Parthasarathy in the \linebreak[4] setting of the many-server Poisson queue, we consider the integral \linebreak[4] $\int_0^{\infty}[\lim_{u\to\infty} E(X(u))-E(X(t))]dt$ as a measure of the speed of convergence towards stationarity of the process $\{X(t), t \geq 0\}$, and evaluate the integral explicitly in terms of the parameters of the process in the case that $\{X(t), t \geq 0\}$ is an ergodic birth-death process on $\{0,1,\ldots\}$ starting in 0. We also discuss the discrete-time counterpart of this result, and examine some specific examples.",
keywords = "MSC-60J80, IR-65741, EWI-3374",
author = "P. Coolen-Schrijner and {van Doorn}, {Erik A.}",
note = "Imported from MEMORANDA",
year = "2000",
language = "Undefined",
series = "Memorandum / Department of Applied Mathematics",
publisher = "University of Twente, Department of Applied Mathematics",
number = "1554",

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Coolen-Schrijner, P & van Doorn, EA 2000, On the convergence to stationarity of birth-death processes. Memorandum / Department of Applied Mathematics, no. 1554, University of Twente, Department of Applied Mathematics, Enschede.

On the convergence to stationarity of birth-death processes. / Coolen-Schrijner, P.; van Doorn, Erik A.

Enschede : University of Twente, Department of Applied Mathematics, 2000. (Memorandum / Department of Applied Mathematics; No. 1554).

Research output: Book/ReportReportOther research output

TY - BOOK

T1 - On the convergence to stationarity of birth-death processes

AU - Coolen-Schrijner, P.

AU - van Doorn, Erik A.

N1 - Imported from MEMORANDA

PY - 2000

Y1 - 2000

N2 - Taking up a recent proposal by Stadje and Parthasarathy in the \linebreak[4] setting of the many-server Poisson queue, we consider the integral \linebreak[4] $\int_0^{\infty}[\lim_{u\to\infty} E(X(u))-E(X(t))]dt$ as a measure of the speed of convergence towards stationarity of the process $\{X(t), t \geq 0\}$, and evaluate the integral explicitly in terms of the parameters of the process in the case that $\{X(t), t \geq 0\}$ is an ergodic birth-death process on $\{0,1,\ldots\}$ starting in 0. We also discuss the discrete-time counterpart of this result, and examine some specific examples.

AB - Taking up a recent proposal by Stadje and Parthasarathy in the \linebreak[4] setting of the many-server Poisson queue, we consider the integral \linebreak[4] $\int_0^{\infty}[\lim_{u\to\infty} E(X(u))-E(X(t))]dt$ as a measure of the speed of convergence towards stationarity of the process $\{X(t), t \geq 0\}$, and evaluate the integral explicitly in terms of the parameters of the process in the case that $\{X(t), t \geq 0\}$ is an ergodic birth-death process on $\{0,1,\ldots\}$ starting in 0. We also discuss the discrete-time counterpart of this result, and examine some specific examples.

KW - MSC-60J80

KW - IR-65741

KW - EWI-3374

M3 - Report

T3 - Memorandum / Department of Applied Mathematics

BT - On the convergence to stationarity of birth-death processes

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -

Coolen-Schrijner P, van Doorn EA. On the convergence to stationarity of birth-death processes. Enschede: University of Twente, Department of Applied Mathematics, 2000. (Memorandum / Department of Applied Mathematics; 1554).