# Families of birth-death processes with similar time-dependent behaviour

R.B. Lenin, P.R. Parthasarathy, Willem R.W. Scheinhardt, Erik A. van Doorn

Research output: Book/ReportReportProfessional

### Abstract

We consider birth-death processes taking values in ${\cal N} \equiv \{0,1,\ldots\}$, but allow the death rate in state $0$ to be positive, so that escape from ${\cal N}$ is possible. Two such processes with transition functions $\{p_{ij}(t)\}$ and $\{{\tilde p}_{ij}(t)\}$ are said to be {\it similar} if, for all $i,j \in {\cal N},$ there are constants $c_{ij}$ such that ${\tilde p}_{ij}(t) = c_{ij}p_{ij}(t)$ for all $t \geq 0.$ We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family.
Original language Undefined Enschede University of Twente, Department of Applied Mathematics 14 0169-2690 Published - 1999

### Publication series

Name Memorandum / Department of Applied Mathematics Department of Applied Mathematics, University of Twente 1486 0169-2690

• EWI-3306
• IR-65675
• METIS-141298
• MSC-60J80

### Cite this

Lenin, R. B., Parthasarathy, P. R., Scheinhardt, W. R. W., & van Doorn, E. A. (1999). Families of birth-death processes with similar time-dependent behaviour. (Memorandum / Department of Applied Mathematics; No. 1486). Enschede: University of Twente, Department of Applied Mathematics.
Lenin, R.B. ; Parthasarathy, P.R. ; Scheinhardt, Willem R.W. ; van Doorn, Erik A. / Families of birth-death processes with similar time-dependent behaviour. Enschede : University of Twente, Department of Applied Mathematics, 1999. 14 p. (Memorandum / Department of Applied Mathematics; 1486).
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title = "Families of birth-death processes with similar time-dependent behaviour",
abstract = "We consider birth-death processes taking values in ${\cal N} \equiv \{0,1,\ldots\}$, but allow the death rate in state $0$ to be positive, so that escape from ${\cal N}$ is possible. Two such processes with transition functions $\{p_{ij}(t)\}$ and $\{{\tilde p}_{ij}(t)\}$ are said to be {\it similar} if, for all $i,j \in {\cal N},$ there are constants $c_{ij}$ such that ${\tilde p}_{ij}(t) = c_{ij}p_{ij}(t)$ for all $t \geq 0.$ We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family.",
keywords = "EWI-3306, IR-65675, METIS-141298, MSC-60J80",
author = "R.B. Lenin and P.R. Parthasarathy and Scheinhardt, {Willem R.W.} and {van Doorn}, {Erik A.}",
note = "Imported from MEMORANDA",
year = "1999",
language = "Undefined",
isbn = "0169-2690",
series = "Memorandum / Department of Applied Mathematics",
publisher = "University of Twente, Department of Applied Mathematics",
number = "1486",

}

Lenin, RB, Parthasarathy, PR, Scheinhardt, WRW & van Doorn, EA 1999, Families of birth-death processes with similar time-dependent behaviour. Memorandum / Department of Applied Mathematics, no. 1486, University of Twente, Department of Applied Mathematics, Enschede.

Families of birth-death processes with similar time-dependent behaviour. / Lenin, R.B.; Parthasarathy, P.R.; Scheinhardt, Willem R.W.; van Doorn, Erik A.

Enschede : University of Twente, Department of Applied Mathematics, 1999. 14 p. (Memorandum / Department of Applied Mathematics; No. 1486).

Research output: Book/ReportReportProfessional

TY - BOOK

T1 - Families of birth-death processes with similar time-dependent behaviour

AU - Lenin, R.B.

AU - Parthasarathy, P.R.

AU - Scheinhardt, Willem R.W.

AU - van Doorn, Erik A.

N1 - Imported from MEMORANDA

PY - 1999

Y1 - 1999

N2 - We consider birth-death processes taking values in ${\cal N} \equiv \{0,1,\ldots\}$, but allow the death rate in state $0$ to be positive, so that escape from ${\cal N}$ is possible. Two such processes with transition functions $\{p_{ij}(t)\}$ and $\{{\tilde p}_{ij}(t)\}$ are said to be {\it similar} if, for all $i,j \in {\cal N},$ there are constants $c_{ij}$ such that ${\tilde p}_{ij}(t) = c_{ij}p_{ij}(t)$ for all $t \geq 0.$ We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family.

AB - We consider birth-death processes taking values in ${\cal N} \equiv \{0,1,\ldots\}$, but allow the death rate in state $0$ to be positive, so that escape from ${\cal N}$ is possible. Two such processes with transition functions $\{p_{ij}(t)\}$ and $\{{\tilde p}_{ij}(t)\}$ are said to be {\it similar} if, for all $i,j \in {\cal N},$ there are constants $c_{ij}$ such that ${\tilde p}_{ij}(t) = c_{ij}p_{ij}(t)$ for all $t \geq 0.$ We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family.

KW - EWI-3306

KW - IR-65675

KW - METIS-141298

KW - MSC-60J80

M3 - Report

SN - 0169-2690

T3 - Memorandum / Department of Applied Mathematics

BT - Families of birth-death processes with similar time-dependent behaviour

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -

Lenin RB, Parthasarathy PR, Scheinhardt WRW, van Doorn EA. Families of birth-death processes with similar time-dependent behaviour. Enschede: University of Twente, Department of Applied Mathematics, 1999. 14 p. (Memorandum / Department of Applied Mathematics; 1486).