scholarly journals A Birth and Death Process Related to the Rogers–Ramanujan Continued Fraction

1998 ◽  
Vol 224 (2) ◽  
pp. 297-315 ◽  
Author(s):  
P.R Parthasarathy ◽  
R.B Lenin ◽  
W Schoutens ◽  
W Van Assche
Keyword(s):  
Continued Fraction ◽  
Death Process ◽  
Birth And Death Process

Excursions of birth and death processes, orthogonal polynomials, and continued fractions

1999 ◽  
Vol 36 (3) ◽  
pp. 752-770 ◽  
Author(s):  
Fabrice Guillemin ◽  
Didier Pinchon
Keyword(s):  
Orthogonal Polynomial ◽  
Continued Fractions ◽  
Spectral Measure ◽  
Continued Fraction ◽  
Transition Probability ◽  
Polynomial System ◽  
Death Process ◽  
Probabilistic Interpretation ◽  
Birth And Death Process ◽  
Birth And Death Processes

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

scholarly journals Exact representations for tumour incidence for some density-dependent models

2005 ◽  
Vol 2005 (16) ◽  
pp. 2655-2667
Author(s):  
P. R. Parthasarathy ◽  
Klaus Dietz
Keyword(s):  
Continued Fraction ◽  
Death Process ◽  
Transition Rates ◽  
Birth And Death Process ◽  
System Of Difference Equations ◽  
Normal Cells ◽  
Death Rates ◽  
Density Dependent ◽  
Proliferation And Differentiation ◽  
Complete Solutions

Carcinogenesis is a multistage random process involving generic changes and stochastic proliferation and differentiation of normal cells and genetically altered stem cells. In this paper, we present the probability of time to tumour onset for a carcinogenesis model wherein the cells grow according to a birth and death process with density-dependent birth and death rates. This is achieved by transforming the underlying system of difference equations which results in a continued fraction. This continued fraction approach helps us to find the complete solutions. The popular Moolgavkar-Venzon-Knudson (MVK) model assumes constant birth, death, and transition rates.

Excursions of birth and death processes, orthogonal polynomials, and continued fractions

1999 ◽  
Vol 36 (03) ◽  
pp. 752-770 ◽  
Author(s):  
Fabrice Guillemin ◽  
Didier Pinchon
Keyword(s):  
Orthogonal Polynomial ◽  
Continued Fractions ◽  
Spectral Measure ◽  
Continued Fraction ◽  
Transition Probability ◽  
Polynomial System ◽  
Death Process ◽  
Probabilistic Interpretation ◽  
Birth And Death Process ◽  
Birth And Death Processes

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

A stochastic process related to language change

1970 ◽  
Vol 7 (01) ◽  
pp. 69-78 ◽  
Author(s):  
Barron Brainerd
Keyword(s):  
Stochastic Process ◽  
Language Change ◽  
Death Process ◽  
Birth And Death Process ◽  
Standard Methods

The purpose of this note is two-fold. First, to introduce the mathematical reader to a group of problems in the study of language change which has received little attention from mathematicians and probabilists. Secondly, to introduce a birth and death process, arising naturally out of this group of problems, which has received little attention in the literature. This process can be solved using the standard methods and the solution is exhibited here.

Multi-Location Inventory Model Considering Bidirectional and Unidirectional Transshipments Simultaneously

2013 ◽  
Vol 694-697 ◽  
pp. 2742-2745
Author(s):  
Jin Hong Zhong ◽  
Yun Zhou
Keyword(s):  
Process Model ◽  
Approximate Calculation ◽  
Inventory Model ◽  
Inventory System ◽  
Death Process ◽  
Birth And Death Process ◽  
Inventory Models ◽  
Continuous Review ◽  
Poisson Demand ◽  
Queue Theory

Abstract. A cross-regional multi-site inventory system with independent Poisson demand and continuous review (S-1,S) policy, in which there is bidirectional transshipment between the locations at the same area, and unidirectional transshipment between the locations at the different area. According to the M/G/S/S queue theory, birth and death process model and approximate calculation policy, we established inventory models respectively for the loss sales case and backorder case, and designed corresponding procedures to solve them. Finally, we verify the effectiveness of proposed models and methods by means of a lot of contrast experiments.

AN AGE-DEPENDENT BIRTH AND DEATH PROCESS

Biometrika ◽  
1955 ◽  
Vol 42 (3-4) ◽  
pp. 291-306 ◽  
Author(s):  
W. A. O'N WAUGH
Keyword(s):  
Death Process ◽  
Birth And Death Process ◽  
Age Dependent

On a property of finite-state birth and death processes

1986 ◽  
Vol 23 (04) ◽  
pp. 859-866
Author(s):  
A. J. Branford
Keyword(s):  
Simple Proof ◽  
Death Process ◽  
Birth And Death Process ◽  
Birth And Death Processes ◽  
Converse Result ◽  
Finite State ◽  
Event Times

A simple proof is given of the result that the ‘overflow' from a finite-state birth and death process is a renewal stream characterized by hyperexponential inter-event times. Our structure is utilized to give a converse result that any hyperexponential renewal stream can be so produced as the overflow from a finite-state birth and death process.

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