On a property of finite-state birth and death processes

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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On a property of finite-state birth and death processes

Journal of Applied Probability ◽

10.2307/3214460 ◽

1986 ◽

Vol 23 (4) ◽

pp. 859-866 ◽

Cited By ~ 4

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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Quasi-stationary distributions of birth-and-death processes

Advances in Applied Probability ◽

10.1017/s0001867800031050 ◽

1978 ◽

Vol 10 (03) ◽

pp. 570-586 ◽

Cited By ~ 21

Author(s):

James A. Cavender

Keyword(s):

State Space ◽

Stationary Distribution ◽

Parameter Family ◽

Death Process ◽

Birth And Death Process ◽

Stationary Distributions ◽

Birth And Death Processes ◽

Finite State ◽

Finite State Space ◽

Quasi Stationary Distribution

Letqn(t) be the conditioned probability of finding a birth-and-death process in statenat timet,given that absorption into state 0 has not occurred by then. A family {q1(t),q2(t), · · ·} that is constant in time is a quasi-stationary distribution. If any exist, the quasi-stationary distributions comprise a one-parameter family related to quasi-stationary distributions of finite state-space approximations to the process.

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Quasi-stationary distributions of birth-and-death processes

Advances in Applied Probability ◽

10.2307/1426635 ◽

1978 ◽

Vol 10 (3) ◽

pp. 570-586 ◽

Cited By ~ 46

Author(s):

James A. Cavender

Keyword(s):

State Space ◽

Stationary Distribution ◽

Parameter Family ◽

Death Process ◽

Birth And Death Process ◽

Stationary Distributions ◽

Birth And Death Processes ◽

Finite State ◽

Finite State Space ◽

Quasi Stationary Distribution

Let qn(t) be the conditioned probability of finding a birth-and-death process in state n at time t, given that absorption into state 0 has not occurred by then. A family {q1(t), q2(t), · · ·} that is constant in time is a quasi-stationary distribution. If any exist, the quasi-stationary distributions comprise a one-parameter family related to quasi-stationary distributions of finite state-space approximations to the process.

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On the parity of individuals in birth and death processes

Advances in Applied Probability ◽

10.1017/s0001867800020607 ◽

1982 ◽

Vol 14 (03) ◽

pp. 484-501

Author(s):

S. K. Srinivasan ◽

C. R. Ranganathan

Keyword(s):

General Model ◽

Death Process ◽

Birth And Death Process ◽

Birth Rates ◽

Birth And Death Processes ◽

Age Dependent ◽

The Mean ◽

Number Of Individuals ◽

Number Of Births

This paper deals with the parity of individuals in an age-dependent birth and death process. A more general model with parity and age-dependent birth rates is also considered. The mean number of individuals with parity 0, 1, 2, ·· ·is obtained for the two models. The first moments of the total number of births in the population up to time t and the sum of the parities of the individuals existing at time t are obtained. A brief discussion on the parity of individuals in a population including ‘twins' is also given.

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Birth and death processes with random environments in continuous time

Journal of Applied Probability ◽

10.1017/s0021900200097576 ◽

1981 ◽

Vol 18 (01) ◽

pp. 19-30 ◽

Cited By ~ 6

Author(s):

Robert Cogburn ◽

William C. Torrez

Keyword(s):

Discrete Time ◽

Random Environment ◽

Continuous Time ◽

Death Process ◽

Random Environments ◽

Birth And Death Process ◽

Time Model ◽

Birth And Death Processes ◽

Discrete Time Model ◽

Extinction Probabilities

A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process.

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Application of Stieltjes theory for S-fractions to birth and death processes

Advances in Applied Probability ◽

10.1017/s0001867800021388 ◽

1983 ◽

Vol 15 (03) ◽

pp. 507-530 ◽

Cited By ~ 3

Author(s):

G. Bordes ◽

B. Roehner

Keyword(s):

Asymptotic Behavior ◽

Lower Bound ◽

Special Class ◽

Jacobi Matrix ◽

General Theorem ◽

Sufficient Conditions ◽

Death Process ◽

Nearest Neighbour ◽

Birth And Death Process ◽

Birth And Death Processes

We are interested in obtaining bounds for the spectrum of the infinite Jacobi matrix of a birth and death process or of any process (with nearest-neighbour interactions) defined by a similar Jacobi matrix. To this aim we use some results of Stieltjes theory for S-fractions, after reviewing them. We prove a general theorem giving a lower bound of the spectrum. The theorem also gives sufficient conditions for the spectrum to be discrete. The expression for the lower bound is then worked out explicitly for several, fairly general, classes of birth and death processes. A conjecture about the asymptotic behavior of a special class of birth and death processes is presented.

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Birth and death processes in randomly fluctuating environments

Nagoya Mathematical Journal ◽

10.1017/s0027763000008278 ◽

2002 ◽

Vol 166 ◽

pp. 93-115

Author(s):

Kanji Ichihara

Keyword(s):

Random Environment ◽

Time Dependent ◽

Death Process ◽

Birth And Death Process ◽

Birth And Death Processes ◽

Fluctuating Environments

AbstractA birth and death process in a time-dependent random environment is introduced. We will discuss the recurrence and transience properties for the process.

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Multivariate birth-and-death processes as approximations to epidemic processes

Journal of Applied Probability ◽

10.2307/3212492 ◽

1973 ◽

Vol 10 (1) ◽

pp. 15-26 ◽

Cited By ~ 29

Author(s):

D. A. Griffiths

Keyword(s):

Transient Process ◽

Branching Process ◽

Large Population ◽

Disease Spread ◽

Death Process ◽

Birth And Death Process ◽

Birth And Death Processes ◽

Epidemic Size

This paper presents the theory of a multivariate birth-and-death process and its representation as a branching process. The bivariate linear birth-and-death process may be used as a model for various epidemic situations involving two types of infective. Various properties of the transient process are discussed and the distribution of epidemic size is investigated. For the case of a disease spread solely by carriers when the two types of infective are carriers and clinical infectives the large population version of a model proposed by Downton (1968) is further developed and shown under appropriate circumstances to closely approximate Downton's model.

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ZIPF AND LERCH LIMIT OF BIRTH AND DEATH PROCESSES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964809990179 ◽

2009 ◽

Vol 24 (1) ◽

pp. 129-144 ◽

Cited By ~ 4

Author(s):

B. Klar ◽

P. R. Parthasarathy ◽

N. Henze

Keyword(s):

Chemical Kinetics ◽

Continued Fractions ◽

Zipf’S Law ◽

Death Process ◽

Speed Of Convergence ◽

Birth And Death Process ◽

Birth And Death Processes ◽

Zipf's Law ◽

Wide Range ◽

Epidemiology Data

Birth and death processes are useful in a wide range of disciplines from computer networks and telecommunications to chemical kinetics and epidemiology. Data from many different areas such as linguistics, music, or warfare fit Zipf's law surprisingly well. The Lerch distribution generalizes Zipf's law and is applicable in survival and dispersal processes. In this article we construct a birth and death process that converges to the Lerch distribution in the limit as time becomes large, and we investigate the speed of convergence. This is achieved by employing continued fractions. Numerical illustrations are presented through tables and graphs.

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