Application of birth-and-death processes to the cascade theory

1968 ◽  
Vol 46 (10) ◽  
pp. S212-S215 ◽  
Author(s):  
K. Kobayakawa ◽  
S. Miono
Keyword(s):  
Complete Solution ◽  
Initial Energy ◽  
Death Process ◽  
Recent Experimental Data ◽  
Birth And Death Process ◽  
Birth And Death Processes ◽  
Death Rates ◽  
Electromagnetic Showers ◽  
Better Than ◽  
Number Of Particles

A birth-and-death process taking immigration into account is considered. The complete solution of the equation governing the generalized birth-and-death process when the birth and death rates λ(t)and μ(t) and also the immigration factor ν(t) may be any specified functions of the time t is given. This solution can be applied to the fluctuation problem in electromagnetic showers. The distribution function of the number of particles at given depth t with given initial energy is derived. The results obtained are compared with recent experimental data, and the agreement is much better than in past theoretical works.

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.

On the parity of individuals in birth and death processes

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.

Birth and death processes with random environments in continuous time

1981 ◽  
Vol 18 (01) ◽  
pp. 19-30 ◽  
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.

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

1983 ◽  
Vol 15 (03) ◽  
pp. 507-530 ◽  
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.

scholarly journals Birth and death processes in randomly fluctuating environments

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.

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

1973 ◽  
Vol 10 (1) ◽  
pp. 15-26 ◽  
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.

A linear birth and death process under the influence of another process

1975 ◽  
Vol 12 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Prem S. Puri
Keyword(s):  
Markov Process ◽  
Closed Form ◽  
Limit Distribution ◽  
Explicit Solution ◽  
Negative Integer ◽  
Death Process ◽  
Birth And Death Process ◽  
Death Rates ◽  
Time To Extinction

Let {X1 (t), X2 (t), t ≧ 0} be a bivariate birth and death (Markov) process taking non-negative integer values, such that the process {X2(t), t ≧ 0} may influence the growth of the process {X1(t), t ≧ 0}, while the process X2 (·) itself grows without any influence whatsoever of the first process. The process X2 (·) is taken to be a simple linear birth and death process with λ2 and µ2 as its birth and death rates respectively. The process X1 (·) is also assumed to be a linear birth and death process but with its birth and death rates depending on X2 (·) in the following manner: λ (t) = λ1 (θ + X2 (t)); µ(t) = µ1 (θ + X2 (t)). Here λ i, µi and θ are all non-negative constants. By studying the process X1 (·), first conditionally given a realization of the process {X2 (t), t ≧ 0} and then by unconditioning it later on by taking expectation over the process {X2 (t), t ≧ 0} we obtain explicit solution for G in closed form. Again, it is shown that a proper limit distribution of X1 (t) always exists as t→∞, except only when both λ1 > µ1 and λ2 > µ2. Also, certain problems concerning moments of the process, regression of X1 (t) on X2 (t); time to extinction, and the duration of the interaction between the two processes, etc., are studied in some detail.

ZIPF AND LERCH LIMIT OF BIRTH AND DEATH PROCESSES

2009 ◽  
Vol 24 (1) ◽  
pp. 129-144 ◽  
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.

Weak convergence of conditioned birth and death processes

1994 ◽  
Vol 31 (1) ◽  
pp. 90-100 ◽  
Author(s):  
G. O. Roberts ◽  
S. D. Jacka
Keyword(s):  
Weak Convergence ◽  
Independent Interest ◽  
Death Process ◽  
Birth And Death Process ◽  
Birth And Death Processes ◽  
Monotonic Properties

We consider the problem of conditioning a non-explosive birth and death process to remain positive until time T, and consider weak convergence of this conditional process as T → ∞. By a suitable almost sure construction we prove weak convergence. The almost sure construction used is of independent interest but relies heavily on the strong monotonic properties of birth and death processes.

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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