Weak convergence of the simple birth-and-death process

The existence of a weak limit birth-and-death process on the natural integers for the simple birth-and-death process conditional on non-extinction up to time t as t→∞ is proved. Starting from the latter a new weak limiting procedure yields a diffusion Markov process on the positive infinite semi-axis.

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On Sieved Orthogonal Polynomials II: Random Walk Polynomials

Canadian Journal of Mathematics ◽

10.4153/cjm-1986-020-x ◽

1986 ◽

Vol 38 (2) ◽

pp. 397-415 ◽

Cited By ~ 13

Author(s):

Jairo Charris ◽

Mourad E. H. Ismail

Keyword(s):

Random Walk ◽

Markov Process ◽

Orthogonal Polynomials ◽

Transition Probabilities ◽

Death Process ◽

Birth And Death Process ◽

Stationary Markov Process ◽

Image Position

A birth and death process is a stationary Markov process whose states are the nonnegative integers and the transition probabilities(1.1)satisfy(1.2)as t → 0. Here we assume βn > 0, δn + 1 > 0, n = 0, 1, …, but δ0 ≦ 0. Karlin and McGregor [10], [11], [12], showed that each birth and death process gives rise to two sets of orthogonal polynomials. The first is the set of birth and death process polynomials {Qn(x)} generated by

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

Journal of Applied Probability ◽

10.2307/3212402 ◽

1975 ◽

Vol 12 (1) ◽

pp. 1-17 ◽

Cited By ~ 18

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.

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Weak convergence of conditioned birth and death processes

Journal of Applied Probability ◽

10.2307/3215237 ◽

1994 ◽

Vol 31 (1) ◽

pp. 90-100 ◽

Cited By ~ 6

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.

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On selecting the most reliable components

Journal of Applied Mathematics and Decision Sciences ◽

10.1155/s1173912698000078 ◽

1998 ◽

Vol 2 (2) ◽

pp. 133-145 ◽

Cited By ~ 2

Author(s):

Dylan Shi

Keyword(s):

Markov Process ◽

Death Process ◽

Series System ◽

Birth And Death Process ◽

Retrograde Motion ◽

The Past ◽

Optimal Policies ◽

Average Proportion ◽

The Mean ◽

Mean Time

Consider a series system consisting of n components of k types. Whenever a unit fails, it is replaced immediately by a new one to keep the system working. Under the assumption that all the life lengths of the components are independent and exponentially distributed and that the replacement policies depend only on the present state of the system at each failure, the system may be represented by a birth and death process. The existence of the optimum replacement policies are discussed and the ε-optimal policies axe derived. If the past experience of the system can also be utilized, the process is not a Markov process. The optimum Bayesian policies are derived and the properties of the resulting process axe studied. Also, the stochastic processes are simulated and the probability of absorption, the mean time to absorption and the average proportion of the retrograde motion are approximated.

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

Journal of Applied Probability ◽

10.1017/s0021900200033040 ◽

1975 ◽

Vol 12 (01) ◽

pp. 1-17 ◽

Cited By ~ 6

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 {X 1 (t), X 2 (t), t ≧ 0} be a bivariate birth and death (Markov) process taking non-negative integer values, such that the process {X 2(t), t ≧ 0} may influence the growth of the process {X 1(t), t ≧ 0}, while the process X 2 (·) itself grows without any influence whatsoever of the first process. The process X 2 (·) is taken to be a simple linear birth and death process with λ 2 and µ 2 as its birth and death rates respectively. The process X 1 (·) is also assumed to be a linear birth and death process but with its birth and death rates depending on X 2 (·) in the following manner: λ (t) = λ 1 (θ + X 2 (t)); µ(t) = µ 1 (θ + X 2 (t)). Here λ i, µi and θ are all non-negative constants. By studying the process X 1 (·), first conditionally given a realization of the process {X 2 (t), t ≧ 0} and then by unconditioning it later on by taking expectation over the process {X 2 (t), t ≧ 0} we obtain explicit solution for G in closed form. Again, it is shown that a proper limit distribution of X 1 (t) always exists as t→∞, except only when both λ 1 > µ 1 and λ 2 > µ 2. Also, certain problems concerning moments of the process, regression of X 1 (t) on X 2 (t); time to extinction, and the duration of the interaction between the two processes, etc., are studied in some detail.

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Weak convergence of conditioned birth and death processes

Journal of Applied Probability ◽

10.1017/s0021900200107351 ◽

1994 ◽

Vol 31 (01) ◽

pp. 90-100 ◽

Cited By ~ 7

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.

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

Journal of Applied Probability ◽

10.1017/s0021900200116031 ◽

1986 ◽

Vol 23 (04) ◽

pp. 851-858 ◽

Cited By ~ 1

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.

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A stochastic process related to language change

Journal of Applied Probability ◽

10.1017/s0021900200026942 ◽

1970 ◽

Vol 7 (01) ◽

pp. 69-78 ◽

Cited By ~ 2

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.

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Multi-Location Inventory Model Considering Bidirectional and Unidirectional Transshipments Simultaneously

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.694-697.2742 ◽

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.

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