On the subcritical Bellman-Harris process with immigration

1974 ◽  
Vol 11 (4) ◽  
pp. 652-668 ◽  
Author(s):  
A. G. Pakes ◽  
Norman Kaplan
Keyword(s):  
Poisson Process ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Compound Poisson Process ◽  
Limiting Distribution ◽  
Batch Arrival ◽  
Compound Poisson ◽  
Queueing Process ◽  
Age Dependent ◽  
Necessary And Sufficient

Some necessary and sufficient conditions are found for the existence of a proper non-degenerate limiting distribution for a Bellman-Harris age-dependent branching process with a compound renewal immigration component. A number of these results are applicable to the batch arrival GI/G/∞ queueing process. Some aspects of the situation when there is no such limiting distribution are considered. The situation when the immigration component is a nonhomogeneous compound Poisson process is briefly considered.

On the subcritical Bellman-Harris process with immigration

1974 ◽  
Vol 11 (04) ◽  
pp. 652-668 ◽  
Author(s):  
A. G. Pakes ◽  
Norman Kaplan
Keyword(s):  
Poisson Process ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Compound Poisson Process ◽  
Limiting Distribution ◽  
Batch Arrival ◽  
Compound Poisson ◽  
Queueing Process ◽  
Age Dependent ◽  
Necessary And Sufficient

Some necessary and sufficient conditions are found for the existence of a proper non-degenerate limiting distribution for a Bellman-Harris age-dependent branching process with a compound renewal immigration component. A number of these results are applicable to the batch arrival GI/G/∞ queueing process. Some aspects of the situation when there is no such limiting distribution are considered. The situation when the immigration component is a nonhomogeneous compound Poisson process is briefly considered.

Some properties of continuous-state branching processes, with applications to Bartoszyński’s virus model

1985 ◽  
Vol 17 (01) ◽  
pp. 23-41
Author(s):  
Anthony G. Pakes ◽  
A. C. Trajstman
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Compound Poisson Process ◽  
Explicit Construction ◽  
General Context ◽  
Compound Poisson ◽  
Continuous State ◽  
Virus Model

It is known that Bartoszyński’s model for the growth of rabies virus in an infected host is a continuous branching process. We show by explicit construction that any such process is a randomly time-transformed compound Poisson process having a negative linear drift. This connection is exploited to obtain limit theorems for the population size and for the jump times in the rabies model. Some of these results are obtained in a more general context wherein the compound Poisson process is replaced by a subordinator.

scholarly journals Bulk input queues with quorum and multiple vacations

1996 ◽  
Vol 2 (2) ◽  
pp. 95-106 ◽  
Author(s):  
Jewgeni H. Dshalalow ◽  
Jay Yellen
Keyword(s):  
Poisson Process ◽  
Preliminary Analysis ◽  
Queueing System ◽  
Compound Poisson Process ◽  
Batch Service ◽  
Single Server ◽  
Compound Poisson ◽  
Multiple Vacations ◽  
Queueing Process ◽  
Multiple Vacation

The authors study a single-server queueing system with bulk arrivals and batch service in accordance to the general quorum discipline: a batch taken for service is not less thanrand not greater thanR(≥r). The server takes vacations each time the queue level falls belowr(≥1)in accordance with the multiple vacation discipline. The input to the system is assumed to be a compound Poisson process. The analysis of the system is based on the theory of first excess processes developed by the first author. A preliminary analysis of such processes enabled the authors to obtain all major characteristics for the queueing process in an analytically tractable form. Some examples and applications are given.

A branching process with disasters

1975 ◽  
Vol 12 (1) ◽  
pp. 47-59 ◽  
Author(s):  
Norman Kaplan ◽  
Aidan Sudbury ◽  
Trygve S. Nilsen
Keyword(s):  
Asymptotic Behavior ◽  
Renewal Process ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Population Process ◽  
Age Dependent ◽  
The Mean ◽  
Necessary And Sufficient

A population process is considered where particles reproduce according to an age-dependent branching process, and are subjected to disasters which occur at the epochs of an independent renewal process. Each particle alive at the time of a disaster, survives it with probability p and the survival of any particle is assumed independent of the survival of any other particle. The asymptotic behavior of the mean of the process is determined and as a consequence, necessary and sufficient conditions are given for extinction.

A branching process with disasters

1975 ◽  
Vol 12 (01) ◽  
pp. 47-59 ◽  
Author(s):  
Norman Kaplan ◽  
Aidan Sudbury ◽  
Trygve S. Nilsen
Keyword(s):  
Asymptotic Behavior ◽  
Renewal Process ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Population Process ◽  
Age Dependent ◽  
The Mean ◽  
Necessary And Sufficient

A population process is considered where particles reproduce according to an age-dependent branching process, and are subjected to disasters which occur at the epochs of an independent renewal process. Each particle alive at the time of a disaster, survives it with probability p and the survival of any particle is assumed independent of the survival of any other particle. The asymptotic behavior of the mean of the process is determined and as a consequence, necessary and sufficient conditions are given for extinction.

Some properties of continuous-state branching processes, with applications to Bartoszyński’s virus model

1985 ◽  
Vol 17 (1) ◽  
pp. 23-41 ◽  
Author(s):  
Anthony G. Pakes ◽  
A. C. Trajstman
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Compound Poisson Process ◽  
Explicit Construction ◽  
General Context ◽  
Compound Poisson ◽  
Continuous State ◽  
Virus Model

It is known that Bartoszyński’s model for the growth of rabies virus in an infected host is a continuous branching process. We show by explicit construction that any such process is a randomly time-transformed compound Poisson process having a negative linear drift.This connection is exploited to obtain limit theorems for the population size and for the jump times in the rabies model. Some of these results are obtained in a more general context wherein the compound Poisson process is replaced by a subordinator.

scholarly journals Continuous branching processes : the discrete hidden in the continuous : Dedicated to Claude Lobry

2008 ◽  
Vol Volume 9, 2007 Conference in... ◽  
Author(s):  
Etienne Pardoux
Keyword(s):  
Poisson Process ◽  
Branching Process ◽  
Branching Processes ◽  
Compound Poisson Process ◽  
Levy Process ◽  
Initial Condition ◽  
Compound Poisson ◽  
International Audience ◽  
Number Of Individuals ◽  
Branching Property

International audience Feller diffusion is a continuous branching process. The branching property tells us that for t > 0 fixed, when indexed by the initial condition, it is a subordinator (i. e. a positive–valued Lévy process), which is fact is a compound Poisson process. The number of points of this Poisson process can be interpreted as the number of individuals whose progeny survives during a number of generations of the order of t × N, where N denotes the size of the population, in the limit N ―>µ. This fact follows from recent results of Bertoin, Fontbona, Martinez [1]. We compare them with older results of de O’Connell [7] and [8]. We believe that this comparison is useful for better understanding these results. There is no new result in this presentation. La diffusion de Feller est un processus de branchement continu. La propriété de branchement nous dit que à t > 0 fixé, indexé par la condition initiale, ce processus est un subordinateur (processus de Lévy à valeurs positives), qui est en fait un processus de Poisson composé. Le nombre de points de ce processus de Poisson s’interprète comme le nombre d’individus dont la descendance survit au cours d’un nombre de générations de l’ordre de t × N, où N désigne la taille de la population, dans la limite N --> µ. Ce fait découle de résultats récents de Bertoin, Fontbona, Martinez [1]. Nous le rapprochons de résultats plus anciens de O’Connell [7] et [8]. Ce rapprochement nous semble aider à mieux comprendre ces résultats. Cet article ne contient pas de résultat nouveau.

Exit problems for the difference of a compound Poisson process and a compound renewal process

2008 ◽  
Vol 59 (3-4) ◽  
pp. 271-296 ◽  
Author(s):  
Victor Kadankov ◽  
Tetyana Kadankova
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Compound Poisson Process ◽  
Compound Poisson ◽  
Compound Renewal Process ◽  
The Difference ◽  
Exit Problems
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