On finite exponential moments for branching processes and busy periods for queues

Using a known fact that a Galton–Watson branching process can be represented as an embedded random walk, together with a result of Heyde (1964), we first derive finite exponential moment results for the total number of descendants of an individual. We use this basic and simple result to prove analogous results for the population size at time t and the total number of descendants by time t in an age-dependent branching process. This has applications in justifying the interchange of expectation and derivative operators in simulation-based derivative estimation for generalized semi-Markov processes. Next, using the result of Heyde (1964), we show that, in a stable GI/GI/1 queue, the length of a busy period and the number of customers served in a busy period have finite exponential moments if and only if the service time does.

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The asymptotic shape of the branching random walk

Advances in Applied Probability ◽

10.2307/1426719 ◽

1978 ◽

Vol 10 (1) ◽

pp. 62-84 ◽

Cited By ~ 40

Author(s):

J. D. Biggins

Keyword(s):

Random Walk ◽

Explicit Formula ◽

Branching Process ◽

Analogous Result ◽

Convex Set ◽

Branching Random Walk ◽

Asymptotic Shape ◽

Age Dependent ◽

Watson Process

In a supercritical branching random walk on Rp, a Galton–Watson process with the additional feature that people have positions, let be the set of positions of the nth-generation people, scaled by the factor n–1. It is shown that when the process survives looks like a convex set for large n. An analogous result is established for an age-dependent branching process in which people also have positions. In certain cases an explicit formula for the asymptotic shape is given.

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Sums of Lifetimes in Age Dependent Branching Processes

Journal of Applied Probability ◽

10.1017/s0021900200032630 ◽

1969 ◽

Vol 6 (01) ◽

pp. 195-200 ◽

Cited By ~ 2

Author(s):

J. Howard Weiner

Keyword(s):

Distribution Function ◽

Branching Process ◽

Past History ◽

Branching Processes ◽

Lifetime Distribution ◽

Parent Cell ◽

Absolutely Continuous ◽

Age Dependent ◽

A Cell ◽

Additional Assumptions

Consider a Bellman-Harris [1] age dependent branching process. At t = 0, a cell is born, has lifetime distribution function G(t), G(0) = 0, assumed to be absolutely continuous with density g(t). At the death of the cell, k new cells are born, each proceeding independently and identically as the parent cell, and independent of past history. Denote by h(s) = Σ k=0 ∞ pk s k and suppose h(1) ≡ m, and assume h”(1) < ∞. Additional assumptions will be added as required.

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The asymptotic shape of the branching random walk

Advances in Applied Probability ◽

10.1017/s0001867800029487 ◽

1978 ◽

Vol 10 (01) ◽

pp. 62-84 ◽

Cited By ~ 3

Author(s):

J. D. Biggins

Keyword(s):

Random Walk ◽

Explicit Formula ◽

Branching Process ◽

Analogous Result ◽

Convex Set ◽

Branching Random Walk ◽

Asymptotic Shape ◽

Age Dependent ◽

Watson Process

In a supercritical branching random walk on R p , a Galton–Watson process with the additional feature that people have positions, let be the set of positions of the nth-generation people, scaled by the factor n –1. It is shown that when the process survives looks like a convex set for large n. An analogous result is established for an age-dependent branching process in which people also have positions. In certain cases an explicit formula for the asymptotic shape is given.

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Malthusian behaviour of the critical and subcritical age-dependent branching processes with arbitrary state space

Journal of Applied Probability ◽

10.2307/3212465 ◽

1976 ◽

Vol 13 (3) ◽

pp. 455-465

Author(s):

D. I. Saunders

Keyword(s):

State Space ◽

Branching Process ◽

Branching Processes ◽

Rates Of Convergence ◽

Expected Number ◽

Subcritical Case ◽

Arbitrary State ◽

Malthusian Parameter ◽

Age Dependent ◽

Number Of Individuals

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.

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A note on a functional equation arising in Galton-Watson branching processes

Journal of Applied Probability ◽

10.2307/3212181 ◽

1971 ◽

Vol 8 (3) ◽

pp. 589-598 ◽

Cited By ~ 18

Author(s):

Krishna B. Athreya

Keyword(s):

Functional Equation ◽

Continuous Function ◽

Generating Function ◽

Branching Process ◽

Branching Processes ◽

Probability Generating Function ◽

Age Dependent ◽

Watson Process ◽

Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

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Coalescence in Subcritical Bellman-Harris Age-Dependent Branching Processes

Journal of Applied Probability ◽

10.1239/jap/1371648962 ◽

2013 ◽

Vol 50 (2) ◽

pp. 576-591

Author(s):

Jyy-I Hong

Keyword(s):

Limit Distribution ◽

Branching Process ◽

Branching Processes ◽

Simple Random Sampling ◽

Single Type ◽

Last Common Ancestor ◽

Death Time ◽

Age Dependent ◽

Lines Of Descent ◽

First Time

We consider a continuous-time, single-type, age-dependent Bellman-Harris branching process. We investigate the limit distribution of the point process A(t)={at,i: 1≤ i≤ Z(t)}, where at,i is the age of the ith individual alive at time t, 1≤ i≤ Z(t), and Z(t) is the population size of individuals alive at time t. Also, if Z(t)≥ k, k≥2, is a positive integer, we pick k individuals from those who are alive at time t by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let Dk(t) be the coalescence time (the death time of the last common ancestor) of these k random chosen individuals. We study the distribution of Dk(t) and its limit distribution as t→∞.

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A note on a functional equation arising in Galton-Watson branching processes

Journal of Applied Probability ◽

10.1017/s002190020003566x ◽

1971 ◽

Vol 8 (03) ◽

pp. 589-598 ◽

Cited By ~ 6

Author(s):

Krishna B. Athreya

Keyword(s):

Functional Equation ◽

Continuous Function ◽

Generating Function ◽

Branching Process ◽

Branching Processes ◽

Probability Generating Function ◽

Age Dependent ◽

Watson Process ◽

Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

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The first- and last-birth problems for a multitype age-dependent branching process

Advances in Applied Probability ◽

10.1017/s0001867800042348 ◽

1976 ◽

Vol 8 (03) ◽

pp. 446-459 ◽

Cited By ~ 20

Author(s):

J. D. Biggins

Keyword(s):

Random Walk ◽

Branching Process ◽

Branching Random Walk ◽

First Birth ◽

Age Dependent

IfBnis the time of the first birth in thenth generation in a supercritical irreducible multitype Crump–Mode process then when there are people in every generationBn/nconverges to a constant; ifDnis the time of the last birth in thenth generation thenDn/nalso converges to a constant on the survival set. Analogous results hold for the extreme members of thenth generation in a branching random walk.

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A limit theorem for a class of supercritical branching processes

Journal of Applied Probability ◽

10.1017/s002190020003610x ◽

1972 ◽

Vol 9 (04) ◽

pp. 707-724 ◽

Cited By ~ 17

Author(s):

R. A. Doney

Keyword(s):

General Class ◽

Branching Process ◽

Branching Processes ◽

Generating Functional ◽

Number Of Children ◽

Supercritical Case ◽

Age Dependent ◽

Watson Process ◽

Supercritical Branching Processes ◽

Convergence In Mean

In the Bellman-Harris (B-H) age-dependent branching process, the birth of a child can occur only at the time of its parent's death. A general class of branching process in which births can occur throughout the lifetime of a parent has been introduced by Crump and Mode. This class shares with the B-H process the property that the generation sizes {ξn } form a Galton-Watson process, and so may be classified into subcritical, critical or supercritical according to the value of m = E{ξ 1}. Crump and Mode showed that, as regards extinction probability, asymptotic behaviour, and for the supercritical case, convergence in mean square of Z(t)/E[Z(t)], as t → ∞, where Z(t) is the population size at time t given one ancestor at t = 0, properties of the B-H process can be extended to this general class. In this paper conditions are found for the convergence in distribution of Z(t)/E{Z(t)} in the supercritical case to a non-degenerate limit distribution. In contrast to the B-H process, these conditions are not the same as those for ξn /mn to have a non-degenerate limit. An integral equation is established for the generating function of Z(t), which is more complicated than the corresponding one for the B-H process and involves the conditional probability generating functional of N(x), x 0, ≧ the number of children born to an individual in the age interval [0, x].

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