If the number of offspring that an organism has is Poisson-distributed, and if the average number of offspring of each organism is no bigger than 1, then the descendants of each individual will ultimately become extinct. The number of descendants that an individual ultimately has in that situation is a random variable distributed according to a Borel distribution.
for n = 1, 2, 3 ....
Derivation and branching process interpretation
Let X be the total number of individuals in a Galton–Watson branching process. Then a correspondence between the total size of the branching process and a hitting time for an associated random walk gives
where Sn = Y1 + … + Yn, and Y1 … Yn are independent identically distributed random variables whose common distribution is the offspring distribution of the branching process. In the case where this common distribution is Poisson with mean μ, the random variable Sn has Poisson distribution with mean μn, leading to the mass function of the Borel distribution given above.
Since the mth generation of the branching process has mean size μm − 1, the mean of X is
Queueing theory interpretation
In an M/D/1 queue with arrival rate μ and common service time 1, the distribution of a typical busy period of the queue is Borel with parameter μ.
If Pμ(n) is the probability mass function of a
Borel(μ) random variable, then the mass function
μ(n) of a sized-biased sample from the distribution (i.e. the mass function proportional to nPμ(n) ) is given by
In words, this says that a Borel(μ) random variable has the same distribution as a size-biased Borel(μU) random variable, where U has the uniform distribution on [0,1].
This relation leads to various useful formulas, including
The Borel–Tanner distribution generalizes the Borel distribution. Let k be a positive integer. If X1, X2, … Xk are independent and each has Borel distribution with parameter μ, then their sum W = X1 + X2 + … + Xk is said to have Borel–Tanner distribution with parameters μ and k. This gives the distribution of the total number of individuals in a Poisson–Galton–Watson process starting with k individuals in the first generation, or of the time taken for an M/D/1 queue to empty starting with k jobs in the queue. The case k = 1 is simply the Borel distribution above.
where Sn has Poisson distribution with mean nμ. As a result, the probability mass function is given by
for n = k, k + 1, ... .
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