Introduction to Stochastic Models

Chapter 6

Branching Models

The theory of branching processes focuses on the study of mathematical models related to the increase or decrease of populations of individuals who procreate and replace one another throughout generations, according to rules where chance plays a major role. At the beginning of this theory, the objects under study were married men and the aim was to see how fast surnames disappeared. In recent applications, the study objects are heterozygous subjects carrying a mutant gene, clients in a queueing system, or neutrons in a nuclear reactor, to cite just three main examples. We can also find other examples throughout this chapter.

6.1. The Bienaymé-Galton-Watson model

6.1.1. Historical considerations

The history of the branching process theory represents one of the most exciting pages of probability theory. Just some time ago, we thought we should date back this theory to 1873 when the British biometrician, Francis Galton (1822–1911), who had been for some time been interested in measuring the decline of aristocratic families (in terms of their decrease and extinction), published the famous Problem 4001 in the London newspaper The Educational Times on the 1st of April 1873:

“A large nation, of whom we will only concern ourselves with adult males, N in number, and who each bear separate surnames colonize a district. Their law of population is such that, in each generation, a₀ per cent of the adult males have no male children who reach adult life; a₁ have one ...