Aspects of Lagrangian probability distributions

Abstract

Lagrangian distributions are reviewed from the viewpoint of the Galton-Watson process. They are related to the busy period in queuing systems and to the first visit in random walks.A property of the distributions is remarked for the application to vacant vehicles in a new transit system. Combinatorial identities of multinomial and binomial coefficients and related recurrences are shown by a probabilistic method. Based on the identities and recurrences, random forests generated by the Poisson and geometric Galton-Watson processes are characterized.