On total progeny of multitype Galton-Watson process and the first passage time of random walk on lattice

In this paper, we form a method to calculate the probability generating function of the total progeny of multitype branching process. As examples, we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and (2-1) random walk. Consequently, we could give the probability generating functions and the distributions of the first passage time of corresponding random walks. Especially, for recurrent random walk which could stay at its position with probability 0 < r < 1, we show that the tail probability of the first passage time decays as (frac{2} {{sqrt {pi (1 - r)} }}frac{1} {{sqrt n }} ) when n → ∞.