Comment on the existence of asymptotic state distribution

Question

Assignment:

1. Consider a BDP with state space S = {0, 1,..., 5} and parameters

λ_i = i + 1, i = 0,1,...4

μ_i= 6 - i, i= 1, 2,....5

Find the asymptotic state distribution.

2. Consider a BDP with infinite state space, constant death rate µ (except for state zero, whose death rate is zero) and state-dependent birth rate given by

λ_j = λ/j+1, j= 0, 1, 2,...

with λ > 0.

a) Comment on the existence of the asymptotic state distribution.

b) Find the asymptotic state distribution vector π.

c) Find the asymptotic mean state lim_t→∞ E[X(t)].