Let $ N (t): t ≥ 0 % be a Poisson process with rate λ. By Example 12.38, the process $ N (t): t ≥ 0 % is a continuous-time Markov chain. Hence it satisfies equations (12.12), the Chapman-Kolmogorov...

Let $ N (t): t ≥ 0 % be a Poisson process with rate λ. By Example 12.38, the process $ N (t): t ≥ 0 % is a continuous-time Markov chain. Hence it satisfies equations (12.12), the Chapman-Kolmogorov equations. Verify this fact by direct calculations.

Equation 12.12

Example 12.38

Let $ N (t): t ≥ 0 % be a Poisson process with rate λ. Then $ N (t): t ≥ 0 % is a stochastic process with state space S = {0, 1, 2,...}, with the property that upon entering a state i, it will remain there for an exponential amount of time with mean 1/λ and then will move to state i + 1 with probability 1. Hence $ N (t): t ≥ 0 % is a continuous-time Markov chain with νi = λ for all i ∈ S, p_i(i+1)
= 1; p_ij
= 0, if j ,= i + 1. Furthermore, for all t, s > 0,