Markov/Poisson Particle System. Consider a particle system in a countable space S similar to the one in Section 3.11 with the following modifications. Each particle moves independently in continuous...

Markov/Poisson Particle System. Consider a particle system in a countable space S similar to the one in Section 3.11 with the following modifications. Each particle moves independently in continuous time according to an ergodic CTMC with transition probabilities pij (t) and stationary distribution pi, i ∈ S. That is, pij (t) is the probability that a particle starting in state i is in state j at time t. Assume the system is empty at time 0 and that particles enter the system according to a space-time Poisson process M on R+ × S, where M((0, t] × B) is the number of arrivals in (0, t] that enter B ⊆ S, and E[M((0, t] × {i})] = λtpi. Let Qi(t) denote the quantity of particles in state i at time t. Show that