Consider a Markov chain {an = 0, 1, - - -} on the state space S = {0, 1, 2}. Suppose that the Markov chain has the transition matrix 011 P=100 it\" 1. Show that the state space is irreducible. 2. Show that the Markov chain is periodic. Find the period of the Markov chain. 3. Let h denote a stationary mass of the Markov chain. Find h(:r) for all a: E S. Consider a Markov chain {Km 11 = O, 1, - - -} on the state space S = {0,1,2}. Suppose that the Markov chain has the transition matrix A A .2. 10 10 10 P = A :4. A 10 10 10 i .2. .4_ 1o 10 10 1. Show that the Markov chain has a unique stationary mass. 2. Let h denote the stationary mass of the Markov chain. Find h(x) for all a: E S. 3. Show that the Markov chain has the steady state mass. 4. Let h" denote the steady state mass of the Markov chain. Find h*(x) for all a: E S. Starting with the R Punkting for the simulation of the junegration - death process, modify it to include "birth's in addition to immigrations and death, Use your modified function to simulate a bug realisation Prom the birth-immigration- death process where the birth, death , and immigration rates are all one, starting from an initial condition of zero.Problem 2: 10 points Consider a birth-and-death process, X = {X(t) : t 2 0} , associated with the service line that consists of N = 10 servers. When all 10 servers are occupied, the new request is refused and not coming into the service line. As there are k k =3. (10 -k) for 0 10, and P [X(t + 4) - X(t) = -1\\X(t) = k] =/k = 2 . k for 0