Hi, please can you answer this stochastic processes question for me, thank you.

2. (a) Let X = (Xn : n e No) be a time homogenous Markov chain on state space S = {1, 2, 3, 4, 5, 6, 7} with transition probability matrix 1/2 2/5 1/10 0 0 0 0 0 4/5 0 0 0 1/10 1/10 0 0 0 0 0 0 1 P = 1/3 0 0 0 2/3 0 0 3/10 0 0 0 7/10 0 0 0 3/10 7/10 0 0 0 0 0 0 0 0 0 4/5 1/5 (i) Determine communication classes for the above chain. [2 marks] (ii) Using the socalled rst step decomposition technique, show which of states are recurrent, which are transient. [7 marks] (iii) Identify aperiodic states for the above chain. [3 marks] (iv) In a long term, which state is most frequently visited by X? [3 marks] (b) A computer is shared by 2 users who send tasks to a computer remotely and work independently. At any minute, any connected user may disconnect with probability 0.3, and any disconnected user may connect with a new task with probability 0.1. Let Xn be the number of concurrent users at minute 71. (i) Determine the state space of the above process. Show that {Xn : n e No} is a time homogeneous irreducible Markov chain. [2 marks] (ii) Calculate the transition probability matrix P. [2 marks] (iii) Calculate the limits of P\" and E[X,,] as n tends to innity. [6 marks]