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- Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.Suppose that a Markov chain has transition probability matrix 1 2 1 P (1/2 1/2 2 1/4 3/4 (a) What is the long-run proportion of time that the chain is in state i, i = 1,2 ? 5. What should r2 be if it is desired to have the long-run average (b) Suppose that ri reward per unit time equal to 9?There are two printers in the computer lab. Printer i operates for an exponential time withrate λi before breaking down, i = 1, 2. When a printer breaks down, maintenance is called to fix it,and the repair times (for either printer) are exponential with rate μ. (a) Can we analyze this as a birth and death process? Briefly explain your answer.(b) Model this as a continuous time Markov chain (CTMC). Clearly define all the statesand draw the state transition diagram.
- Let X, be the Markov chain with state space Z and transition probability Pa,z+1 = P, Pa,2-1 = 1- p, where p > 1/2. Assume X, = 0. (a) Let Y = min{Xo, X1,...}. What is the distribution of Y? (b) For positive integer k, let T = min{n : X, = k} and let e(k) = E[TR]. Explain why e(k) = ke(1). (c) Find e(1). Hint: part (b) might be helpful. (d) Use (c) to give another proof that e(1) = o if p = 1/2.Suppose that {Xn} is a Markov chain with state space S = {1, 2}, transition matrix （1/5 4/5 2/5 3/5）, and initial distribution P (X0 = 1) = 3/4 and P (X0 = 2) = 1/4. Compute the following: P(X2 =2)Let P = (0.9 0.9 0.1 0.2 0.8 with two states A and B. be a transition matrix for a Markov Chain 1. What proportion of the state A population will be in state B after two steps Number 2. What proportion of the state B population will be in state B after two steps Number 3. Find the steady state vector x x1= Number X2= Number Write the results accurate to the 3rd decimal place
- 8. At each time n = 0, 1, 2, ... a number Yn of particles enters a chamber, where {Yn n ≥ 0} are independent and Poisson distributed with parameter λ. Lifetimes of particles are independent and geometrically distributed with parameter p. Let Xn be the number of particles in the chamber at time n. Show that X is a Markov chain, and find its stationary distribution.Let X be a random variable with sample space {1,2, 3} and probability distribu- (G 1 ). Find a transition matrix P such that the Markov chain {X„} tion T = simulates X.A Continuous-Time Markov Chain Consisting of Two States. Consider a machine that works for an exponential amount of time having mean 1/A before breaking down; and suppose that it takes an exponential amount of time with mean 1/µ to repair the machine. If the machine is in the operational condition at time 0, what is the probability that it will be working at time t = 5 ? HINT: write down the backward Kolmogorov differential equations and solve it.