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416-4 - the number of items has fallen under the value s...

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Denker Spring 2010 416 Stochastic Modeling - Assignment 4 Due Date: Monday, February 15, 2010 Problem 1: Let X 0 be a rv with values in Z and let Z n ( n 1) be independent, identically distributed rv, also independent of X 0 , taking values in { 1; - 1 } . Prove or disprove whether X n , defined by X n +1 = X n + Z n +1 n = 0 , 1 , 2 ,..., is a Markov chain. Problem 2: (Repair shop.) During day n , Z n +1 machines break down and they enter the repair shop on day n + 1. Every day one machine waiting for service is repaired. Model the number of machines waiting for service by a Markov chain. Is the Markov chain irreducible? Draw the graph of the associated Markov chain. Problem 3: (Inventory.) A given commodity is stocked in order to satisfy a continuing demand. The aggregated demand between time n and n + 1 is a random quantity which can be assumed to form an i.i.d. sequence of rv. Let S denote the initial number of items in the stock and assume that the stock is filled up to S after each time unit provided
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Unformatted text preview: the number of items has fallen under the value s . Immediate demand which cannot be satis±ed from the existing stock, will be satis±ed immediately after re-stocking. Model the number of items in the stock as a Markov chain. Discuss the properties of irreducibility and aperiodicity. Problem 4: A Markov chain ( X n ) n ≥ with states 0, 1 and 2, has the transition matrix 1 2 1 3 1 6 1 3 2 3 1 2 1 2 If P ( X = 0) = P ( X = 1) = 1 4 , compute E [ X 3 ]. Problem 5: Consider the matrices in Exercise 14 on page 265. 1. Show that they are transition matrices of Markov chains. Determine the state spaces of these Markov chains. 2. Determine the classes of these Markov chains. 3. Which states are recurrent and which are transient? 4. Which of the 4 Markov chains are recurrent and which are transient?...
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