ps2 (1) - MS&E 221 Problem Set 2 Ramesh Johari...

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Unformatted text preview: MS&E 221 Problem Set 2 Ramesh Johari Due: Weds., February 2, 2011, 5:00 PM in the basement of HEC Reading. 4.1-4.4, 4.5.1, 4.7 in Ross. Problem 1. Ross, Chapter 4, problem 14: Problem 2. (A queueing model) Consider a queue (or a waiting room) that can hold at most N customers at a time, including the customer currently being served. At each time step, one of three things happens: A new customer arrives with probability p . If the queue is already full, then this new cus- tomer is blocked, i.e., turned away. Whenever the queue is nonempty, with probability q a customer completes service and leaves the queue. With the remaining probability ( 1- p if no one is present in the queue, and 1- p- q if the queue is nonempty), nothing happens to the state of the queue. (a) . Draw the graph representing the Markov chain describing the number of customers in the queue (with states , 1 ,...,N ), and label the probability of each transition. (b) . Which states are recurrent? (c) . Suppose the queue starts with i > customers. What is the probability that at least one customer is blocked before the queue empties? (Hint: A customer is blocked if and only if the queue is in state N , and a new arrival occurs.) (d) . Suppose the queue starts empty. After the first customer arrives, what is the probabil- ity that exactly four customers are blocked before the queue returns to being completely empty? (Hint: Use the Strong Markov Property to note that, conditional on one customer just having been 1 blocked, the process behaves like a queue starting with N customers.) Problem 3. Bertsekas and Tsitsiklis, Chapter 6, Problem 11 (note that steady state means the current distribution of the state is the invariant distribution): 2 Problem 4.Problem 4....
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ps2 (1) - MS&E 221 Problem Set 2 Ramesh Johari...

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