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ex-11 - Spring 2011 OR3510/5510 Problem Set 11 When this is...

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Spring 2011 OR3510/5510 Problem Set 11 When this is due: We are back to the usual, non-crisis-prelim schedule. This problem set is due noon Monday May 2. Reading: We are finishing our work on renewal and regenerative theory in Chapter 7. (1) Kegs of beer at Happy Harry’s bar hold 24 liters. Only one is kept on the bar at a time; the rest are in the back. Customers arrive according to a renewal process with finite mean interarrival times. Each customer orders half a liter. When a keg is empty, it is instantaneously replaced with a full keg from the back. Let ξ ( t ) be the level in the keg on the bar at time t. Assume the state space is S = { . 5 , 1 , 1 . 5 , . . . , 24 } . (a) Is ξ ( t ) regenerative? Why or why not? (b) Compute lim t →∞ P [ ξ ( t ) = k ] . (c) What is the the long run percentage of time the keg on the bar is full. (2) Smith works on a temporary basis. The mean length of each job he gets is three months. If the amount of time he spends between jobs is exponentially distributed with mean 2, then at what rate does Smith get new jobs?
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