416-11 - λ i = ( i + 1) λ and death rates μ i = i , i...

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Denker Spring 2010 416 Stochastic Modeling - Assignment 11 Due Date: Monday, April 19, 2010 Problem 1: (Problem 2, p. 407) Suppose a one-celled organism can be in two states – either A or B. An individual in state A will change to state B at an exponential rate α ; and individual in state B divides into two new individuals of type A at an exponentail rate β . DeFne an appropriate continuous-time Markov chain for a population of such organisms and determine the appropriate parameters of this model. Problem 2: (Problem 3, p. 407) Consider two machines that are maintained by a single repairman. Machine i functions for an exponential time with rate μ i before breaking down. The repair times (for either machine) are exponential with rate μ . Can we analyze this as a birth and death process? If not, how can we analyze it? Problem 3: (Problem 6, p. 408) Consider a birth and death process with birth rates
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Unformatted text preview: λ i = ( i + 1) λ and death rates μ i = i , i ≥ 0. 1. Determine the expected time to go from state 0 to state 4. 2. Determine the expected time to go from state 2 to state 5. Problem 4: (Problem 9, p. 408) The birth and death process with parameters λ n = 0 and μ n = μ , ( n > 0), is called a pure death process. ±ind p ij ( t ). Problem 5: (Problem 14, p. 409) Potential customers arrive at a full-service , one-pump gas station at a Poisson rate of 20 cars per hour. However, customers will enter the station only if there are no more than two cars at the pump (one being served and one waiting).Suppose the amount of time required to service a car is exponentially distributed with expectation of Fve minutes. 1. What fraction of the attendants time will be spent servicing cars? 2. What fraction of potential customers are lost?...
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This note was uploaded on 04/15/2010 for the course STAT 416 taught by Professor Denker during the Spring '08 term at Penn State.

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