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Section 3

# Section 3 - ORIE3510 Introduction to Engineering Stochastic...

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ORIE3510 Introduction to Engineering Stochastic Processes Spring 2010 Section 3 Review Recurrence (positive and null), transience, periodicity and ergodicity Limiting and stationary distributions & long-run proportions. Problem 4.29 We have N employees, with N large. There are 3 job classifications and employees change job classifications (independently) according to a DTMC with probability matrix 0 . 7 0 . 2 0 . 1 0 . 2 0 . 6 0 . 2 0 . 1 0 . 4 0 . 5 We want to know what percentage of employees are in each job classification. Solution: The chain is finite state and irreducible which implies that all states are recurrent. To see why all states must be recurrent (for general number of states M ), assume that all states are transient. Then, since state 0 is transient, there must exist a time T 0 after which 0 is never visited. Likewise, for any other state i , there must exist a time T i after which i is never visited. Hence, there exists a time T = max { T 0 , . . . , T M } after which no state is visited. This is a contradiction, so all states must be recurrent. Note, by Hw3, that P ij > 0 for all i, j S , so we have P ( n ) ij > 0 for all n 1 and all i, j S . This implies that the chain is aperiodic, and hence ergodic. Therefore, we know that the chain has a unique stationary distribution, π , which is also the limiting distribution.

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Section 3 - ORIE3510 Introduction to Engineering Stochastic...

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