Stochastic

This construction is useful because yn is simpler to

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Unformatted text preview: en the process will want to leave i immediately, so we will always suppose that each state i has i < 1. If i = 0, then Xt will never leave i. So suppose i > 0 and let r(i, j ) = q (i, j )/ i Here r, short for “routing matrix,” is the probability the chain goes to j when it leaves i. Informal construction. If Xt is in a state i with i = 0 then Xt stays there forever and the construction is done. If i > 0, Xt stays at i for an exponentially distributed amount of time with rate i , then goes to state j with probability r(i, j ). 122 CHAPTER 4. CONTINUOUS TIME MARKOV CHAINS Formal construction. Suppose, for simplicity, that i > 0 for all i. Let Yn be a Markov chain with transition probability r(i, j ). The discrete-time chain Yn , gives the road map that the continuous-time process will follow. To determine how long the process should stay in each state let ⌧0 , ⌧1 , ⌧2 , . . . be independent exponentials with rate 1. At time 0 the process is in state Y0 and should stay there for an amount of time that is exponential with rate (Y0 ), so we let the time the p...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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