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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 discretetime chain Yn ,
gives the road map that the continuoustime 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.
 Spring '10
 DURRETT
 The Land

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