Lecture 12

# 0 t i which implies ii q 1 q i cs 522 v 094

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Unformatted text preview: 1 + ::::: + q i CS 522, v 0.94, d.medhi, W’99 ik ii + ::::: + q ik + ::: = 0: 7 X qij + qii = 0: In short, we have j =i 3 6 Typically, we denote qii = ,qi , thus, we have X qij = qi: j =i 6 The matrix, Q, is known as the transition density matrix, or, inﬁnitesimal generator, or a rate matrix. If the state space S is ﬁnite, then 0 ,q0 q01 ::: q0m 1 Q = B q10 ,q1 ::: q1m C : @ A ::: qm0 qm1 ::: ,qm This is another way to write the C-K equation in terms of rate: P t = dP t = QP t: dt 4 dP t = P tQ: or, dt Now, consider the vector  t := f0 t 1t :::g: This is the probability vector of the state of the system at time t, i.e., probability of being at state 0 in time t is 0 t etc. 0 t! Now, we move to the steady-state situation, i.e, what is the system going to be like in steady-state as 1. It is easy to see that  t = 0P t: Also, dt =  0P t =  0P tQ = tQ: dt If we denote the steady-state probability vector by  , i.e. 0 5  = tlim t !1 then, from the relation (5)...
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## This document was uploaded on 03/19/2014 for the course CS 6030 at Western Michigan.

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