Unformatted text preview: probability matrix: 6 4. Chapman-Kolmogorov equation
So far we have mentioned one-step transition probabilities, i.e., probability of An given
equation provides a relation for multiple steps as follows: p n + m =
ij where n m=0 12 n C-K np m:
kj 1 sp t: 2 ik k :::. For continuous time, this can be written as: p s + t =
ij where s Xp A ,1 . Xp ik kj k t 0: In matrix notation, P s + t = P sP t: Set the initial transition probability matrix as P 0 = I
i.e., p ij 0 Deﬁne, q ij =1 (identity matrix) if i = j else 0 if i 6= j: lim p t , p 0 = lim p tt for j 6= i:
q := lim p t , p 0 = lim p tt , 1 :
t := ij ij ij t t ii ii ii ii t t Note that q ’s are instantaneous rate. In matrix notation, Q = lim P tt , I :
t where, Q = q : Now, recall the relation
ij Xp t = 1: ij j If we move 1 from RHS to LHS of this equation and then divide by t and let t ! 0, we get lim fp 1t=t + ::::: + p t , 1=t + :::: + p t=t + :::g = 0
t i which implies ii q...
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This document was uploaded on 03/19/2014 for the course CS 6030 at Western Michigan.
- Fall '08
- Computer Networks