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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 CK equation in terms of rate: P t = dP t = QP t:
dt
4
dP t = P tQ:
or,
dt
Now, consider the vector t := f0 t 1t :::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 steadystate situation, i.e, what is the system going to be like in steadystate as
1.
It is easy to see that t = 0P t: Also, dt = 0P t = 0P tQ = tQ:
dt
If we denote the steadystate probability vector by , i.e.
0 5 = tlim t
!1 then, from the relation (5)...
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 Fall '08
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 Computer Networks

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