Unformatted text preview: ure
process are independent.
Since there is more than a little handwaving going on in the proof of Theorem 4.11 and its application here, it is comforting to note that one can simply
verify from the deﬁnitions that
Lemma 4.12. If ⇡ (m, n) = c m+n /(µm µn ), where c = (1
/µ1 )(1
/µ2 )
12
is a constant chosen to make the probabilities sum to 1, then ⇡ is a stationary
distribution.
Proof. The ﬁrst step in checking ⇡ Q = 0 is to compute the rate matrix Q. To
do this it is useful to draw a picture which assumes m, n > 0 (m 1, n + 1) (m, n + 1) I
@ @ (a) @ @ µ1 @ µ2 @ @ (m 1, n) @ ?  (m, n) @  (m + 1, n)
I
@ µ2
?
(m, n (c) @ 1) @ µ1
@
@
@
(b) (m + 1, n 1) The rate arrows plus the ordinary lines on the picture, make three triangles.
We will now check that the ﬂows out of and into (m, n) in each triangle balance.
In symbols we note that
(a) µ1 ⇡ (m, n) = c m+n
= ⇡ (m
µm 1 µn
1
2 (b) µ2 ⇡ (m, n) = c m+n
= µ1 ⇡ (m + 1, n
µm µn 1
12 (c) ⇡ (m, n) = c m+n+1 µm µn
12 1, n)
1) = µ2 ⇡ (m, n + 1) This shows that ⇡ Q = 0 when m, n > 0. There are three other cases to
conside...
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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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