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Unformatted text preview: nd r2 = 2 + p1 r1 (4.28) 146 CHAPTER 4. CONTINUOUS TIME MARKOV CHAINS Plugging in the values for this example and solving gives
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1
1
1
1
r1 = 1 + r2 and r2 = 2 + r1 = 2 +
1 + r2
4
2
2
4
So (7/8)r2 = 5/2 or r2 = 20/7, and r1 = 1 + 20/28 = 11/7. Since
r1 = 11/7 < 4 = µ1 and r2 = 20/7 < 3.5 = µ2 this analysis suggests that there will be a stationary distribution.
To prove that there is one, we return to the general situation and suppose
that the ri we ﬁnd from solving (4.28) satisfy ri < µi . Thinking of two independent M/M/1 queues with arrival rates ri , we let ↵i = ri /µi and guess:
mn
Theorem 4.13. If ⇡ (m, n) = c↵1 ↵2 where c = (1
stationary distribution. ↵1 )(1 ↵2 ) then ⇡ is a 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. Here, we have assumed that m and n are
both positive. To make the picture slightly less cluttered, we have only labeled
half of the arrows and have used qi = 1 pi . (m 1, n + 1) (a) (m (m, n + 1) @@
I
6
µ2 q2
@@
@
µ2 p2 @
@@
R@ ?
@
1, n) @  (m, n)
1 @ (c)
@
@
@ µ1 q1  (m + 1, n) 6 @@...
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 Spring '10
 DURRETT
 The Land

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