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@@ µ1 p1
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2
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(m, n 1)
(m + 1, n
(b) 1) The rate arrows plus the dotted lines in 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 need to show that
(a) µ1 ⇡ (m, n) = µ2 p2 ⇡ (m (b) µ2 ⇡ (m, n) = µ1 p1 ⇡ (m + 1, n (c) ( 1 + 2 )⇡ (m, n) 1, n + 1) + = µ2 (1 1) + 1 ⇡ (m 1, n) 2 ⇡ (m, n 1) p2 )⇡ (m, n + 1) + µ1 (1 p1 )⇡ (m + 1, n) mn
Filling in ⇡ (m, n) = c↵1 ↵2 and canceling out c, we have
m
mn
µ1 ↵1 ↵2 = µ2 p2 ↵1
mn
µ2 ↵1 ↵2 = ( 1 + mn
2 )↵1 ↵2 = 1 n+1
m
n
↵2 + 1 ↵1 1 ↵2
m
n
mn
µ1 p1 ↵1 +1 ↵2 1 + 2 ↵1 ↵2 1
m
mn
n
µ2 (1 p2 )↵1 ↵2 +1 + µ1 (1 p1 )↵1 +1 ↵2 147 4.6. QUEUEING NETWORKS* Canceling out the highest powers of ↵1 and ↵2 common to all terms in each
equation gives
µ1 ↵1 = µ2 p2 ↵2 +
( 1 1 µ2 ↵2 = µ1 p1 ↵1 + 2 + 2) = µ2 (1 p2 )↵2 + µ1 (1 p1 )↵1 Filling in µi ↵i = ri , the three equations become
r 1 = p2 r2 + 1 r2 = p1 r1 +...
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 Spring '10
 DURRETT
 The Land

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