Let va mint xt 2 a be the time of the rst visit to a

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Unformatted text preview: I @@ µ1 p1 @@ 2 @@ ? R @@ (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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