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Unformatted text preview: 4.28), we investigate stability by solving the system of equations for the rj that represent the arrival rate at station j . As remarked earlier,
the departure rate from station j must equal the arrival rate, or a linearly growing queue would develop. Thinking about the arrival rate at j in two di↵erent
ways, it follows that
K
X
rj = j +
ri p(i, j )
(4.30)
i=1 This equation can be rewritten in matrix form as r =
r = (I p) + rp and solved as (4.31) 1 By reasoning in Section 1.9, where unfortunately r is what we are calling p here:
r= 1
X n=0 pn = 1K
XX ip n (i, j ) n=0 i=1 The answer is reasonable: pn (i, j ) is the probability a customer entering at i is
at j after he has completed n services. The sum then adds the rates for all the
ways of arriving at j . 148 CHAPTER 4. CONTINUOUS TIME MARKOV CHAINS Having found the arrival rates at each station, we can again be brave and
guess that if rj < µj , then the stationary distribution is given by
⇡ (n1 , . . . , nK ) = K
Y ✓ rj ◆nj ✓ µj j =1 1 rj
µj ◆ (4.32) This is true, but the proof is more complicated than for the two s...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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