assignment9

For the chain to be positive recurrent 0 c assume

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Unformatted text preview: r i ≥ m; πi /πi−1 = (λ/µ)i π0 /i, πi /πi−1 = ρi π0 mm /m!, Since � i πi = 1, we have 3 for i &lt; m; for i ≥ m; � ... � π0 = 1 + m−1 � i (λ/µ) /i! + i=1 ∞ � �−1 i m ρ m /m! i=m � = 1+ i=m−1 � i=1 (mρ)m (λ/µ) /i! + m!(1 − ρ) �−1 i M/M/∞: Setting m = ∞ in the M/M/m result, we get: � �i λ π0 πi = , for all i ≥ 0 µ i! � Using the Taylor sries expansion of eλ/mu = i (λ/µ)i /i!, we see that π0 = e−λ/µ . Thus, � �i λ exp(−λ/µ) , for all i ≥ 0 πi = i! µ b) M/M/1: For the chain to be transient, we need λ/µ &gt; 1, for null recurrent, λ/µ = 1, and for positive recurrent λ/µ &lt; 1. M/M/m: For the chain to be transient, we need λ/mµ &gt; 1, for null-recurrent, λ/mµ = 1, for positive recurrent λ/mµ &lt; 1. M/M/∞: For the chain to be transient, we need λ &gt; 0 and µ = 0 (i.e., customers arrive but they do not depart.) We can not have null-recurrence. For µ &gt; 0, we show that the expected queue length is ﬁnit...
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This document was uploaded on 03/19/2014 for the course EECS 6.262 at MIT.

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