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Unformatted text preview: e, which implies steady state probabilities. For the chain to
be positive recurrent, µ > 0.
c) Assume positive recurrence for each queue.
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M/M/1: L = i iπi = ρ/(1 − ρ) = λ/(µ − λ). To ﬁnd Lq , we observe that L is Lq plus
the expected number of customers in the service, i.e., L = Lq + (a − π0 ). Thus,
Lq = L − (1 − π0 ) = ρ2 /(1 − ρ) = λ2 /[µ(µ − λ)]. Using Little’s theorem, W = L/λ = a/[µ(1 − ρ)] = 1/(µ − λ)
Wq = Lq /λ = ρ/[µ(1 − ρ)] = λ/[µ(µ − λ)]
M/M/m:
There are customers in the queue only if all the servers are busy, i.e., if there are more
customers than servers in the system: 4 Lq = �
i>m (i − m)πi = � (i − m)ρi π0 mm /m! = ρπ0 (ρm)m /[(1 − ρ)2 m!] i>m Where π0 is given in part (a). The expected delay Wq in the queue is then given by
Little’s formula of a customer in the system as Wq = Lq /λ. The delay W in the system is
the queuing delay plus service delay, so W = Lq /λ + 1/µ. Finally, the expect...
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This document was uploaded on 03/19/2014 for the course EECS 6.262 at MIT.
 Spring '11
 RobertGallager

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