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Unformatted text preview: ) for j s,
for j s. From this we conclude that
8 ✓ ◆k
>c
>
<
k! µ
⇡ (k ) =
✓ ◆k
>c
>
:
s!sk s µ ks
k (4.26) s where c is a constant that makes the sum equal to 1. From the last formula we
P1
see that if < sµ then j =0 ⇡ (j ) < 1 and it is possible to pick c to make the
sum equal to 1. From this it follows that 141 4.5. MARKOVIAN QUEUES
If < sµ, then the M/M/s queue has as stationary distribution. The condition < sµ for the existence of a stationary distribution is natural
since it says that the service rate of the fully loaded system is larger than the
arrival rate, so the queue will not grow out of control. Conversely,
If > sµ, the M/M/s queue is transient. Why is this true? An M/M/s queue with s rate µ servers is less e cient than
an M/M/1 queue with 1 rate sµ server, since the single server queue always
has departures at rate sµ, while the s server queue sometimes has departures
at rate nµ with n < s. An M/M/1 queue is transient if its arrival rate is larger
than its service rate.
Formulas for the...
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 Spring '10
 DURRETT
 The Land

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