Then as along as there is a stationary distribution

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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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