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Unformatted text preview: stationary distribution ⇡ (n) for the M/M/s queue are
unpleasant to write down for a general number of servers s, but it is not hard
to use (4.26) to ﬁnd the stationary distribution in a concrete cases: If s = 3,
= 2 and µ = 1, then
X k=2 so P1 k=0 1
⇡ (k ) = · 2
(2/3)j = 6c
j =0 ⇡ (k ) = 9c and we have
⇡ (0) = ,
⇡ (1) = ,
⇡ (k ) =
9 ✓ ◆k
3 2 for k 2 Our next result is a remarkable property of the M/M/s queue.
Theorem 4.8. If
< µs, then the output process of the M/M/s queue in
equilibrium is a rate Poisson process.
Your ﬁrst reaction to this should be that it is crazy. Customers depart at rate
0, µ, 2µ, . . ., sµ, depending on the number of servers that are busy and it is
usually the case that none of these numbers = . To further emphasize the
surprising nature of Theorem 4.8, suppose for concreteness that there is one
server, = 1, and µ = 10. If, in this situation, we have just seen 30 departures
in the last 2 hours, then it seems reasonable to guess that the server is busy
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
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