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Unformatted text preview: hine formula (3.7), since a key
ingredient in the derivation is false: arriving customers who enter the system
do not see the time average queue length.
By our fourth queueing equation, (3.6), the server’s busy periods have mean
✓
◆
✓
◆
1
1 65
19
1
1=
1=
EB =
⇡ (0)
2 27
27
which agrees with the computation in Example 4.21. Multiple servers
Our next example is queue with s servers with an unlimited waiting room, a
system described more fully in Example 4.3.
Example 4.25. M/M/s queue. Imagine a bank with s
1 tellers that
serve customers who queue in a single line if all servers are busy. We imagine
that customers arrive at the times of a Poisson process with rate , and each
requires an independent amount of service that has an exponential distribution
with rate µ. As explained in Example 1.3, the ﬂip rates are q (n, n + 1) = and
(
µn
if n s
q (n, n 1) =
µs
if n s
The conditions that result from using the detailed balance condition are
⇡ (j
⇡ (j 1) = µj ⇡ (j )
1) = µj ⇡ (j...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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