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Unformatted text preview: the empty state. Thus the server experiences
alternating busy periods with duration Bn and idle periods with duration In .
In the case of Markovian inputs, the lack of memory property implies that In
has an exponential distribution with rate . Combining this observation with
our result for alternating renewal processes we see that the limiting fraction of
time the server is idle is
= ⇡ (0)
1/ + EBn
by (3.5). Rearranging, we have
EBn = 1 ✓ 1
⇡ (0) 1 ◆ (3.6) Note that this is
not a . For the fourth and ﬁnal formula we will have
to suppose that all arriving customers enter the system so that we have the
following special property of Poisson arrivals is:
PASTA. These initials stand for “Poisson arrivals see time averages.” To be
precise, if ⇡ (n) is the limiting fraction of time that there are n individuals in
the queue and an is the limiting fraction of arriving customers that see a queue
of size n, then
Theorem 3.7. an = ⇡ (n).
Why is this true? If we condition on there being arrival at time t, then the times
of the previous arrivals are a Poisson process with rate . Thus knowing that
there is an arrival at time t does not a↵ect the distribution of what happened
before time t.
Example 3.7. Workload i...
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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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