# Let at t tn t and z t tn t1 t at gives the

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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 1/ = ⇡ (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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