Stochastic

# On the other hand if we imagine that customers pay

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Unformatted text preview: V (t) ! t Combining this with the result of (a), we see that the fraction of visits on which bulbs are replaced N (t) 1/(µF + 1/ ) 1/ ! = V (t) µF + 1/ This answer is reasonable since it is also the limiting fraction of time the bulb is o↵. 106 3.2 CHAPTER 3. RENEWAL PROCESSES Applications to Queueing Theory In this section we will use the ideas of renewal theory to prove results for queueing systems with general service times. In the ﬁrst part of this section we will consider general arrival times. In the second we will specialize to Poisson arrivals. 3.2.1 GI/G/1 queue Here the GI stands for general input. That is, we suppose that the times ti between successive arrivals are independent and have a distribution F with mean 1/ . We make this somewhat unusual choice of notation for mean so that if N (t) is the number of arrivals by time t, then Theorem 3.1 implies that the long-run arrival rate is N (t) 1 lim = = t!1 t Eti The second G stands for general service times. That is, we assume that the ith customer requires an amount of service si , where the si are independent and have a distribution G with mean 1/µ. Again, the notation for the mean is chosen so that the service rate is µ. The ﬁnal 1 indicates there is one server. Our ﬁrst result states t...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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