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
longrun 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.
 Spring '10
 DURRETT
 The Land

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