Unformatted text preview: the state times the amount of time we spend
there. Similarly the last three are
35
·1
60 16 1
·
60 2 91
·
60 3 where again these are the probability we visit the state times the amount of
time we spend there. 137 4.5. MARKOVIAN QUEUES 4.5 Markovian Queues In this section we will take a systematic look at the basic models of queueing
theory that have Poisson arrivals and exponential service times. The arguments
in Section 3.2 explain why we can be happy assuming that the arrival process
is Poisson. The assumption of exponential services times is hard to justify, but
here, it is a necessary evil. The lack of memory property of the exponential is
needed for the queue length to be a continuous time Markov chain. We begin
with the simplest examples. Single server queues
Example 4.23. M/M/1 queue. In this system customers arrive to a single
server facility 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. From the description it should be clear that the transition rates are
q (n, n + 1) =
q (n, n 1) = µ if n
if n 0
1 so we have a birth and death chain with birth rates n = and death rates
µn = µ. Plugging into our formula for the stationary distribution, (4.18), we
have
✓ ◆n
n 1 ··· 0
⇡ (n) =
· ⇡ (0) =
⇡ (0)
(4.22)
µn · ...
View
Full
Document
This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

Click to edit the document details