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Unformatted text preview: ous customers, so it is
independent of her service time, i.e., E (si qi ) = Esi · WQ and we have
Y = (Esi )WQ + E (s2 /2)
i PASTA implies that arriving customers see the long run average behavior so
the workload they see Z = WQ , so we have
WQ = (Esi )WQ + E (s2 /2)
i
Solving for WQ now gives E (s2 /2)
i
(3.7)
1
E si
the socalled PollaczekKhintchine formula. Using formula (3.3), and Theorem 3.6, we can now compute
WQ = W = WQ + Esi L= W Example 3.8. We see a number of applications of the equations from this
section to Markovian queues in Chapter 4. Customers arrive at the CIT help
desk at rate 1/6 per minute, i.e., the mean time between arrivals is 6 minutes.
Suppose that each service takes a time with mean 5 and standard deviation
p
59.
(a) In the long run what is the fraction of time, ⇡ (0), that the server is idle?
= 1/6, Esi = 5 = 1/µ, so by (3.5) ⇡ (0) = 1 (1/6)/(1/5) = 1/6.
(b) What is the average waiting W time for a customer (including their service
time)? Es2 = 52 + 59 = 84, so (3.7) implies
i
WQ = E s2 /2
(1/6) · 84/2
i
=
= 42
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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