CS/EE 147
HW 6: Transform world
Guru: Lina
Assigned: 05/13/10
Due: 05/26/10, Raga’s mailbox, 1pm
We encourage you to discuss these problems with others, but you need to write up the actual solutions alone.
At the top of your homework sheet, list all the people with whom you discussed. Crediting help from other
classmates will not take away any credit from you. Start early and come to office hours with your questions!
Note: This homework is your another chance to make up for lost points! The last extra credit question
is quite easy.
1
Busy periods [20 points]
A busy period starts when the server becomes busy and ends when it goes idle. It turns out that understanding
the distribution of the length of a busy period is fundamental to the study of many scheduling policies, as
we will see in the next few lectures.
Consider an M/GI/1 queue and let
B
denote the length of a busy period and let
B
(
x
)
denote the length
of a busy period conditional on the first job in the busy period having size
x
.
(a) Use renewalreward arguments to derive the mean length of a busy period,
E
[
B
]
.
(b) Derive the Laplace transform of
B
(
x
)
in terms of the Laplace transform of
B
.
(c) Use the Laplace transform of
B
(
x
)
to derive the following recursive formula for the Laplace transform
of
B
:
e
B
(
s
) =
e
X
(
s
+
λ

λ
e
B
(
s
))
.
(d) Take derivatives of the Laplace transform in order to calculate
E
[
B
]
and
E
[
B
2
]
.
(e) Contrast
E
[
B
]
with
E
[
T
FCFS
Q
]
when
E
[
X
2
]
is large/small. (Use the phrase “inspection paradox”.)
2
The distribution of Excess [20 points]
In class we used renewalreward theory to derive the mean excess. It turns out we can use the same ideas to
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 Fall '09
 Variance, Probability theory, probability density function, Cumulative distribution function

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