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UCLA Electrical Engineering Department
EE132B
HW Set Solution #4
Professor Izhak Rubin
Solution to Problem 1
(
29
(
29
(
29
(
29
{
}
(
29
0
0
(1)
Let
1 denote the number of retransmissions. Then, the random variable
1 has a
geometric distribution with parameter
. Thus, we have
1
1
1
1
(2)
Let
denote
R
R
n
R
R
n
n
k
R
N
N
p
p
E N
nP N
n
n
p p
p
N
∞
∞
=
=
=
=
=

=

∑
∑
(
29
{
}
1
the number of retransmissions for the
data frame,
1,.
..,
. Then,
. Note that
,
1,.
..,
is a set of independent and identically
distributed random variables. These random variab
th
M
k
k
R
R
R
k
k
k
M
N
M
N
N
k
M
=
=
=
=
∑
(
29
{
}
les are governed by a geometric
distribution with parameter
. For
to be equal to 2, there are two possible cases:
(i)
One of the random variables in
,
1,.
..,
is equal to 2, and other rando
R
k
R
p
N
M
N
k
M
=
{
}
(
29
(
29
1
1
m
variables are all equal to 0.
(ii) Any two of the random variables
,
1,.
..,
are each equal to 1, and the other
random variables are all equal to 0.
Hence, we have:
2
2
2
k
R
R
M
k
R
k
R
N
k
M
P N
M
P
N
P N
=
=
=
=
=
=
=
∑
(
29
(
29
(
29
(
29
(
29
(
29
(
29
2
1
1
1
2
3
1
2
1
1
2
1
2
3
2
,
0,.
..,
0
...
0,.
..,
0,
2
1,
1,
0,.
..,
0
...
0,.
..,
0,
1,
1
2,
0,.
..,
0
1,
1,
0,.
..,
0
1
2
1
1
1
M
M
M
R
R
R
R
R
M
M
M
M
R
R
R
R
R
R
R
R
M
M
R
R
R
R
R
R
R
N
N
P N
N
N
P N
N
N
N
P N
N
N
N
M
M
P N
N
N
P N
N
N
N
M
p p



=
=
+
+
=
=
=
+
=
=
=
=
+
+
=
=
=
=
=
=
=
=
+
=
=
=
=
=


(
29
(
29
(
29
(
29
(
29
(
29
1
2
2
1
1
1
2
1
1
.
2
M
M
M
M
p
p p
p p
p
M M
p
p


+



+
=

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View Full DocumentSolution to Problem 2
(1)
Note that when station A transmits a data frame, it may not receive an acknowledgement if the frame is
delivered to station B with errors or if the acknowledgement for the frame which is received without errors is
lost. Let
r
denote the probability that station A does not receive an acknowledgement. Let
A
and
B
denote the
events that a frame is received with errors and that an acknowledgement is lost, respectively. Then, we have
(
29
(
29
(
29
(
29
(
29
1
c
c
r
P A
A
B
P A
P A
B
p
p q
=
=
+
=
+

I
U
I
Whenever an acknowledgement is not received, the data frame is retransmitted. Therefore, the number of
retransmissions for a dataframe
N
R
(1) has a geometric distribution with parameter
r
. The average number of
retransmissions is given by:
(
29
(
29
(
29
(
29
(
29
1
1
1
1
1
R
p
p q
r
E N
r
p
q
+

=
=



(2)
(
29
(
29
(
29
(
29
(
29
(
29
2
2
1
2
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 Spring '09
 IzhakRubin
 Electrical Engineering

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