Prob. 1 (20 points) Let
γ
be the expected number of transmitted frames from node
A
to
node
B
per successfully accepted packet at node
. Let
β
be the expected number of
transmitted frames from node
to node
between the transmission of a given frame and
the reception of feedback about that frame (including the frame in transmission when the
feedback arrives). Let
p
be the probability that a frame arriving at node
contains errors
(with successive frames assumed independent). Assume that node
i
s
a
lway
s
busy
transmitting frames, that
n
is large enough that node
never goes back in the absence of
feedback, and that node
always goes back on the next frame after hearing that the
awaited frame contained errors. Find
as a function of
and
.
Sol. 1
1.) The success probability in 1st packet transmission =
1(
1 )
p
×
−
= The expected number of
transmitted frames.
2.) The success probability in 2
nd
packet transmission:
(1
)
p
p
−
The expected number of transmitted frames:
(1 1
) (1
)
p
p
+
+−
3.) The success probability in ith packet transmission:
1
)
i
p
p
−
−
The expected number of transmitted frames:
1
((1
)
) (
i
ii
p
p
−
+
−−
(10 points)
1
1
1
{(
(
1) )
)}
1
i
i
p
p
p
p
γβ
∞
−
=
+
∴
=+
−
−
=
−
∑
(10 points)
Sol. 2
[
]
[#of transmitted frames when an error occurs]
+ (1 [
])
[#of transmitted frames when an error does not occurs]
P error
E
P error
E
=×
×
(10 points)
)
)
p
p
βγ
+
+
−
1
1
p
p
+
∴
=
−
(10 points)
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View Full DocumentProb. 2. (15 points) Consider the M/G/1 system with the difference that each busy period is
followed by a single vacation interval. Once this vacation is over, an arriving customer to an
empty starts service immediately. Assume that vacation intervals are independent, identically
distributed, and independent of the customer interarrival and service times. Prove that the
average waiting time in queue is
22
2(1
)
2
X
V
W
I
λ
ρ
=+
−
, where
I
is the average length of an idle
period, and show how to calculate
.
We have
1
R
W
=
−
(2 points)
where
()
11
1
1
lim
Mt
Lt
ii
t
R
XV
tt
→∞
==
⎧⎫
⎨⎬
⎩⎭
∑∑
(4 points)
where
is the number of vacations (or busy periods) up to time
t
. The average length of an
idle period is
00
V
V
I
pv
v e d
v e d dv
λτ
λττ
∞∞
−−
⎡⎤
⎢⎥
⎣⎦
∫∫
∫
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 Spring '09
 ChoiSunghyun
 Probability theory

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