UCSD ECE153
Handout #23
Prof. YoungHan Kim
Tuesday, May 3, 2011
Solutions to Midterm (Spring 2008)
1.
First available teller (20 points).
Consider a bank with two tellers.
The service times for
the tellers are independent exponentially distributed random variables
X
1
∼
Exp(
λ
1
) and
X
2
∼
Exp(
λ
2
), respectively. You arrive at the bank and find that both tellers are busy but
that nobody else is waiting to be served.
You are served by the first available teller once
he/she becomes free. Let the random variable
Y
denote your waiting time. Find the pdf of
Y
.
Solution:
This problem is very similar to Question 4 in Homework Set #4. First observe
that
Y
= min(
X
1
, X
2
). Since
P
{
Y > y
}
= P
{
X
1
> y, X
2
> y
}
= P
{
X
1
> y
}
P
{
X
2
> y
}
=
e

λ
1
y
×
e

λ
2
y
=
e

(
λ
1
+
λ
2
)
y
for
y
≥
0,
Y
is an exponential random variable with pdf
f
Y
(
y
) =
braceleftBigg
(
λ
1
+
λ
2
)
e

(
λ
1
+
λ
2
)
y
,
y
≥
0
,
0
,
otherwise.
2.
Sum of packet arrivals (40 points).
Consider a network router with two types of incoming
packets, wireline and wireless. Let the random variable
N
1
(
t
) denote the number of
wireline
packets arriving during time (0
, t
] and let the random variable
N
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 Spring '11
 YoungHanKim
 Probability theory, Prof. YoungHan Kim, fY M

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