ISyE 3232
Spring 2009
Section A, Monday, April 27 8am
Professor Jim Dai
Final
No notes, no books, and no calculators are allowed.
Georgia Tech student honor code
applies to this test.
Solution to this test will be posted by 9pm on Tuesday.
Please do
not discuss this test with any students who are not in the classroom until after
that time.
The worksheet has a total of 6 pages.
1. (15 points) Assume that call arrival to a call center follows a nonhomogeneous Pois
son process.
The call center opens from 9am to 5pm.
During the first hour, the
arrival rate increases linearly from 0 at 9am to 60 calls per hour at 10am.
After
10am, the arrival rate is constant at 60 calls per hour.
(a) Plot the arrival rate function
λ
(
t
) as a function of time
t
; indicate clearly the
time unit used.
(b) Find the probability that exactly 5 calls have arrived by 9:10am.
(c) What is the probability that the 1st call arrives after 9:20am?
(d) What is the probability that there are exactly one call between 11:00am and
11:05am and at least two calls between 11:03am and 11:06am?
Solution.
(a) If we use the time units of mins:
0
≤
t
≤
60 min
s
⇒
λ
(
t
) =
1
60
t
60
≤
t
≤
480 min
s
⇒
λ
(
t
) = 1
(b) The arrival process N(t) during 9am and 9:10 am is a nonhomogeneous arrival
process, the average rate is 1/12 call per min. So we can view N(9am,9:10am)
as possion random variable with mean 10
/
12 = 5
/
6call.
The mean of poisson random variable can also be calculated through:
10
Z
0
λ
(
t
)
dt
=
10
Z
0
1
60
tdt
=
1
60
·
1
2
t
2

t
=10
t
=0
=
1
60
·
1
2
·
100 =
5
6
So the probability of exactly 5 calls during 9am and 9:10 am should be
P
(
N
(9
am,
9 : 10
am
) = 5) =
(
5
6
)
5
5!
e

5
6
1
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(c) It’s equivalent to show the probability of there’s no arrival between 9am to
9:20am.
Similar as part b,the average rate is 1/6 call per min, thus we can
view N(9am,9:20am) as possion random variable with mean 10/3 call. Thus the
required probability should be
P
(
N
(9
am,
9 : 20
am
) = 0) =
e

10
3
(d) the rate after 11 am is always 1 call per min. We are calculating the intersec
tion of two events. There’re two possibilities of the arrival time of the one call
in 11am11:05 am: either it arrives between 11am11:03m,or it arrives between
11:03am11:05am.
P(there are exactly one call between 11:00am and 11:05am and at least two calls
between 11:03am and 11:06am) =P(exactly one call in 11am11:03am and no call
in 11:0311:05 am and at least two calls between 11:05am and 11:06am)+P(no
call in 11am11:03am and one call in 11:03am11:05am and at least one call in
11:0511:06am)
Denote
N
(
x
) as possion random variable with mean x. By independent incre
ment property of nonhomogenous possion process, we know the above identity
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 Spring '07
 Billings
 Probability theory, Markov chain, longrun average number

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