IEOR 4106: Introduction to Operations Research: Stochastic Models
Spring 2011, Professor Whitt, March 10
1. Money Withdrawn from an ATM Machine
Customers arrive at an automated teller machine (ATM) at the times of a Poisson process
with a rate of
λ
= 10 per hour.
Suppose that the amount of money withdrawn on each
transaction has a mean of $30 and a standard deviation of $20.
(a) Find the mean and variance of the total amount of dollars withdrawn in 8 hours.
(b) What is the approximate probability that the total amount of money withdrawn in the
first 8 hours exceeds $3
,
400?
(c) How do the answers change if the Poisson arrival process is a
nonhomogeneous
Poisson
process witharrival rate function
λ
(
t
) = 4
t
,
t
≥
0?
———————————————————————
(a) Assuming that successive withdrawals are IID, this is a
compound Poisson process
;
see Section 5.4.2. Let
X
(
t
) be the total amount withdrawn in the time interval [0
, t
]. Let
N
(
t
)
be the number of customers to come to the ATM in the interval [0
, t
]. Let
Y
n
be the amount
of the
n
th
withdrawal. Then
X
(
t
) can be represented as the following random sum of random
variables
X
(
t
) =
N
(
t
)
X
i
=1
Y
i
.
Hence, from
§
5.4 (p. 346 in the last edition),
E
[
X
(
t
)] =
λtE
[
Y
1
]
and
var
(
X
(
t
)) =
λtE
[
Y
2
1
]
,
(1)
so that
E
[
X
(8)] = 10
×
8
×
30 = 2400
and
var
(
X
(
t
)) = 10
×
8
×
((30
2
+ (20)
2
) = 104
,
000
.
The standard deviation is
√
104
,
000
≈
322
.
49.
But why are those the correct formulas? To see why, look at Examples 3.10 and 3.17 in
Chapter 3. We discuss the harder variance formula. Let
Y
=
N
X
i
=1
X
i
,
where
N
is a nonnegativeintegervalued random variable and
X
i
are IID random variables.
(We are using new notation here.) Then, by the conditional variance formula in Proposition
3.1,
V ar
(
Y
) =
E
[
V ar
(
Y

N
)] +
V ar
(
E
[
Y

N
]) =
E
[
N
]
V ar
(
X
1
) +
E
[
X
1
]
2
V ar
(
N
)
.
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 Spring '11
 WhitWard
 Operations Research, Probability theory, x3, Professor Whitt

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