IEOR 4106, Spring 2011, Professor Whitt
Introduction to Renewal Theory: Tuesday, April 5
1. Painful Memories: Visit to a Museum
Recall:
A popular museum is open for 11 hours each day, but only admits new visitors
during the first 9 hours. All visitors must leave at the end of the elevenhour period. Suppose
that visitors (which may be individuals or small groups, which we treat as individuals) arrive
at the museum according to a Poisson process at rate 100 per hour. Suppose that each visitor,
independently of all other visitor, spends a random time in the museum that is uniformly
distributed between 0 and 2 hours. Suppose that 25% of these visitors visit the museum gift
shop while they are in the museum. (Visiting the gift shop is assumed not to alter the total
length of stay in the museum.) Statistics have revealed that the dollar value of the purchases
by each visitor to the gift shop has, approximately, a gamma distribution with mean $40 and
standard deviation $30.
I. Formalizing what has been assumed: random variables and stochastic pro
cesses.
We can formalize the information problem by defining several stochastic processes,
some sequence of random variables and some continuoustime processes. We now review the
structure, based on the assumptions above:
continuoustime stochastic processes:
Let
N
(
t
) be the number of visitors to come to
the museum in the first
t
hours, i.e., during the time interval [0
, t
]. (Here 0
≤
t
≤
9, but ignore
the termination time; think of it as a stochastic process with
t
≥
0.) Let
M
(
t
) be the number
of visitors to come to the museum during the time interval [0
, t
] that will go to the gift shop
sometime during their visit. Let
D
(
t
) be the dollar value of the purchases from the gift shop
by all the visitors that initially arrived at the museum in the interval [0
, t
].
associated random variables;
Let
X
i
be the interarrival time between the (
i

1)
st
visitor
and the
i
th
visitor to the museum. Let
U
j
be the interarrival time between the (
j

1)
st
visitor
and the
j
th
visitor to the museum, counting only those that eventually go to the gift shop
during their visit.. Let
Z
i
= 1 if the
i
th
visitor to the museum goes to the gift shop sometime
during his visit; otherwise
Z
i
= 0. Let
Y
j
be the dollar value of all purchases by the
j
k
rmth
visitor to arrive among those that go to the gift shop.
discretetime stochastic processes:
For the random variables above, we get the as
sociated stochastic process (sequence of these random variables):
X
i
:
i
≥
1
}
,
U
j
:
j
≥
1
}
,
Z
i
:
i
≥
1
}
,
Y
j
:
j
≥
1
}
.
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 Spring '11
 WhitWard
 Probability theory, zi

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