IEOR 4106: Introduction to Operations Research: Stochastic Models
Spring 2005, Professor Whitt, Second Midterm Exam
Chapters 56 in Ross, Thursday, March 31, 11:00am1:00pm
Open Book: but only the Ross textbook plus one
8
×
11
page of notes
Justify your answers; show your work.
1. The IEOR Department Ricoh Printer (
30
points)
The Columbia IEOR Department has a versatile Ricoh printer that can rapidly print one
sided and twosided copies, but unfortunately it often goes down.
Ricoh is alternately up
(working) and down (waiting for repair or under repair).
The average up time (time until
breakdown) is 4 days, while the average down time (time until repair) is 3 days.
Assume
continuous operation.
Let
X
(
t
) = 1 if the Ricoh is working at time
t
, and let
X
(
t
) = 0
otherwise.
(a) What do we need to assume about the successive up and down times in order to make
the stochastic process
{
X
(
t
) :
t
≥
0
}
a continuoustime Markov chain (CTMC)?
Henceforth assume that these extra assumptions are in place, so that indeed the stochastic
process
{
X
(
t
) :
t
≥
0
}
is a CTMC.
(b) Construct the CTMC; i.e., specify the model.
(c) Assuming that Ricoh has been working continuously for 7 days, what is the probability
that it will remain working at least 8 more days?
(d) Suppose that Ricoh has been working continuously for 12 days.
From that moment
forward, let
T
be the time until the
second
breakdown. What is the expected value
E
[
T
]?
(e) What is the longrun proportion of time that Ricoh is up?
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 Spring '08
 Whitt
 Operations Research, Probability theory, Markov chain, Ricoh, taxi stand

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