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Class Problems: DTMC
1. A company uses two forecasting tools to predict the demand of its product. Tool
i
is eﬀective
w.p.
p
i
(for
i
= 1
,
2). If the
n
th
prediction uses tool
i
and it is observed to be eﬀective, then
the (
n
+ 1)
st
prediction is also done using the same tool; if it is observed to be ineﬀective,
then the (
n
+ 1)
st
prediction is made using the other tool. Let
X
n
be the tool used for the
n
th
prediction. Show that
{
X
n
, n
≥
0
}
is a DTMC, ﬁnd its transition probability matrix, and
draw the transition diagram.
2. There are two identical routes for messages to traverse from node
A
to
B
. If a route was
congested when a message needed to be sent, then it will be congested w.p.
q
when the next
message needs to be sent. Likewise, if a route was congestionfree when a message needed to
be sent, then it will be congestionfree w.p.
p
when the next message needs to be sent. The
routes behave independently of each other. Let
X
n
be the number of routes available when
the
n
th
message is attempted. Show that
{
X
n
, n
≥
0
}
is a DTMC.
3. Consider a machine tool that is used on an NCmachine. The time to produce a part and
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 Spring '08
 Klutke

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