IND ENG 173, Practice First Midterm
3. A certain computer can crash on any day with probability
p
∈
(0
,
1), independently of previous
days. If it crashes, it gets sent at the end of the day to receive maintenance. If the computer
goes
m
days without crashing it will be sent for maintenance at the end of that day regardless
of whether it crashed during the day or not.
Let
X
n
denote the number of days that the computer has operated without maintenance at
the beginning of its
n
th day of operation;
X
n
∈ {
0
,
1
,
2
, . . . , m

1
}
. The state 0 corresponds
to the machine just having received maintenance.
(a)
[10 pts]
Write down the onestep transition probabilities (in terms of
p
and
m
).
(b)
[10 pts]
Write down a system of linear equations (in terms of
p
and
m
) that when solved
will give the stationary probabilities. Do not solve it.
IND ENG 173, Practice First Midterm
5
(c)
[15 pts]
Solve the system of equations from part (a) and compute (
π
0
, π
1
, π
2
, . . . , π
m

1
).