Consider a drilling machine in a factory. It could be in one of
three different states: G, in which it is working normally (making
good parts); B in which it is working but producing bad parts; or
D, in which it is under repair (down") and not making parts.
Transitions can occur at times 0, 30 seconds, 60 seconds, etc.
Assume that all transitions are governed by Bernoulli
distributions. That is, each time at which a transition can occur
it occurs with a probability determined by the origin and
destination state, and not dependent on how long the system was in
the origin state. When it is working or making bad parts, the
machine performs an operation in exactly 30 seconds. Assume that
when it is in one of these states and it changes state, it changes
state before it makes the next part and that the change takes no
time. That is, when it goes from G to B, it makes one bad part; and
when it goes from G to D, it does not make a part. When the machine
is making good parts, it could go to the bad part state or it could
go to the downstate. The mean time until it leaves the good state
is 20 minutes. When it leaves G, it goes to D 90% of the time and
it goes to B 10% of the time. After it reaches the bad state, it
stays in that state until it produces an average of 30 parts, and
then it always goes to D. After the machine enters D, it stays
there for a random length of time whose mean is 3 minutes, and then
it always goes to G. Part 1: What are the probabilities of going
from each state to each other state in one operation time? (Please
answer with 3 decimals) (pij is the probability of going from state
j to state I.)
PART 2
What is the steady-state probability of being in each state?
(Please answer with 3 decimals)
P(G)
P(B)
P(D)