Q: 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.) The
solution will be released after this problem is finished. You will
need the solution to complete the rest of the problem.
pGG =
pGB =
pGD =
pBG =
pBB =
pBD =
pDG =
pDB =
pDD =
Part 2
What is the steady-state probability of being in each state?
(Please answer with 3 decimals)
P(G) =
P(B) =
P(D) =
Part 3
What is the production rate of good parts in parts/min? (Please
answer with 3 decimals) =
What is the production rate of bad parts in parts/min? (Please
answer with 3 decimals) =
The yield is the number of good parts produced divided by the
total number of parts produced (or the production rate of good
parts divided by the total production rate). What is the yield?
(Please answer with 3 decimals) =
