Question

Problem 2.2 Consider a production system consisting of three Bernoulli
machines and a controller, which also obeys the Bernoulli reliability model. This
production system is considered up if the controller and at least two machines are
up. During each cycle, the controller is up with probability 0.8 and each machine
is up with probability 0.9. The controller and the machines fail independently.
(a) Calculate the probability that the production system is up.
(b) Assume that a second controller, identical to the first one, is added to the
system, and the re-designed system is up if at least one controller and two
machines are up. Calculate the probability that the re-designed production
system is up.
(c) Explain the reason for the difference between the two numbers you calculated.

Answer 1

a) The production system is considered up when one controller and atleast two machines are up.

So the probability that the system is up is equal to:

P(controller is up)*P(atleast 2 machines are up)

P(C) = 0.8

P(M) = P(alteast 2 machines are up)

= P(2 machines are up) + P(3 machines are up)

P(2 machines are up) = 3*0.9*0.9*0.1 = 0.243

P(3 machines are up) = 0.9³ = 0.729

P(M) = 0.972

P(system is up) = 0.8*0.972 = 0.7776

b) The system is up when atleast 1 controller is up and atleast 2 machines are up.

P(atleast 2 machines are up) = 0.972 [as calculated in (a)

P(C) = P(atleast 1 controller is up)

P(C) = P(1 controller is up) + P(2 controllers are up)

P(1 controller is up) = 2*0.8*0.2 = 0.32

P(2 controllers are up) = 0.8*0.8 = 0.64

P(C) = 0.96

Hence the required probability that

P(system is up) = 0.96*0.972 = 0.93312

c) We can see that the probability that the redesigned system is up is much higher than the probability that the system will work with only one controller. The difference in the probabilities is as large as 0.15552 or 15.55%.

This difference is due to the fact the additional controller provides a backup to the original controller and hence in case the original controller fails it is highly probable that the new controller will work to keep the system up.

This as a result leads to an increased probability that the system will stay up.