Assignment:
In an assembly line of a factory, a robot is capable of fixing a bolt every T seconds, when working properly. However, from time to time the robot breaks down and needs repairing. Suppose that the probability that the robot breaks down after a bolt fixing is p, independently of the past history. Also suppose that each repair takes a random number of periods T with geometric distribution of mean 1/q. Determine the following quantities:
a) the distribution of the number of bolts fixed in each run, that is, in the interval between two consecutive repairs;
b) the percentage of time in which the robot is out of order;
c) the probability that the robot is out of order after k = 10 000 periods of length T.