A system is made up of two components. We suppose that the lifetime (in years) of each component has an exponential distribution with parameter λ = 2 and that the components operate independently. When the system goes down, the two components are then immediately replaced by new ones. We consider three cases:
1. the two components are placed in series (so that both components must function for the system to work);
2. the two components are placed in parallel (so that a single operating component is sufficient for the system to function) and the two components operate at the same time;
3. the two components are placed in parallel, but only one component operates at a time and the other is in standby.
Let N(t), for t > 0, be the number of system failures in the interval [0, t]. Answer the following questions for each of the cases above.
(a) Is {N(t), t ≥ 0} a Poisson process? If it is, give its rate λ. If it's not, justify.
(b) What is the average time elapsed between two consecutive system failures?