6.1. A certain type component has two states: 0 = OFF and 1 = OPERATING.

In state 0, the process remains there a random length of time, which is exponentially distributed with parameter

a, and then moves to state 1. The time in state 1 is exponentially distributed with parameter 13, after which the process returns to state 0.

The system has two of these components, A and B, with distinct parameters:

In order for the system to operate, at least one of components A and B must be operating (a parallel system). Assume that the component stochastic processes are independent of one another. Determine the long run probability that the system is operating by

(a) Considering each component separately as a two-state Markov chain and using their statistical independence;

(b) Considering the system as a four-state Markov chain and solving equations (6.11).

Component Operating Failure Rate Repair Rate A BA B BB