7.2.2 A certain type component has two states: 0 D OFF and 1 D OPERATING. In state 0, the process remains there a random length of time, which is exponentially distributed with parameter , and then moves to state 1. The time in state 1 is exponentially distributed with parameter , 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. Consider the successive instants that the system enters the failed state from an operating state. Use the memoryless property of the exponential distribution to argue that these instants form a renewal process.

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Component Operating Failure Rate Repair Rate A B BA BB A