Question

A certain type component has two states: 0=OFF and 1=OPERATING. In state 0, the process remains there for 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: Component Operating Failure Rate Repair Rate A ?_A ?_A B ?_B ?_B In order for systems 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;

          A certain type component has two states: 0=OFF and 1=OPERATING. In state 0, the process remains there for 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:

Component    Operating Failure Rate    Repair Rate
A            ?_A                       ?_A
B            ?_B                       ?_B

In order for systems 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;

A certain type component has two states: 0=OFF and 1=OPERATING. In state 0, the process remains there for 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:

Component Operating Failure Rate Repair Rate
A ?A ?A
B ?B ?B

In order for systems 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;

Added by Mitchell M.

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