A system is subject to failure and repair. The conditions of components of the system are monitored by sensors. When the sensor measurement of the performance of a component is outside of tolerance, a message is generated. When a message is generated, the component is repaired. 40% of the messages are false alarms; the other 60% of the messages are due to impending failure. Whether or not a message is a false alarm is independent from message to message. Once the system is repaired, it is as good as new.
The time from when the system is repaired until a new message is generated exponential distribution with mean of 30 days. If the message is a false alarm, the repair time has an exponential distribution with mean of 3 days. If the message is due to an impending failure, the repair time has a gamma distribution with shape parameter 2 and mean of 6 days. Each repair of the false alarm costs $300. Each repair of an impending failure costs $3000.
Compute the long run average repair cost.
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