Product Rule
The validity of the Product Rule is of particular significance when considering the use of redundancy to improve the reliability of a system. It is shown in Parallel (or Active) Redundancy Group that if two independent items, A and B, are in parallel redundancy, then their combined reliabilities are given by:
 | By putting the two items in parallel and using the Product Rule, a considerable gain in reliability has been achieved for independent items. In practice, however, it may sometimes be questionable whether such independence is really valid. |
Example
Consider a situation where the two items are subject to a severe mechanical loading, such as shock through handling or transportation. Assuming that both items are of the same strength and are subject to the same loading, then, if the load exceeds the strength of one to produce failure, it will probably exceed the strength of the other. Such failures are termed common mode. In such cases, the Product Rule is invalid.
If both items fail together, then the system reliability (RS)is the same as if it comprised only one item and would be equal to R. Therefore, if the items were in series, system reliability would be higher than indicated by the Product Rule (i.e., Rs>R2 ), and it would be lower than indicated by the Product Rule if the items were in redundancy (i.e., Rs<(2R-R2)).
Thus, it can be seen that the methods for calculating reliability described in Parallel (or Active) Redundancy Group depend crucially on the assumption of independence of failure occurrence. Where dependence exists, in the form of common mode failures for example, then calculations become more difficult, and in fact are the subject of much current research in the reliability field. As stated earlier in Combining Reliabilities (Without Repair), such analysis is outside the scope of this guide.
| | The reader is also reminded that there are now two or more independent items to fail, and hence to maintain and so on. The apparent improvement in reliability comes with |