MTBF calculation for redundant system

Lambda = failures per million hours

Serial Failures:

Let's do the serial failure case first. In this case, the failure of
any one component will cause failure of the system. This may sometimes
be illustrated as:
A -> B -> C -> D
So if you have four components, A, B, C, and D each with MTBF of
20,000, 10,000, 15,000, and 30,000 hours respectively, using this
method, the MTBF of the system is calculated as:
Lambda A = 1,000,000 / 20,000 = 50.0
Lambda B = 1,000,000 / 10,000 = 100.0
Lambda C = 1,000,000 / 15,000 = 66.67
Lambda D = 1,000,000 / 30,000 = 33.33
Lambda (composite system) = 50+100+66.67+33.33 = 250
MTBF (composite system) = 1,000,000 / 250 = 4,000 hours

Redundant Components:

If you have two components in parallel (e.g., dual power supplies)
where a failure of both components is required to fail the system, the
failure rate of the system is MUCH less than either component.

Assume A and B both have MTBF of 100 hours or Lambda = 10,000. The
failure rate F for A and B would then be 0.01 for each. For
comparison, the serial solution has Lambda = 20,000 failures per
million or MTBF = 50 hours.

For the redundant case, the probability (F) that both items are failed
at the same time is:
F = FA * FB
F = 0.01 * 0.01
F = 0.0001
Solving for lambda gets
Lambda = 100
or
MTBF = 10,000 hours
So there is a substantial improvement in reliability when using
redundant components. Note that if you have serial components before /
after the redundant components, you still need to handle those in
series with the redundant components.

A composite system:

If you have both serial / parallel components, break up the system
into pieces and do the lambda calculations as serial or parallel. I
usually end up with several serial items to add at the end and then
compute the overall system MTBF value.