Prajakta, I see no one answered your question yet. My approach to reliability is to think in terms of mean time between failures (MTBF) and mean time to repair (MTTR), and approach the question using probabilities. The overall system reliability will be determined as system availability (A), which can also be stated as MTBF and MTTR. These are the parameters used to express "reliability." In short, being more explicit, when defining "reliability" as you have done.

In your case, you have some idea of the operating conditions of the switch, as in, switching operations per unit time. For example, if this is a battery backup, you might have some idea of the expected power failures per unit time. With that information, you should be able to estimate the MTBF of that switch. The manufacturer should have that information. Number of switching cycles before failure, which you can convert to a time figure, knowing the frequency of this switching operation.

MTBFsw = cycles specified before failure / switching operations/unit time

Similarly, you should be able to determine the mean time between failures of systems A and sb, from their manufacturers.

Then you should also be able to estimate how much time it takes to detect that a fault occurred, and replace the switch or other component. That's MTTR.

Availability of any component is A = MTBF / (MTBF + MTTR)

So, using this approach, you can compute AA, Asb, and Asw, which of course are respectively the availability of the main system A, the standby system sb, and the switch sw.

Now you compute the probability of these systems being down, which is 1-A, which will lead to the probability of the entire system being down.

Prob. of system being down = (1 - AA) * (1 - Asb) * (1 - Asw)

That's just the combined probability of everything having failed. So, it follows that the probability of everything being operational is 1 minus that total failure:

Availability of system (As) = 1 - [(1 - AA) * (1 - Asb) * (1 - Asw)]

You have computed overall system availability. This availability figure is often described as a "string of nines," as in As = 0.9993. But that number doesn't mean much to the average mortal. So reliability is frequently expressed in terms of MTBF. Now you can do some algebra and compute system MTBF, from the availability of each component.

First, since you already have the MTTR for each component, you should be able to estimate a system MTTR, call it MTTRs. The algebra to compute system MTBFs is straightforward.

As = MTBFs / (MTBFs + MTTRs)

MTBFs = As * MTBFs + As * MTTRs

MTBFs - As * MTBFs = As * MTTRs

MTBFs * (1 - As) = As * MTTRs

MTBFs = MTTRs * As / (1 - As)

As a quick check, if availability of the system gets close to 1, always available, MTBF of the system will approach infinity. Which makes sense. Conversely, if availability is zero, then MTBF is also zero, which also makes sense.

Albert Manfredi Thank you very much for the words of advice. I really appreciate that you explained me to the depth. It has given me a different insight to look at reliabilty in all.

Prajakta, you are very welcome. But I also obsessed about your problem some more, and there's another way to compute this, assuming that the switch could potentially fail at any time. In other words, the switch is a single point of failure, and we could allow it to fail even if not during a switching operation.

In that case, using the same introductory stuff on availability of A and of sb, we could proceed like this.

Prob. of both A and sb being down simultaneously = (1 - AA) * (1 - Asb)

Prob. of at least A or sb being available = 1 - (1 - AA) * (1 - Asb)

The above equation gives the probability of either the main system A, or the backup system sb, or both together, being fully operational.

Now you compute the availability of the whole system, with that switch as a single point of failure:

As = Asw * [1 - (1 - AA) * (1 - Asb)]

What this equation says is that the availability of the system can never be greater than the availability of the switch. This is probably the better answer.

Of course, the MTBFs would then be calculated same as in my previous answer.