MTBF of system with 2 units and 1 standby

[sarikrish]

:confused: Goodmorning everyone,
I need your help in Calculating MTBF of a system with 2 primary units and i standby to support a load. If one primary unit fails then the standby kicks in to compensate for the failure. I have the MTBFs of the 3 units. How do you work with the MTBF's for standby redundancy. All articles do calculations using Reliability. I need some help here.
Thanks

[Rui Assis]

If I correctly understood your case, the two duty pumps “a” and “b” in parallel form a series association in what reliability is concerned (n = 2; k = 0), as if any one of them fails, the load cannot be sustained (and a stand-by has to come into play). In this case the MTTF of the system formed by these two duty pumps is simply given by MTTFdt = 1 / (1/MTTFa + 1/MTTFb), where “dt” stands for duty.

Now, if the system is unrepairable – meaning that when a pump fails, it is left in place as it is –, then, the MTTF of the system formed by the two duty pumps and the stand-by pump is given by MTTFus = MTTFdt + MTTFsb, where “rs” stands for unrepairable system, “dt” for duty and “sb” for stand-by.

If the system is repairable, then, my advice is to use simulation. BlockSim can handle this.

Rui

[sarikrish]

Hello Rui,

Thanks for your response. I did the calculations as follows.
1). I solved for the MTBF(1) of the systems in series. Because both units have to work to support the load.
2). I then solved for the MTBF(1) in parallel with the standby unit. But here I used the formula:
MTBF = n/lambda (this is the formula for MTBF of the system with standby units where n is the number of units in standby.)

I hope this is correct. Please advice.
Thanks,

[Tarik El-Azzouzi]

What kind of standby system do you have? Cold, warm, hot?

-Cold: Standby unit cannot fail in inactive mode
-Warm: The failure rate of the standby unit during the inactive mode is less the failure rate when it becomes active.
-Hot: The failure rate of the standby unit during the inactive mode is the same as the failure rate when it becomes active. This is a simple parallel configuration.

Treating the standby unit as a unit that is simply in parallel with the other 2 active units is not correct in case of a warm or cold standby. It’s only correct if you have hot standby.

For comparison, let’s assume 2 active units in series with MTBF1=10,000 , MTBF2=500 and a standby unit with MTBF3=5000.

-For a (cold) standby configuration, the MTBFsystem=3670.77.
-For 2 units in series and a standby in parallel, the MTBFsystem=5038.98.
(Calculations done is BlockSim). Here you are treatign the standby unit as a unit that has been operating (and aging) since the begining.

For the cold or warm standby cases, analyzing the system is a bit complicated, the following link describes the proper ways of analyzing standby configurations:
http://www.weibull.com/SystemRelWeb/standby_components.htm. BlockSim can handle this type of analysis

[sarikrish]

Thanks for the response. The unit is in cold standby. So it will kick in only when either of the 2 units fails (sorry, I did my best to put it in a RBD diagram below, hope you understand it)

--|----(unit1)--------(unit2)---|------>load
| |
|--------(standby)---------- |

All three units have the same MTBF of 1700 hours.
Thanks,

[Tarik El-Azzouzi]

I presume you are assume all the units follow the exponential distribution (usually not a good assumption) and that there is no problem switching to the standby unit and that 2units are required for a successful operation. Then the MTBF of the system according to your description is MTBF=1699.4567 (as calculated by BlockSim).

Note that you can download a free evaluation version of BlockSim at http://www.reliasoft.com/BlockSim/evaluate.htm