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Reliability Phase Modeling

Reliability Phase Modeling

Reliability Phase Modeling

(OP)
Question on how to find reliability when you have two different time periods rolled up into the overall reliability requirement. I'm having a doozy of a time trying to create a reliability model for a range safety system that has multiple independent strings.

For illustrative purposes, let's say there is a requirement to be .999 reliable over a 3 hour period. Let's say the system has 3 independent strings (i.e. broadcasting transmitters). We will call them Transmitter 1 (T1), Transmitter 2 (T2), and Transmitter 3 (T3).

During phase 1 (0 to 1 hour), there is no hit against reliability as long as T3 is working along with either T1 and/or T2.

So a conditional table has 3 passing possibilities for phase 1 (0 to 1 hour):

1)T1=T2=T3 are working
2)T1 not working; T2=T3 are working
3)T1=T3 are working; T2 not working

Adding up each probability gives you R(1) for phase 1 reliability. Easy enough so far...

During phase 2 (1 to 3 hours), T2 and T3 are required however T1 is not. Here is the kicker that confuses me: What if you could repair failed transmitters during phase 1 before they entered phase 2?

I believe I need to multiply the transmitter dependability variables (MTBF/MTBF+MTTRF) so that you don't encounter a failure condition as soon as you transition from phase 1 to phase 2. There are two acceptable entry conditions into Phase 2:

1)T1(D)*T2(D)*T3(D)
2)(1-T1(D))*T2(D)*T3(D)

Adding these two values would give Dependability (D).

Now, how do I create the conditional table for phase 2?

Do you create a separate Phase 2 conditional table for each of the two passable dependability conditions above?

If so, how would you roll all of these variables up into your overall reliability for the 3 hour period?

Phase 1 Reliability (R1), Dependability (D), and Phase 2 Reliability from two different conditional tables R2 and R3....

Do you just multiply all of these together? R1*D*R2*R3? Or would it be R1*D*(R2+R3)

If this doesn't make any sense...I apologize. It's late and my head hurts!

RE: Reliability Phase Modeling

Reliability of repairable items is the probability that failure will not occur within the time period of interest = 1 - P(failure)

Reliability of two transmitters is the probability of one transmitter not failing during the total time of phase 1 and phase 2, squared.

If a transmitter is repaired before entering phase 2, then it essentially becomes a new unit again at the start of phase 2

Say the probability of failure of one transmitter during it's first 1 hour of service is 0.05, then the probability of 2 failing is .05^2 = 0.025; reliability = 1-0.025 = 97.5% With three units, 0.05^3 = .000125

If the probability of a transmitter failing during its first 2 hours of service is 0.1, then the probability of two failing during that time is 0.1^2 = 0.01, reliability of two units for 2 hours of service is 90%
If one is repaired before entering the 2nd hour, then it starts anew with its P(f) = 0.05 and P(other failing)= 0.1 P(of only 1 failing) is 0.05 * 0.01 = .005; reliability is 99.5%

Independent events are seldomly independent.

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