# Converting Probabilistic requirements to MTBF

#### Converting Reliability between Probabilistic Mission Duration & Mean Time Between Failure

One can translate requirements or statements of reliability between probabilistic mission duration and MTBF using the exponential distribution.  The cumulative distribution function for the exponential distribution (cumulative probability of failure in a reliability context) is:

$$F(t) = 1 – e^\frac{-t}{\theta}$$

Where t is the mission duration and θ is the MTBF for a repairable system if you assume that repairs are perfect.

Thus, the probability that a howitzer with a MTBF of 62.5 hours will fail an 18 hour mission is:

$$F(18) = 1 – e^{\frac{-18}{62.5}} = .25$$

Conversely, the system has a 75% chance of completing the mission successfully.

There are several problems with making this conversion.  First, a mean may not be the correct metric to summarize mission reliability.  Additionally, the requirement is now based on the assumption that the exponential distribution is reasonable and that failures are independent of one another.  Finally, in some cases this translation can result in MTBF requirements that exceed the expected use of the system.  For example, consider a bomb that travels on a fighter aircraft.  If we need extremely high probabilities of completing a standard 2-hour mission without an in-flight failure we could use this translation to conclude we need a weapon with an extremely large MTBF.  The table below shows several translations.  For extremely high reliabilities we quickly generate large values of MTBF.  In our bomb reliability example, we might not expect an individual weapon to ever exceed 50 hours of flight time, which makes having a nearly 200 hour mean requirement unreasonable.

Probability of Mission Completion (Mission Duration)Mean Time Between Failure
99% (2-Hour Mission)199 Hours
95% (2-Hour Mission)39 Hours
95% (4-Hour Mission)78 Hours

Despite the problems with making this translation, it is a common practice in Defense applications.  In cases where the translation is reasonable, it can provide a meaningful way of planning tests for continuous reliability metrics that is more efficient than strictly planning around the pass/fail requirement at the mission level.