MTBF, Failure Rate, and Annualized Failure Rate Again

Quote of the Day

The sowing is behind; now is the time to reap. The run has been taken; now is the time to leap. Preparation has been made; now is the time for the venture of the work itself.

— Theologian Karl Barth describing midlife. Some days I think he is right – some days I am not so sure.

Introduction

Figure 1: Relationship Between Failure Rate,
MTBF, and Annualized Failure Rate.

I just had another meeting where folks thought that specifications for Annualized Failure Rate (AFR), failure rate (λ), and Mean Time Between Failures (MTBF) were three different things – folks, they are mathematically equivalent. I have given up writing the formulas down as a way to explain the concept (like here). Maybe a graphic will illustrate the relationship better? I have tried this approach before – the most successful was about component temperatures. That graphic has saved me hours trying to explain how temperature limits are specified in hardware.

Figure 1 is my attempt at showing the equivalence of these three specifications. This graphic assumes that the units of these specifications are fixed with

• MTBF is in hours.
• λ is in failures per 1E9 hours (AKA FIT)
• AFR is in % per year.

Figure 2 shows a worked example.

Figure 2: Example of the Calculations Illustrated in Figure 1.

I should mention that many computer algebra systems have the ability to handle any units you choose. Figure 3 shows an example from Mathcad.

Figure 3: Example Calculation Using Mathcad's Unit System.

You will note that the answers in Figure 3 are slightly different than in Figure 2. This is because Mathcad defines a year as 365.242… days. The formulas in Figure 2 assume 365 days in a year, which is a common assumption in reliability work.