**Product Reliability** is defined as the probability that a device will perform its required function, subjected to stated conditions, for a specific period of time. Product Reliability is quantified as MTBF (Mean Time

Between Failures) for repairable product and MTTF (Mean Time To Failure) for non-repairable product.

Now that we’ve defined it, how can we measure it, or better yet, how can we predict it?

**The Famous Bathtub Curve**

Figure 1 shows the reliability “bathtub curve” which models the cradle to grave instantaneous failure rate vs. time, which we would see if we were to wait long enough and keep good records for a given lot of devices. This curve is modeled mathematically by exponential functions. More on this later.

**Figure 1. Reliability Bathtub Curve**

The life of a population of devices (a group of devices of the same type) can be divided into three distinct periods:

**Early Life**

If we follow the slope from the leftmost start to where it begins to flatten out this can be considered the first period. The first period is characterized by a decreasing failure rate. It is what occurs during the “early life” of a population of units. The weaker units fail leaving a population that is more rigorous.

**Useful Life**

The next period is the flat bottom portion of the graph. It is called the “useful life” period. Failures occur more in a random sequence during this time. It is difficult to predict which failure mode will occur, but the rate of failures is predictable. Notice the constant slope.

**Wearout**

The third period begins at the point where the slope begins to increase and extends to the rightmost end of the graph. This is what happens when units become old and begin to fail at an increasing rate. It is called the “wearout” period.

**The formula for calculating the MTBF is**

MTBF= T/R where T = total time and R = number of failures

MTTF stands for Mean Time To Failure. To distinguish between the two, the concept of suspensions must first be understood. In reliability calculations, a suspension occurs when a destructive test or observation has been completed without observing a failure. MTBF calculations do not consider suspensions whereas

MTTF does. MTTF is the number of total hours of service of all devices divided by the number of devices.

It is only when all the parts fail with the same failure mode that MTBF converges to MTTF.

MTTF= T/N where T = total time and N = Number of units under test.

Example: Suppose 10 devices are tested for 500 hours. During the test 2 failures occur.

The estimate of the MTBF is:

MTBF= (10*500)/2 = 2,500 hours / failure.

Whereas for MTTF

MTTF= (10*500)/10 = 500 hours / failure.

If the MTBF is known, one can calculate the failure rate as the inverse of the MTBF. The formula for

Failure rate is:

failure rate= 1/MTBF = R/T where R is the number of failures and T is total time.

Once an MTBF is calculated, what is the probability that any one particular device will be operational at time equal to the MTBF?

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