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Reliability Estimation for Products with Random Usage



For some products, usage is not constant. For example, within a given time period, different printers will print different numbers of pages, and different vehicles will accumulate different numbers of miles. So, if the warranty time is in years but the usage is random, how should the reliability be estimated? This article explains how this type of question can be answered.

Calculate the Usage Distribution

First, we need to collect customer usage data and fit it to a distribution. This can be done by either conducting a customer survey, or using service data in the warranty database. In this example, let's suppose we have collected mileages for 500 vehicles that have been used for three years. Some of the data set is shown next.

Mileages for 500 vehicles

The data set is fitted to a lognormal distribution, and the results show that the three-year mileage follows the distribution with a log-mean of 10.2 and log-std of 0.6. The probability plot is shown next.

Probability plot for mileages

Calculate Reliability as a Function of Usage

The second step is to get the failure time distribution. Assume we conducted an operational test for 40 vehicles. The time to the first failure of each vehicle and the suspension times, as well as the results of fitting the data to a Weibull distribution, are shown next.

Failure time data and calculation

The probability plot is shown next.

Probability plot for failure data

The above plot shows a couple of early failures. After checking the record, nothing abnormal about these early failures was found, so they will be included in the analysis.

From the obtained reliability model, we can calculate the reliability for a given mileage. For example, the reliability at 50,000 miles is 0.9719, as given below.

Reliability at 50,000 miles

Calculate Reliability at a Given Time

Once both the mileage and failure time distributions are obtained, we are ready to calculate the reliability by the end of the warranty time.

If 100,000 vehicles were sold in January 2015 and the warranty time is three years, what percentage of the vehicles will fail by the end of the warranty? At three years, the mileage accumulated by each vehicle is different. Based on the mileage data we collected, we know the mileage is a random variable that follows a lognormal distribution. We can use the estimated mileage distribution to estimate the mileage at any percentile. For example, its 90th percentile is shown next.

Mileage at 90th percentile

According to this result, 90% of the vehicles will accumulate a mileage of less than 58,213 miles in three years. A simple and conservative way to calculate the reliability at three years is to use this 90th percentile as the mileage in the calculation. The estimated reliability is calculated as shown next.

Reliability equation

The above method is similar to the so-called safety factor method. If the average mileage in three years is x miles, then we can use
y = s * x to calculate the reliability, where s is the safety factor. From the collected mileage data above, we know x is 32,301. The safety factor used in the above calculation is 58,213/32,301 = 1.8. The calculation shows that about 5% of the vehicles will fail by the end of the warranty. The drawback of using a safety factor is clear. If s is too big, then the estimated reliability will be too low. If it is too small, then the estimated reliability will be too high.

There is a better way to calculate the reliability by considering the randomness of the usage, since we already know the distribution of the mileage. The reliability can be calculated using this distribution:

Stress-strength equation

where fusage(x) is the three-year mileage distribution and Rfailure(x) is the reliability in terms of miles. This equation is the same as the commonly used stress-strength equation. Therefore, we can use the stress-strength tool in Weibull++ to calculate the reliability. For this example, the stress and strength are set as shown next.

Data sets for strength and stress

The final result is shown next.

Pdf of stress vs. pdf of strength

As you can see on the control panel, the estimated reliability for three years of use is 97.97%.

Estimated reliability

Both mileage and failure time distributions are estimated from sample data. If the sample size is small, then the uncertainty of the estimated results will be large. The confidence bounds for the estimated reliability will reflect the effect of the sample size. Weibull++ can also calculate the confidence bounds for the reliability, as given below.

Confidence bounds for reliability

The lower bound rounds to 95.96%. This means that less than 4.04% of the vehicles will fail in three years, at a one-sided confidence level of 95% (the same as a two-sided confidence level of 90%).


In this article, we illustrated how to accurately estimate a product's reliability when its usage is random. The commonly used safety factor method is very easy to apply. However, it is difficult to quantify just how "safe" the selected safety factor is, and so it is difficult to determine whether the value is within an appropriate range. The stress-strength method has a sound statistical background, and the estimated result will be more accurate by considering the randomness of the usage in the calculation. Tools such as Weibull++ can help engineers perform the required calculations.