Reliability HotWire: eMagazine for the Reliability Professional
Reliability HotWire

Issue 82, December 2007

Hot Topics
Estimating Reliability Based on Multiple Random Stress Types

Products are affected by the stresses applied to them. This is an essential concept in quantifying and designing-in reliability, as emphasized by many design methodologies, such as Design for Reliability (DfR) analysis, Robust Design and Engineering Design By Reliability (EDBR). This article describes an approach to estimating the reliability of a product affected by multiple random stress types. ALTA 7 and Weibull++ 7 are used in the analysis.

In a previous issue of Reliability Edge, an article described an approach for estimating reliability based on a distribution of usage data. In that article, only one stress (usage) was considered in the analysis. In another Reliability Edge article, a product's reliability was estimated based on the distribution of two parameters (load and diameter). These two parameters were used to calculate a single stress type. In both of these articles, the concept of stress-strength analysis was used to estimate the overall reliability of the product. Stress-strength analysis can be used easily when dealing with one stress type. However, when dealing with multiple independent stress types, that concept becomes harder to apply. In this article, we describe an approach that can be applied when multiple stress types (either customer usage or environmental factors) are to be considered in the analysis along with their respective distributions. This approach is based on:

  • Understanding the life-stress relationship combined with the distribution. This model describes the probability of failure based on particular combinations of stress values. This analysis is performed with the help of ALTA 7.
  • Using simulation as the tool to incorporate the randomness of the stresses. This analysis is performed using the Monte Carlo simulation tool in Weibull++ 7 (RENO can also be used).


We will consider a product affected by two stresses. The approach we will follow, however, can be generalized to more than two stresses. An electronic product is affected by temperature and voltage. In the field, the values of these two stresses are random. Data showing the randomness of these stresses in the field have been collected. Based on these data, distributions that describe the stresses have been estimated. Temperature (in units of K) follows a normal distribution with μ = 323 and σ = 20. Voltage (in units of V) follows a normal distribution with μ = 12 and σ = 3. The manufacturer wants to estimate the B10 life of the product (i.e. the age by which 10% of the units fail) with 95% confidence.


To download the *.rso7 file for this example, right-click here to save the file to your computer. Please note that this file contains both ALTA and Weibull++ Folios. You must have both applications installed in order to be able to view both types of Folios; if you have only one of the applications, you will be able to view only the Folio(s) specific to that software. Free demonstration copies of the software are available for download from


Establishing the Life-Stress Relationship and Distribution Model

The life-stress relationship combined with the distribution provides a model that describes the expected life of a product given a certain combination of the stress levels exerted on the product. The product is assumed to follow a Weibull distribution. The temperature-nonthermal life-stress relationship is used:

where U represents temperature, V represents voltage and B, C and n are parameters.

Substituting the above life-stress relationship into the characteristic life parameter (which is considered to be stress-dependent) of the Weibull distribution yields the Inverse Power-Weibull model.


The above model can be built from quantitative accelerated tests. The following table shows the failure times, in hours, of units tested under different levels of temperature and voltage.

The estimated model parameters are calculated in ALTA 7 (as shown in the above figure)

  • β = 5.8744

  • B = 3282.0013

  • C = 30.5492

  • n = 2.4511

Note: To generalize this approach to more than two stress types, the general log-linear and proportional hazards models can be used for the life-stress relationship. ALTA 7 PRO allows up to eight simultaneous stress types.

Incorporating the Stress Distributions

The above model can be used to calculate reliability (or other metrics, such as B10, MTTF, median life, conditional reliability, etc.) by specifying certain values of voltage and temperature. The calculations should reflect the random nature of the stresses. Simulation is a simple, straightforward tool to handle this otherwise complex problem. The algorithm of the simulation is as follows:

  • Step 1 - Generate N random combinations of the temperature and voltage based on their respective distributions.
  • Step 2 - For each of the N combinations of temperature and voltage, calculate the metric of interest (n this case B10 life) based on the life-stress relationship combined with the distribution for each of the N combinations of temperature and voltage.
  • Step 3 - We now have N values of B10 life, which are random because of the randomness of temperature and voltage. The set of N B10 values can be fitted with a distribution and the B10 can be estimated with 95% confidence.

Step 1 is performed in Weibull++ 7 using the Monte Carlo simulation tool. N = 1000 random values of temperature and voltage are generated. Weibull++ 7 displays the randomly generated numbers in a Standard Folio.

Generating 1000 random temperature values

Generating 1000 random voltage values

Step 2 is performed in ALTA 7 using the General Spreadsheet. The random (unsorted) values of temperature and voltage are entered in different columns. In the third column, the B10 life is calculated (using the Function Wizard's TIMEATPF_S function). The Function Wizard and calculated General Spreadsheet are shown next.

In Step 3, the B10 results obtained in the previous step are entered into a Standard Folio in Weibull++ 7 and fitted with the best fit model. The generalized gamma distribution is found to be a good fit for the B10 data, as seen in the following probability plot.

Now, we are ready to calculate the B10 life of the product with a 95% confidence. As shown in the next figure, the 95% 1-sided lower bound on the B10 life of the product is 345.39 hours.

Note: Your results may vary from the ones shown here, as the results are based on simulation.


In this article, we presented a simple approach to estimate reliability when multiple random stresses affect the life of the product. The procedure was demonstrated with an example using two stress types. This approach can be generalized to more than two stress types.

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