 Reliability Basics

Predicting the Number of Failures for Repairable Systems

[Editor's Note: This article has been updated since its original publication to reflect a more recent version of the software interface.]

Many complex systems, such as military vehicles and mining equipment, are repairable. For repairable systems, the number of failures at a given operation period is one of the most important reliability metrics. Based on the predicted number of failures, proper resources can be allocated. ReliaSoft’s RGA (reliability growth analysis) software package provides tools for repairable system modeling and prediction. In this article, we will explain how to predict the number of failures, with confidence bounds, for repairable systems.

Modeling

The model used in RGA is a power law non-homogeneous Poisson process (NHPP) model. This model assumes that the rate of occurrence of failure (ROCOF) is a power function of time. For NHPP, the ROCOFs are different at different time periods. The ROCOF for a power law NHPP is: (1)

where λ(t) is the ROCOF at time t, and β and λ are the model parameters.

From Eqn. (1), the expected number of failures from time 0 to t is calculated by: (2)

Therefore, the expected number of failures from time t1 to t2 is: (3)

where Δt = t2-t1.

For an NHPP, if a time interval is given, the number of failures in this given interval follows a Poisson process with parameter of (E(N(Δt)). Defining m = (E(N(Δt)), the probability of observing exactly i failures in this interval is: (4)

Please note that Eqns. (2) and (3) calculate the expected number of failures, while Eqn. (4) calculates the probability of obtaining i failures for a Poisson process with parameter of (E(N(Δt)). The "expected number of failures" and the "number of failures" are different-- the "number of failures" value is derived from a Poisson process that uses the "expected number of failures" value as a parameter.

Parameter Estimation

Once we have failure data, the model parameters and their variance can be estimated using maximum likelihood estimation (MLE). For details, please refer to the Reliability Basics article in last month's issue of HotWire. In this article, we will focus on calculating the confidence bounds for the number of failures and for the expected number of failures. An example is used to illustrate the calculations.

Example

The following figure shows failure data for a repairable system. Figure 1 - Failure data for a repairable system

The estimated β = 1.3009 and λ = 0.0052. The variance and covariance matrix for these two parameters can be obtained from a general spreadsheet, as shown next. Figure 2 - Variance/covariance matrix for model parameters

Using the calculated results from RGA, we can calculate the bounds for the expected number of failures. Based on these calculated bounds, which are then used as the parameter for the Poisson distribution, we can further calculate the confidence bounds for the number of failures. These calculations are not available in RGA; all of the calculations are done in Excel®.

Predicting the Confidence Bounds for the Expected Number of Failures

Assume we want to calculate the expected number of failures and its confidence bounds for the next 200 hours. The current time is 410 hours.

Applying Eqn. (3), the expected number of failures is: (5)

The variance of is: (6)

Once we have obtained the value and variance of the expected number of failures, we can calculate the bounds using the following equation: (7)

where Z1-α/2 is the 1-alpha/2 percentile of a standard normal distribution. 1-alpha is the confidence level. Here we use 90% two-sided confidence bounds.

For this example, the upper bound and the lower bound for the expected number of failure are: (8)

Many engineers mistakenly think that these two values are the bounds for the number of failures in the next 200 hours. In fact, they are the bounds for the expected number of failures. In other words, they are the bounds for the mean value of the number of failures. One may therefore ask how to obtain the bounds for the number of failures. This is shown in the following section.

Predicting the Confidence Bounds for the Number of Failures

As we discussed in Eqn. (4), the number of failures is a random number from a Poisson distribution with a parameter of m(Δt). Using Eqn. (4) and = 8.8185 from Eqn. (5), we can calculate the probability of observing i failures in the interval time of Δt. The results are given in the next table.

Table 1 - Predicting the number of failures using the estimated mean

 Number of Failures = 8.8185 Probability of Failure Cumulative Probabilityof Failure 0 0.00015 0.00015 1 0.0013 0.00145 2 0.00575 0.00721 3 0.01691 0.02412 4 0.03729 0.0614 5 0.06576 0.12717 6 0.09665 0.22382 7 0.12176 0.34558 8 0.13422 0.4798 9 0.13151 0.61131 10 0.11597 0.72728 11 0.09297 0.82026 12 0.06832 0.88858 13 0.04635 0.93493 14 0.02919 0.96412 15 0.01716 0.98128 16 0.00946 0.99074 17 0.00491 0.99565 18 0.0024 0.99805 19 0.00112 0.99917 20 0.00049 0.99966 21 0.00021 0.99987 22 0.00008 0.99995 23 0.00003 0.99998 24 0.00001 0.99999

The first column is the number of failures; the second column is the probability of getting this number of failures; the third column is the cumulative probability of failure. For the 90% two-sided confidence bounds for the number of failures, we can see from the above table that the lower bound is 3 failures and the upper bound is 14 failures. Due to the discrete nature of the Poisson random variable, the lower bound is the largest integer whose cumulative probability is less than 5%; the upper bound is the smallest integer whose cumulative probability is greater than 95%.

Table 1 uses , the mean value of the expected number of failures, for the calculation. However, as given in Eqn. (6), there is uncertainty associated with the estimation of the mean value. Eqn. (7) provides the upper and lower bounds for the mean value of the expected number of failures. To get more conservative results, instead of using the mean value, we can use the upper bound of the expected number of failures to calculate the probability of failures. The calculation is similar to the calculation in Table 1. In Table 2, we calculate the probability of failures using .

Table 2 - Predicting the number of failures using the estimated upper bound Number of Failures = 18.61 Probability of Failure Cumulative Probabilityof Failure 0 0 0 1 0 0 2 0.000001 0.000002 3 0.000009 0.00001 4 0.000041 0.000052 5 0.000154 0.000206 6 0.000477 0.000682 7 0.001268 0.00195 8 0.00295 0.0049 9 0.0061 0.011001 10 0.011354 0.022354 11 0.01921 0.041564 12 0.029794 0.071359 13 0.042656 0.114014 14 0.056707 0.170721 15 0.070361 0.241082 16 0.081846 0.322928 17 0.089606 0.412533 18 0.092651 0.505184 19 0.090757 0.595942 20 0.084458 0.680399 21 0.074852 0.755251 22 0.063324 0.818576 23 0.051242 0.869818 24 0.039738 0.909556 25 0.029584 0.939139 26 0.021177 0.960316 27 0.014598 0.974914 28 0.009703 0.984617 29 0.006227 0.990844 30 0.003863 0.994708

Table 2 shows that the upper and lower bounds for the number of failures are 26 and 11, respectively. The results in Table 2 can be summarized as: the probability of getting no more than 26 failures is 0.960316 with a confidence level of 95% for the expected number of failures. This is because = 18.61 is the one-sided upper bound for the expected number of failures with a confidence level of 95%.

Conclusion

This article explains the differences between the confidence bounds for the expected number of failures and the confidence bounds for the number of failures. The number of failures is a Poisson random variable with its parameters estimated from the data. Due to the limited sample size, the estimated parameter, which is the expected number of failures, has uncertainty associated with it. To calculate the probability of obtaining a certain number of failures, both the uncertainty of the parameter and the uncertainty due to the random Poisson process should be considered. Results in Table 2 include both uncertainties. Copyright ® 2012-2014 ReliaSoft Corporation, ALL RIGHTS RESERVED