# Software Engineering | Goel-Okumoto Model

The Goel-Okumoto model (also called as exponential NHPP model) is based on the following assumptions:

- All faults in a program are mutually independent of the failure detection point of view.
- The number of failures detected at any time is proportional to the current number of faults in a program. This means that the probability of the failures for faults actually occurring, i.e., detected, is constant.
- The isolated faults are removed prior to future test occasions.
- Each time a software failure occurs, the software error which caused it is immediately removed, and no new errors are introduced.

This is shown in the following differential equation:

where a is the expected total number of faults that exist in the software before testing and b is the failure detection rate or the failure intensity of a fault.

- Theorem:

The mean value function solution of the differential equation 1 is given by

This model is known as the **Goel-Okumoto model**

For Type-I data, the estimate of parameters a and b of the Goel-Okumoto model using the MLE method can be obtained by solving the following equations simultaneously:

Similarly, for Type-II data, the estimate of parameters a and b using the MLE

method can be obtained by solving the following equations:

Let and be the MLE of parameters a and b, respectively. We can then obtain the MLE of the mean value function (MVF) and the reliability function as follows:

It is of interest to determine the variability of the number of failures at time t, N(t). One can approximately obtain the confidence intervals for N(t) based on the Poisson distribution as

where is percentile of the standard normal distribution, i.e., N(0, 1).