Alternative Parameter Estimation Methods for the Compound Poisson Software Reliability Model with Clustered Failure Data

摘要

The ‘compound Poisson’ (CP) software reliability model was proposed previously by the first named author for time-between-failure data in terms of CPU seconds, using the ‘maximum likelihood estimation’ (MLE) method to estimate unknown parameters; hence, CPMLE. However, another parameter estimation technique is proposed under ‘nonlinear regression analysis’ (NLR) for the compound Poisson reliability model, giving rise to the name CPNLR. It is observed that the CP model, with different parameter estimation methods, produces equally satisfactory or more favourable results as compared to the Musa–Okumoto (M–O) model, particularly in the event of grouped or clustered (clumped) software failure data. The sampling unit may be a week, day or month within which the failures are clumped, as the error recording facilities dictate in a software testing environment. The proposed CPNLR and CPMLE yield comparatively more favourable results for certain software failure data structures where the frequency distribution of the cluster (clump) size of the software failures per week displays a negative exponential behaviour. Average relative error (ARE), mean squared error (MSE) and average Kolmogorov–Smirnov (K–S Av.Dn) statistics are used as measures of forecast quality for the proposed and competing parameter-estimation techniques in predicting the number of remaining future failures expected to occur until a target stopping time. Comparisons on five different simulated data sets that contain weekly recorded software failures are made to emphasize the advantages and disadvantages of the competing methods by means of the chronological prediction plots around the true target value and zero per cent relative error line. The proposed generalized compound Poisson (MLE and NLR) methods consistently produce more favourable predictions for those software failure data with negative exponential frequency distribution of the failure clump size versus number of weeks. Otherwise, the popularly used competing M–O log-Poisson model is a better fit for those data with a uniform clump size distribution to recognize the log-Poisson effect while the logarithm of the Poisson equation is a constant, hence uniform. The software analyst is urged to perform exploratory data analysis to recognize the nature of the software failure data before favouring a particular reliability estimation method. © 1997 by John Wiley & Sons, Ltd.