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Statistical properties of a system reliability estimator using the Littlewood software reliability model

Part of: Survival analysis and censored data Stochastic processes Parametric inference

Published online by Cambridge University Press:  14 July 2016

Marcus A. Agustin
Marcus A. Agustin*
Affiliation:
Southern Illinois University, Edwardsville
*
∗Postal address: Department of Mathematics and Statistics, Southern Illinois University, Edwardsville, IL 62026, USA. Email address: magusti@siue.edu
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Abstract

This paper considers a competing risks system with p pieces of software where each piece follows the model by Littlewood (1980) described as follows. The failure rate of a piece of software relies on the residual number of bugs remaining in the software where each bug produces failures at varying rates. In effect, bugs with higher failure rates tend to be observed earlier in the testing period. Tasks are assigned to the system and the task completion times as well as the software failure times are assumed to be independent of each other. The system is observed over a fixed testing period and the system reliability upon test termination is examined. An estimator of the system reliability is presented and its asymptotic properties as well as finite-sample properties are obtained.

Keywords

Counting processseries systemsoftware reliability

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 60G55: Point processes 62F12: Asymptotic properties of estimators
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 3 , September 2003 , pp. 766 - 778
Copyright
Copyright © Applied Probability Trust 2003 

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References

Agustin, M. (1999). Asymptotic and finite-sample properties of a reliability estimator for a competing risks system. J. Statist. Comput. Simul. 64, 221–234.Google Scholar
Agustin, M. and Peña, E. (1999). A dynamic competing risks model. Prob. Eng. Inf. Sci. 13, 333–358.CrossRefGoogle Scholar
Andersen, P., Borgan, Ø., Gill, R., and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.Google Scholar
Barendregt, L., and van Pul, M. (1995). On the estimation of the parameters for the Littlewood model in software reliability. Statistica Neerlandica 49, 165–184.Google Scholar
Billingsley, P. (1961). Statistical Inference for Markov Processes. University of Chicago Press.Google Scholar
Crowder, M., Kimber, A., Smith, R., and Sweeting, T. (1991). Statistical Analysis of Reliability Data. Chapman and Hall, London.Google Scholar
Fakhre-Zakeri, I., and Slud, E. (1995). Mixture models for software reliability with imperfect debugging: identifiability of parameters. IEEE Trans. Reliab. 44, 104–113.CrossRefGoogle Scholar
Ferdous, J., Uddin, M., and Pandey, M. (1995). Reliability estimation with Weibull inter failure times. Reliab. Eng. System Safety 50, 285–296.CrossRefGoogle Scholar
Goel, A., and Okumoto, K. (1979). Time-dependent error-detection rate model for software reliability and other performance measures. IEEE Trans. Reliab. 28, 206–211.CrossRefGoogle Scholar
Jacod, J. (1975). Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.CrossRefGoogle Scholar
Jelinski, Z., and Moranda, P. (1972). Software reliability research. In Statistical Computer Performance Evaluation, ed. Freiberger, W., Academic Press, New York, pp. 465–484.Google Scholar
Kurtz, T. (1983). Gaussian approximations for Markov chains and counting processes. Bull. Internat. Statist. Inst. 50, 361–375.Google Scholar
Kvam, P., and Singh, H. (1998). Estimating reliability of components with increasing failure rate using series system data. Naval Res. Logistics 45, 115–123.Google Scholar
Littlewood, B. (1980). Theories of software reliability: how good are they and how can they be improved? IEEE Trans. Software Eng. 6, 489–500.CrossRefGoogle Scholar
Mazzuchi, T., and Singpurwalla, N. (1988). Software reliability models. In Handbook of Statistics, Vol. 7, Quality Control and Reliability, eds Krishnaiah, P. R. and Rao, C. R., North-Holland, Amsterdam, pp. 73–98.Google Scholar
Moek, G. (1984). Comparison of some software reliability models for simulated and real failure data. Internat. J. Model. Simul. 4, 29–41.Google Scholar
Mok, A., and Chen, D. (1997). A multiframe model for real-time tasks. IEEE Trans. Software Eng. 23, 635–645.CrossRefGoogle Scholar
Serfling, R. (1980). Approximation Theorems of Mathematical Statistics. John Wiley, New York.CrossRefGoogle Scholar
Singpurwalla, N., and Wilson, S. (1999). Statistical Methods in Software Engineering: Reliability and Risk. Springer, New York.CrossRefGoogle Scholar
Slud, E. (1997). Testing for imperfect debugging in software reliability. Scand. J. Statist. 24, 555–572.CrossRefGoogle Scholar
Sun, J., Gardner, M., and Liu, J. (1997). Bounding completion times of jobs with arbitrary release times, variable execution times, and resource sharing. IEEE Trans. Software Eng. 23, 603–615.Google Scholar
Van Pul, M. (1992). Asymptotic properties of a class of statistical models in software reliability. Scand. J. Statist. 19, 1–23.Google Scholar
Van Pul, M. (1993). Statistical Analysis of Software Reliability Models. CWI, Amsterdam.Google Scholar