A New Mathematical Logistic Model and Its Applications

10.6186/IJIMS.2014.25.2.1

Hoang Pham；David H. Pham；Hoang Pham Jr.

Logistic reliability growth ； predictive-ratio risk ； loglog distribution ； normalized criteria distance

International Journal of Information and Management Sciences

25卷2期（2014 / 07 / 01）

79 - 99

英文

We present a new 4-parameter logistic growth model where the rate of change of quantity function is directly proportional to its remaining quantity for growth by a time-dependent logistic function per quantity per unit time. The model can be used to determine the expected number of quantities at time t. Several real world applications are discussed to illustrate the usefulness of the new model including the earthquake occurrence events, the student population growth, and the software modeling. Examples are included to illustrate the goodness-of-fit of the proposed model and existing logistic growth models based on real data sets collected from software applications, earthquake events in the US, and a high school senior class. Three goodness-of-fit test criteria and a recent normalized criteria distance method are used to illustrate the model comparisons. The results show that the proposed model fit significantly better than other existing growth models.

1. Verhulst, P. (1845). Recherches mathé matiques sur la loi d'accroissement de la population, Nouv. mém. de l'Academie Royale des Sci. et Belles-Lettres de Bruxelles, Vol.18, 1-41
2. http://latino.foxnews.com/latino/news/2014/04/17/how-safe-is-california-earthquakes-in-chile-and-nicaragua-raise-questions-about/
3. Goel, A. L.,Okumoto, K.(1979).Time-dependent fault-detection rate model for software and other performance measures.IEEE Trans. on Reliability,28,206-211.
4. Kapur, P. K.,Pham, H.,Aggarwal, A. G.,Kaur, G.(2012).Two dimensional multi-release software reliability modeling and optimal release planning.IEEE Transactions on Reliability,61,758-768.
5. Kapur, P. K.,Pham, H.,Anand, S.,Yadav, K.(2011).A unified approach for developing software reliability growth models in the presence of imperfect debugging and error generation.IEEE Transactions on Reliability,60,331-340.
6. Keilman, N.(2001).Uncertain population forecasts.Nature,412,490-491.
7. Ohba, M.(1984).Inflexion S-shaped software reliability growth models.Stochastic Models in Reliability Theory
8. Ohba, M.,Yamada, S.(1984).S-shaped software reliability growth models.Proc. 4th Int. Conf. Reliability and Maintainability
9. Persona, A.,Pham, H.,Sgarbossa, F.(2010).Age replacement policy in random environment using systemability.International Journal of Systems Science,41,1383-1397.
10. Pham, H.(2006).System Software Reliability.Springer.
11. Pham, H.(2013).A software reliability model with vtub-shaped fault-detection rate subject to operating environments.Proc. 19th ISSAT Int'l Conf. on Reliability and Quality in Design,Hawaii:
12. Pham, H.(2014).A new software reliability model with Vtub-shaped fault-detection rate and the uncertainty of operating environments.Optimization,63
13. Pham, H.(1996).A software cost model with imperfect debugging, random life cycle and penalty cost.International Journal of Systems Science,27,455-463.
14. Pham, H.(2007).An imperfect-debugging fault-detection dependent-parameter software.International Journal of Automation and Computing,4,325-328.
15. Pham, H.(2011).Modeling U.S. mortality and risk-cost optimization on life expectancy.IEEE Transactions on Reliability,60,125-133.
16. Pham, H.(1993).,Idaho National Engineering Laboratory.
17. Pham, H.,Deng, C.(2003).Predictive-ratio risk criterion for selecting software reliability models.Proc. 9th ISSAT Int'l Conf. on Reliability and Quality in Design
18. Pham, H.,Lai, C. D.(2007).On recent generalization of the Weibull distribution.IEEE Transactions on Reliability,56,454-458.
19. Pham, H.,Nordmann, L.,Zhang, X.(1999).A general imperfect software debugging model with s-shaped fault detection rate.IEEE Transactions on Reliability,48,169-175.
20. Pham, H.,Zhang, X.(1997).An NHPP software reliability model and its comparison.International Journal of Reliability, Quality and Safety Engineering,4,269-282.
21. Pham, H.,Zhang, X.(2003).NHPP software reliability and cost models with testing coverage.European Journal of Operational Research,145,443-454.
22. Pham, L.,Pham, H.(2000).Software reliability models with time-dependent hazard function based on Bayesian approach.IEEE Transactions on Systems, Man, and Cybernetics - Part A,30,25-35.
23. Sgarbossa, F.,Pham, H.(2010).A cost analysis of systems subject to random field environments and reliability.IEEE Transactions on Systems, Man, and Cybernetics - Part C,40,429-437.
24. Teng, X.,Pham, H.(2006).A new methodology for predicting software reliability in the random field environments.IEEE Transactions on Reliability,55,458-468.
25. Teng, X.,Pham, H.(2004).A software cost model for quantifying the gain with considerations of random field environments.IEEE Transactions on Computers,53,380-384.
26. Weibull, W.(1951).A statistical distribution function of wide applicability.Journal Applied Mechanics,18,293-297.
27. Wilson, D. L.(1994).The analysis of survival (mortality) data: Fitting Gompertz, Weibull and logistic functions.Mechanisms of Ageing and Development,74,15-33.
28. Xiao, X.,Dohi, T.(2011).Wavelet shrinkage estimation for non-homogeneous Poisson process based software reliability models.IEEE Transactions on Reliability,60,211-225.
29. Yamada, S.,Ohba, M.,Osaki, S.(1983).S-shaped reliability growth modeling for software fault detection.IEEE Trans. on Reliability,12,475-484.
30. Yamada, S.,Osaki, S.(1985).Software reliability growth modeling: models and applications.IEEE Transactions on Software Engineering,11,1431-1437.
31. Yamada, S.,Tokuno, K.,Osaki, S.(1992).Imperfect debugging models with fault introduction rate for software reliability assessment.International Journal of Systems Science,23,2241-2252.
32. Zhang, X.,Pham, H.(2006).Software field failure rate prediction before software deployment.Journal of Systems and Software,79,291-300.
33. Zhang, X.,Teng, X.,Pham, H.(2003).Considering fault removal efficiency in software reliability assessment.IEEE Transactions on Systems, Man, and Cybernetics - Part A,33,114-120.