#### Date of Award

Summer 1989

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### Program/Concentration

Statistics

#### Committee Director

Ram C. Dahiya

#### Committee Member

Larry Lee

#### Committee Member

Edward Markowski

#### Committee Member

N. Rao Chaganty

#### Abstract

The problem considered here is the building of Non-homogeneous Poisson Process (NHPP) model. Currently existing popular NHPP process models like Goel-Okumoto (G-O) and Yamada *et al* models suffer from the drawback that the probability density function of the inter-failure times is an improper density function. This is because the event no failure in (0, oo] is allowed in these models. In real life situations we cannot draw sample(s) from such a population and also none of the moments of inter-failure times exist. Therefore, these models are unsuitable for modelling real software error data. On the other hand if the density function of the inter-failure times is made proper by multiplying with a constant, then we cannot assume finite number of expected faults in the system which is the basic assumption in building the software reliability models.

Taking these factors into consideration, we have introduced an extra parameter, say c, in both the G -0 and Yamada *et al* models in order to get a new model. We find that a specific value of this new parameter gives rise to a proper density for inter-failure times. The G -0 and Yamada *et al* models are special cases of these models corresponding to c = 0. This raises the question - “Can we do better than existing G -0 and Yamada *et al* models when 0 < c < 1 ?”. The answer is ‘yes’.

With this objective, the behavior of the software failure counting process { N ( t ) , t > 0} has been studied. Several measures, such as the number of failures by some prespecified time, the number of errors remaining in the system at a future time, distribution of remaining number of faults in the system and reliability during a mission have been proposed in this research. Maximum likelihood estimation method was used to estimate the parameters. Sufficient conditions for the existence of roots of the ML equations were derived. Some of the important statistical aspects of G -0 and Yamada *et al* models, like conditions for the existence and uniqueness of the ML equations, were not worked out so far in the literature. We have derived these conditions and proved uniqueness of the roots for these models. Finally four different sets of actual failure time data were analyzed. ii

#### DOI

10.25777/40ft-9g76

#### Recommended Citation

Hossain, Syed A..
"Software Reliability Models"
(1989). Doctor of Philosophy (PhD), dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/40ft-9g76

https://digitalcommons.odu.edu/mathstat_etds/79