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
","abstract_html":"<p>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 <em>et al</em> 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.</p> <p>Taking these factors into consideration, we have introduced an extra parameter, say c, in both the G -0 and Yamada <em>et al</em> 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 <em>et al</em> 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 <em>et al</em> models when 0 < c < 1 ?”. The answer is ‘yes’.</p> <p>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 <em>et al</em> 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</p>","abstract_has_math":false,"creators":["Hossain, Syed Afzal"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics & Statistics","degree_department":null,"school":null,"contributors":["Ram C. Dahiya","Larry Lee","Edward Markowski","N. Rao Chaganty"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1989,"date_issued":"1989-07-01T07:00:00Z","date_published":"1989-07-01T07:00:00Z","updated_at":"2026-07-24T03:35:15Z","subjects":["Non-homogeneous poisson process (NHPP)","Model","Software reliability","Applied Statistics","Computer Sciences"],"languages":[],"rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
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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
"]},{"key":"dc:title","label":"Title","values":["Software Reliability Models"]}]}],"canonical_facts":{"dc:contributor":["Ram C. Dahiya","Larry Lee","Edward Markowski","N. Rao Chaganty"],"dc:creator":["Hossain, Syed Afzal"],"dc:date.available":["2019-10-02T07:00:00Z"],"dc:description.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
"],"dc:identifier":["https://digitalcommons.odu.edu/mathstat_etds/79"],"dc:rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
"],"dc:subject":["Non-homogeneous poisson process (NHPP)","Model","Software reliability","Applied Statistics","Computer Sciences"],"dc:title":["Software Reliability Models"],"thesis:degree_discipline":["Mathematics & Statistics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:35:15Z"}