Statistical Maintenance Time Estimation Based on Stochastic Differential Equation Models in OSS Development Project



Maintenance Effort, Reliability, Stochastic Differential Equation, Earned Value Management, Transition Probability Distribution, Open Source Project


At present, the method of earned value management is often applied to the actual software projects in various IT companies. Also, open source software (OSS) are used under the various situations, because the OSS are useful for many users to make a cost reduction, standardization of systems, and quick delivery. Many OSS are developed under the peculiar development style known as bazaar method. According to the bazaar method, many faults are detected and fixed by developers around the world, and the fixed result will be reflected in the next release. In this paper, we discuss an OSS effort estimation model by using a conventional stochastic differential equation model. Moreover, we propose an optimal maintenance problem based on the proposed effort estimation model. Then, we discuss the optimal maintenance problem minimizing the maintenance effort and satisfying the earned value requirement, simultaneously. In addition, we also propose a method of judging whether the optimal maintenance time is an appropriate time from the viewpoint of the transition probability distribution of the cumulative number of maintenance effort, because proper management of maintenance effort affects software quality. Furthermore, several numerical examples of optimal maintenance time problem with earned value requirement are shown by using the effort data under actual OSS project


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Author Biographies

Hironobu Sone, Tokyo City University

Tokyo City University, 1-28-1 Tamazutsumi, Setagaya-ku, Tokyo, 158-8557, Japan

Yoshinobu Tamura, Tokyo City University

Tottori University, 4-101 Minami, Koyama, Tottori-shi, 680-8552, Japan


Raymond, S. E.: The Cathedral and the Bazzar: Musings on Linux and Open Source by an Accidental Revolutionary”, O’Reilly and Associates, Sebastopol, California, 1999.

Fleming, Q. E., Koppelman, J. M.: Earned Value Project Management (4th Ed.), PMI, Newton Square, U.S.A., 2010.

Yamada, S., Tamura, Y.: OSS Reliability Measurement and Assessment, Springer International Publishing, Switzerland, 2016.

Norris, J.: Mission-critical development with open source software, IEEE Software Magazine, 21(1), pp. 42-49, 2004.

Zhou, Y., Davis, J.: OSS reliability model: an empirical approach, Proceedings of the Fifth Workshop on OSS Engineering, pp. 67-72, 2005.

Sun, C., Lo, D., Wang, X,. Jiang, J., Khoo, S. A.: discriminative model approach for accurate duplicate bug report retrieval, Proceedings of the 32nd ACM/IEEE International Conference on Software Engineering (ICSE’10), Cape Town. South Africa. 2-8 May, pp.45-54, 2010.

Hooimeijer, P., Weimer, W.: Modeling bug report quality, Proceedings of the Twenty-Second IEEE/ACM International Conference on Automated Software Engineering(ASE ’07), Georgia. USA. 5-9 November, pp. 34-43, 2010.

Nurolahzade, M., Nasehi, M. S., Khandkar, H. S., Rawal, S.: The role of patch review in software evolution: an analysis of the mozilla firefox, Proceedings of the joint international and annual ERCIM workshops on Principles of software evolution (IWPSE) and software evolution (Evol) workshops(IWPSE-Evol ’09), Amsterdam. The Netherlands. 24-25 August, pp.9-18, 2009.

Tamura, Y., Sone, H., Yamada, S.: OSS Project Stability Assessment Support Tool Considering EVM Based on Wiener Process Models, Applied System Innovation, 2(1), pp.1-12, 2019.

Robles, G., Gonzälez-Barahona, M. J., Cervigön, C., Capiluppi, A., Izquierdo-Cortäzar, D.: Estimating development effort in Free/OSS projects by mining software repositories: a case study of OpenStack, Proceedings of the 11th Working Conference on Mining Software Repositories, Hyderabad. India. 31 May-1 June, pp. 222-231, 2014.

Mishra, R., Sureka, A.: Mining Peer Code Review System for Computing Effort and Contribution Metrics for Patch Reviewers, Proceedings of the 2014 IEEE 4th Workshop on Mining Unstructured Data, Victoria. Canada. 30-30 September, pp. 11-15, 2014.

Rakha, S. M., Shang, W., Hassan, E. A.: Studying the needed effort for identifying duplicate issues, Empirical Software Engineering, Springer Science+Business Media: Berlin, Germany, 2015.

Giger, E., Pinzger, M., Gall, H.: Predicting the fix time of bugs, Proceedings of the 2nd International Workshop on Recommendation Systems for Software Engineering, Cape Town. South Africa. 4-4 May, pp.52-56, 2010.

Bougie, G., Treude, C., German, M. D., Storey, A. M.: A comparative exploration of freeBSD bug lifetimes, Proceedings of the 7th IEEE Working Conference on Mining Software Repositories (MSR 2010), Cape Town. South Africa. 2-3 May, pp.106-109, 2010.

Akbarinasaji, S., Caglayan, B., Bener, A.: Predicting Bug-Fixing Time: A Replication Study Using An OSS Project, Systems and Software, 136, pp.173-186, 2018.

Marks, L., Zou, Y., Hassan, E. A.: Studying the fix-time for bugs in large open source projects, Proceedings of the 7th International Conference on Predictive Models in Software Engineering (Promise ’11), Alberta. Canada. 20-21 September, pp.11:1–11:8, 2011.

Yamada, S., Ohba, M., Osaki, S.: S-Shaped Reliability Growth Modeling for Software Error Detection, IEEE Transactions on Reliability, R-32(5), pp.475-484, 1983.

Yamada, S.: Software Reliability Modeling: Fundamentals and Applications, Springer-Verlag; Tokyo/Heidelberg, 2014.

Lyu, M.R. Ed.: Handbook of Software Reliability Engineering. IEEE Computer Society Press; Los Alamitos.; CA, U.S.A., 1996.

Musa, J.D., Iannino, A., Okumoto, K.: Software Reliability: Measurement, Prediction, Application. McGraw-Hill; New York, 1987.

Kapur, P.K., Pham, H., Gupta, A., Jha, P. C.: Software Reliability Assessment with OR Applications, Springer-Verlag; London, 2011.

Wong, E.: Stochastic Processes in Information and Systems. McGraw-Hill; New York, 1971.

Arnold L.: Stochastic Differential Equations-Theory and Applications. John Wiley & Sons; New York, 1971.

Yamada, S., Kimura, M., Tanaka, H., Osaki, S.: Software reliability measurement and assessment with stochastic differential equations, IEICE Transactions on Fundamentals, E77-A(1), 109-116, 1994.

The OpenStack Foundation, The OpenStack project,



How to Cite

Sone, H., Tamura, Y., & Yamada, S. (2019). Statistical Maintenance Time Estimation Based on Stochastic Differential Equation Models in OSS Development Project. Computer Reviews Journal, 5, 126-140. Retrieved from



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