在軟體開發的過程中，移除軟體錯誤是一項高成本且耗時的任務。因此，對專案管理者而言，除錯過程的評估與預測很重要。在過去，專家學者使用軟體可靠度成長模型來描述軟體錯誤的移除過程。他們假設錯誤偵測是非齊次卜瓦松過程，並且為了簡化模型，大部分的研究忽略錯誤偵測到移除之間的時間延遲。近年來，許多學者意識到除錯所需的時間不應該被忽略，因此將排隊理論應用至軟體除錯的研究。除錯人員需要時間來尋找軟體出錯的原因，並移除錯誤的根源。在考慮除錯時間的情況下，許多學者提出無限伺服器排隊模型來評估、分析軟體專案。有一部分的研究甚至考慮除錯人員的數目限制，進而提出有限伺服器排隊模型。然而，過去的這些研究僅關注在除錯的時間延遲與除錯人員的數量。關於軟體錯誤分類的研究很少。若能將軟體錯誤進行分類，則錯誤將能交派於特定的除錯人員並得到更適當的處理。本篇論文的目的是應用快捷排隊理論方法來降低軟體錯誤的平均修復時間，並協助管理軟體專案，如人力配置、成本控制、資源使用率等議題。軟體除錯的流程可以套用排隊理論，而偵測到的軟體錯誤會進行分類並分配至快捷佇列或是一般佇列。實驗結果顯示，使用此方法可以提升軟體除錯系統的效能。此外，我們開發的模擬程序不僅驗證快捷排隊模型的結果，也提供專案管理者一個實用的工具以仔細監控專案變化。

Fault correction is costly and time-consuming in the software development process [1-5]. Thus the measurement and prediction of the fault correction process is important for the software project manager. In the past, researchers used software reliability growth models to describe the fault correction process. They assumed that fault detection is a non-homogeneous Poisson process, and many studies have ignored the time lag between fault detection and fault correction to simplify the models. Recently, the queueing theory has been introduced to the software debugging process, since many researchers are aware that the fault correction time should not be ignored. The engineers need time to identify the reasons for the failures and remove the root causes. Considering the correction time, several infinite server queueing models are proposed to measure and analyze the software project. A small number of research has even considered the limited number of debuggers and has presented finite server queueing models. However, the previous work focused only on the debugging time lag and the number of servers. There is little research about fault classification. If the faults would be classified, the fault can be assigned to a specific engineer to get better service. The aim of this study is to apply the express queueing theoretic approach for reducing the mean resolution time of faults and to support the management of the software project, such as staffing, cost, utilization, etc. The debugging process is modeled by the queueing theory, and the detected faults will be classified and dispatched into either express queue or regular queue. The experiments show that the performance of the debugging system improved while using the proposed approach. Besides, the simulation procedures we developed not only verify the results of the express queueing model, but also provide a useful tool for project managers to monitor the change of the system in more detail.

Abstract in Chinese I
Abstract II
Acknowledgement III
Contents IV
List of Tables VI
List of Figures VII
Notation IX
Chapter 1 Introduction 1
Chapter 2 Express Queueing Modeling 6
2.1 Defect resolution process 6
2.2 Queueing theory 7
2.3 The proposed model 10
Chapter 3 Numerical Examples 15
3.1 Data sets 15
3.2 DS1 16
3.3 DS2 26
Chapter 4 Simulation Procedures with the Queueing Theoretic Approach 30
4.1 Data generating simulation procedure 31
4.2 Traditional queue debugging simulation procedure 34
4.3 Express queue debugging simulation procedure 38
4.4 Simulation results and performance analysis 42
4.5 Priority debugging simulation procedure 46
4.6 Validity Considerations 51
Chapter 5 Conclusions and Future Work 53
Appendixes 55
Appendix A. Original data set of DS1 55
Appendix B. Original data set of DS2 56
References 57

[1] R. S. Pressman, Software Engineering: A Practitioner’s Approach, McGrow-Hill International Edition, 2005.
[2] J. Martin and C. McClure, Software maintenance, Prentice-Hall, 1983.
[3] T. M. Pigoski, Practical Software Maintenance: Best Practices for Managing Your Software Investment, Wiley: New York NY, 1997.
[4] V. Basili, et al., “Understanding and Predicting the Process of Software Maintenance Releases,” Proceedings of the 18th International Conference on Software Engineering, Vol. 464, No. 474, pp. 25-29, March 1996.
[5] F. Calzolari, P. Tonella, and G. Antoniol, “Dynamic Model for Maintenance and Testing Effort,” Proceedings of the International Conference on Software Maintenance, IEEE, pp. 104-112, 1998.
[6] S. S. Gokhale and R. E. Mullen, “Queuing models for field defect resolution process,” Proceedings of ISSRE’06 17th International Symposium on Software Reliability Engineering, IEEE, pp. 353-362, November 2006.
[7] G. Antoniol, A. Cimitile, G. A. Di Lucca, and M. Di Penta, “Assessing staffing needs for a software maintenance project through queuing simulation,” IEEE Transactions on Software Engineering, Vol. 30, No. 1, pp. 43-58, 2004.
[8] H. Zhang, B. Kitchenham, and D. Pfahl, “Reflections on 10 years of software process simulation modeling: a systematic review,” Proceedings of the Software process, 2008 international conference on Making globally distributed software development a success story (ICSP'08), Springer-Verlag, Berlin, Heidelberg, pp. 345-356, 2008.
[9] M. I. Kellner, R. J. Madachy, and D. M. Raffo, “Software process simulation modeling: why? what? how?,” Journal of Systems and Software, Vol. 46, No. 2, pp. 91-105, 1999.
[10] G. Antoniol, G. Casazza, G. A. Di Lucca, M. Di Penta, and F. Rago, “A queue theory-based approach to staff software maintenance centers,” Proceedings of the IEEE International Conference on Software Maintenance (ICSM'01), pp.510, 2001.
[11] J. Jacod, “Two Dependent Poisson Processes Whose Sum Is Still a Poisson Process,” Journal of Applied Probability, Vol. 12, No. 1, pp. 170-172, March 1975.
[12] http://www.defectmanagement.com/defectmanagement/modelmain.htm (Accessed: 9 September 2014)
[13] W. Whitt, “Partitioning customers into service groups” Management Science, Vol. 45, No.11, pp. 1579-1592, 1999.
[14] H. Sakasegawa, “An approximation formula L q≃ α• ρ β/(1-ρ),” Annals of the Institute of Statistical Mathematics, Vol. 29, No. 1, pp. 67-75, 1977.
[15] J. D. Little, “A proof for the queuing formula: L= λ W,” Operations research, Vol. 9, No. 3, pp. 383-387, 1961.
[16] J. D. Musa, A. Iannino, and K. Okumoto, Software Reliability, Measurement, Prediction, and Application, McGraw-Hill, 1987.
[17] K. Z. Yang, ”An Infinite Server Queueing Model for software Readiness Assessment and Related Performance Measure,” Ph.D. Dissertation, Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY, 1996.
[18] D. Gross and C. Haris, The Fundamentals of queueing theory, John Wiley and Sons, 1988.
[19] Å. Björck, Numerical Methods for Least Squares Problems, Society for Industrial and Applied Mathematics (SIAM), 1996.
[20] M. Ohba, “Software reliability analysis models,” IBM Journal of Research and Development, Vol. 28, No 4, pp. 428-443, July 1984.
[21] S. S. Yau and T. J. Tsai, “A Survey of Software Design Techniques,” IEEE Transactions on Software Engineering, Vol. 12, No. 6, pp. 713-721, June 1986.
[22] V. Gibson and J. Senn, “System Structure and Software Maintenance Performance,” Communications of the ACM, Vol. 27, No. 3, pp. 347-358, March 1989.
[23] G. Papapanagiotakis and P. Breuer, “A software maintenance management model based on queueing networks,” Journal of Software Maintenance: Research and Practice, Vol. 6, pp. 73–97, 1994.
[24] R. Ramaswamy, “How to Staff Business Critical Maintenance Projects,” IEEE software, Vol. 17, No. 3, pp. 90-94, 2000.
[25] N. Kalló and T. Koltai, “A review of management issues related to express line systems,” Social and Management Sciences, Vol. 16, No. 1, pp. 21-32, 2009.
[26] D. M. Raffo and M. I. Kellner, “Empirical Analysis in Software Process Simulation Modeling,” Journal of Systems and Software, Vol. 53, No. 1, pp. 31-41, 2000.
[27] T. K. Abdel-Hamid, “The dynamics of software project staffing: a system dynamics based simulation approach,” IEEE Transactions on Software Engineering, Vol. 15, No. 2, pp.109-119, February 1989.
[28] M. Höst, B. Regnell, J. Nedstam, and C. Nyberg, “Exploring bottlenecks in market-driven requirements management processes with discrete event simulation,” Journal of Systems and Software, Vol. 59, No. 3, pp. 323-332, December 2001.
[29] S. S. Gokhale and M. R. Lyu, “A Simulation Approach to Structure-Based Software Reliability Analysis,” IEEE Transactions on Software Engineering, Vol. 31, No. 8, pp. 643-656, 2005.
[30] S. S. Gokhale, M. R. Lyu, K. S. Trivedi, “Incorporating fault debugging activities into software reliability models: a simulation approach,” IEEE Transactions on Reliability, Vol. 55, No. 2, pp. 281-292, 2006.
[31] A. L. Goel and K. Okumoto, “Time-dependent Error-detection Rate Model for Software Reliability and Other Performance Measures,” IEEE Transactions on Reliability, Vol. 28, No. 3, pp. 206-211, August 1979.
[32] S. Yamada, M. Ohba, and S. Osaki, “S-shaped reliability growth modeling for software error detection,” IEEE Transactions on Reliability, Vol. 32, No. 5, pp. 475-478, 1983.
[33] K. S. Trivedi, Probability & statistics with reliability, queuing and computer science applications, John Wiley & Sons, 2008.
[34] M. Schimmel, “Deployment of express checkout lines at supermarkets,” Research paper Business Analytics, April 2013.
[35] J. D. Musa, Software Reliability Data, Cyber Security and Information Systems Information Analysis Center, January 1980.
[36] L. Kleinrock, Queuing systems, Wiley, 1975.
[37] D. S. Johnson, “A Theoretician's Guide to the Experimental Analysis of Algorithms”, Data Structures, Near Neighbor Searches, and Methodology: Proceedings of 5th & 6th DIMACS Implementation Challenges, American Mathematical Society, Providence, pp. 215-250, 2002.
[38] D. G. Kendall, “Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain,” The Annals of Mathematical Statistics, pp. 338-354, 1953.
[39] S. Yamada and S. Osaki, “Software reliability growth modeling : Models and applications,” IEEE Transactions on Software Engineering, Vol. 11, No.12, pp. 1431-1437, December 1985.
[40] C. Y. Huang, and C. T. Lin, “Software reliability analysis by considering fault dependency and debugging time lag,” IEEE Transactions on Reliability, Vol. 55, No.3, pp.436-450, September 2006.
[41] N. F. Schneidewind, “Fault correction profiles,” Proceeding of ISSRE 2003. 14th International Symposium on Software Reliability Engineering, IEEE, pp. 257-267, November 2003.
[42] S. Inoue and S. Yamada, “A Software Reliability Growth Modeling Based on Infinite Server Queueing Theory,” Proceeding of the 9th ISSAT International Conference on Reliability and Quality in Design, pp.305-309, Honolulu, Hawaii, USA, August 2003.
[43] T. Dohi, T. Matsuoka, and S. Osaki, “An Infinite Server Queueing Model for Assessment of the Software Reliability,” Electronics and Communications in Japan (Part III: Fundamental electronic Science), Vol. 85, No.3, pp. 43-51, 2002.
[44] T. Dohi, S. Osaki, and KS. Trivedi, “An Infinite Server Queueing Approach for Describing Software Reliability Growth: Unified Modeling and Estimation Framework,” Proceedings of the 11th Asia-Pacific Software engineering Conference (APSEC 2004) , pp. 110-119, Busan, Korea, Dec. 2004.
[45] W.C. Huang, C.Y. Huang, and C.C. Sue, “Software Reliability Prediction and Assessment Using both Finite and Infinite Server Queueing Approaches,” Proceedings of the 12th IEEE International Symposium on Pacific Rim Dependable Computing (PRDC 2006) , pp. 194-201, Riverside, CA, USE, Dec. 2006.