The phenomenon of software aging refers to the continuing degradation of software system performance with the operation time and is usually caused by the aging-related bugs such as the memory leak and the accumulation of round-off errors. Software rejuvenation acts as one of the proactive fault management techniques against the software aging. In this paper, we evaluate the interval reliability for two basic stochastic models with periodic software rejuvenation by Garg et al. (1995) and Suzuki et al. (2002a, b). The interval reliability is one of the most generalized dependability measures that involve commonly used reliability function and steady-state availability as the special cases, and is helpful for the system design during a fixed mission period. From the mathematical point of view, the interval reliability for the software rejuvenation models leads to the transient analysis. We focus on the phase expansion approach for solving the transient solutions for the basic software rejuvenation models. The phase expansion is an approximate technique that replaces arbitrary probability distributions by the phase-type (PH) distributions. Benefiting from the phase expansion, we can numerically derive the transient interval reliability for two software rejuvenation models. In numerical examples, we discuss the accuracy of the phase expansion and also reveal quantitative properties of the interval reliability measures.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Asmussen, S., & Koole, G. (1993). Marked point processes as limits of Markovian arrival streams. J. Appl. Prob., 30, 365–372.
Bao, Y., Sun, X., Trivedi, K.S. (2005). A workload-based analysis of software aging, and rejuvenation. IEEE Trans. Rel., 54(3), 541–548.
Barlow, R. E., & Proschan, F. (1965). Mathematical theory of reliability. New York: Wiley.
Bobbio, A., Horváth, A., Telek, M. (2005). Matching three moments with minimal acyclic phase type distributions. Stochastic Models, 21(2-3), 303–326.
Bruneo, D., Distefano, S., Longo, F., Puliafito, A. (2013). Workload-based software rejuvenation in cloud systems. IEEE Trans. Computer, 62(6), 1072–1085.
Dohi, T., Goševa-Popstojanova, K., Trivedi, K.S. (2000a). Analysis of software cost models with rejuvenation. In: Proceedings of the 5th International Symposium on High Assurance Systems Engineering (HASE ’00), pp. 25-34.
Dohi, T., Goševa-Popstojanova, K., Trivedi, K.S. (2000b). Statistical non-parametric algorithms to estimate the optimal software rejuvenation schedule. In: Proceedings of IEEE 2000 pacific rim international symposium on dependable computing (PRDC ’00), pp. 77-84.
Dohi, T., Goševa-Popstojanova, K., Trivedi, K.S. (2001). Estimating software rejuvenation schedule in high assurance systems. Computer J., 47, 473–485.
Dohi, T., & Okamura, H. (2016). Dynamic software availability model with rejuvenation. Journal of The Operations Research Society of Japan, 59(4), 270–290.
Dohi, T., Okamura, H., Osaki, S. (2010). Transient analysis of software availability models with rejuvenation. In: Proceedings of the 4th Asia-pacific international symposium on advanced reliability and maintenance modeling (APARM ’10), pp. 169-176.
Dohi, T., Okamura, H., Trivedi, K.S. (2012). Optimizing software rejuvenation policies under interval reliability criteria. In: Proceedings of the 9th IEEE international conference on autonomic and trusted computing (ATC ’12), pp. 478-485.
Dohi, T., Zheng, J., Okamura, H., Trivedi, K. S. (2018). Optimal periodic software rejuvenation policies based on interval reliability criteria. Rel. Eng. Syst. Saf., 180, 463–475.
Dubner, R., & Abate, J. (1968). Numerical inversion of Laplace transform by relating them to the finite Fourier cosine transform. Journal of the ACM, 15, 115–123.
Durbin, F. (1974). Numerical inversion of Laplace transform: an efficient improvement to Dubner and Abate’s method. Computer J., 17(4), 371–376.
Garg, S., Telek, M., Puliafito, A., Trivedi, K.S. (1995). Analysis of software rejuvenation using Markov regenerative stochastic Petri net. In: Proceedings of the 6th international symposium on software reliability engineering (ISSRE ’95), pp. 24-27.
Hosford, J. E. (1960). Measures of dependability. Operations Research, 8(1), 53–64.
Huang, Y., Kintala, C., Kolettis, N., Fulton, N.D. (1995). Software rejuvenation: analysis, module and applications. In: Proceedings of the 25th international symposium on fault tolerant computing (FTC ’95), pp. 381-390.
Kulkarni, V. G. (1995). Modeling and analysis of stochastic systems. New York: Chapman and Hall.
Machida, F., Kim, D. S., Trivedi, K. S. (2013). Modeling and analysis of software rejuvenation in a server virtualized system with live VM migration. Performance Evaluation, 70(3), 212–230.
Okamura, H., & Dohi, T. (2015). Mapfit: An R-based tool for PH/MAP parameter estimation. In Campos, J., & Haverkort, B.R (Eds.) Proceedings of the 12th international conference on quantitative evaluation of systems (QEST ’15), LNCS 9295 (pp. 105–112): Springer-Verlag.
Okamura, H., & Dohi, T. (2016a). Fitting phase-type distributions and Markovian arrival processes: Algorithms and tools. In Fiondella, L., & Puliafito, A (Eds.) Performance and reliability modeling and evaluation (pp. 49–75): Springer.
Okamura, H., & Dohi, T. (2016b). A phase expansion approach for transient analysis of software rejuvenation model. In: Proceedings of the 8th international workshop on software aging and rejuvenation (WoSAR ’16), pp. 98–103.
Okamura, H., Miyahara, S., Dohi, T., Osaki, S. (2001). Performance evaluation of workload-based software rejuvenation scheme. IEICE Trans. Info. Syst (D), E84-D(10), 1368–1375.
Okamura, H., Dohi, T., Trivedi, K. S. (2011). A refined EM algorithm for PH distributions. Performance Evaluation, 68(10), 938–954.
Okamura, H., Yamamoto, K., Dohi, T. (2014). Transient analysis of software rejuvenation policies in virtualized system: Phase-type expansion approach. Quality Technology and Quantitative Management, 11(3), 335–352.
Osogami, T., & Harchol-Balter, M. (2006). Closed form solutions for mapping general distributions to minimal PH distributions. Performance Evaluation, 63(6), 524–552.
Rezaei, A., & Sharifi, M. (2010). Rejuvenation high available virtualized systems. In: Proceedings of the 5th international conference on availability, reliability and security (ARES ’10), pp. 289–294.
Rubino, G., & Sericola, B. (1992). Interval availability analysis using operational periods. Performance Evaluation, 14, 257–272.
Suzuki, H., Dohi, T., Goševa-Popstojanova, K., Trivedi, K.S. (2002a). Analysis of multistep failure models with periodic software rejuvenation. In Artalejo, J. R., & Krishnamoorthy, A (Eds.) Advanced in stochastic modelling. Notable Publications, Inc. (pp. 85–108).
Suzuki, H., Dohi, T., Okamura, H. (2002b). Cost-effective analysis of software systems with periodic rejuvenation. IEICE Trans. Fundamentals of Electronics. Communications and Computer Sciences (A), E85-A(12), 2923–2932.
Thein, T., & Park, J. (2009). Availability analysis of application server using software rejuvenation and virtualization. J. Computer Science and Technology, 24(2), 339–346.
Trivedi, K.S., Vaidyanathan, K., Goševa-Popstojanova, K. (2000). Modeling and analysis of software aging and rejuvenation. In: Proceedings of the 33rd annual on simulation symposium, pp. 270–279.
Vaidyanathan, K., & Trivedi, K. S. (2005). A comprehensive model for software rejuvenation. IEEE Trans. Dependable and Secure Comput., 2(2), 124–137.
Xie, W., Hong, Y., Trivedi, K. S. (2005). Analysis of a two-level software rejuvenation policy. Rel. Eng. Syst. Saf., 87(1), 13–22.
Zheng, J., Okamura, H., Li, L., Dohi, T. (2017). A comprehensive evaluation of Software rejuvenation policies for transaction systems with Markovian arrivals. IEEE Trans. Rel., 66(4), 1157–1177.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This paper is an extension of work originally reported at The 8th International Workshop on Software Aging and Rejuvenation (WoSAR 2016) (Okamura and Dohi 2016b).
About this article
Cite this article
Zheng, J., Okamura, H. & Dohi, T. A transient interval reliability analysis for software rejuvenation models with phase expansion. Software Qual J (2019). https://doi.org/10.1007/s11219-019-09458-1
- Software rejuvenation
- Transient analysis
- Interval reliability
- Phase expansion
- Markov regenerative process
- Pointwise availability