Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-06T02:31:47.674Z Has data issue: false hasContentIssue false

13 - Non-Homogeneous Continuous-Time Markov Chains

from Part IV - State-Space Models with Non-Exponential Distributions

Published online by Cambridge University Press:  30 August 2017

Kishor S. Trivedi
Affiliation:
Duke University, North Carolina
Andrea Bobbio
Affiliation:
Università degli Studi del Piemonte Orientale, Italy
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Reliability and Availability Engineering
Modeling, Analysis, and Applications
, pp. 489 - 508
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] K., Trivedi, Probability and Statistics with Reliability, Queueing and Computer Science Applications, 2nd edn. John Wiley & Sons, 2001.
[2] A., Rindos, S., Woolet, I., Viniotis, and K., Trivedi, “Exact methods for the transient analysis of nonhomogeneous continuous time Markov chains,” in Computations with Markov Chains. Springer, 1995, pp. 121–133.
[3] T. E., Fortmann and K. L., Hitz, An Introduction to Linear Control Systems (Control and System Theory). CRC Press, 1977.
[4] W., Feller, An Introduction to Probability Theory and its Applications, Vols. I and II. John Wiley & Sons, 1968.
[5] G., Cosulich, P., Firpo, and S., Savio, “Power electronics reliability impact on service dependability for railway systems: A real case study,” in Proc. IEEE Int. Symp. on Industrial Electronics, ISIE '96, vol. 2, Jun. 1996, pp. 996–1001.Google Scholar
[6] F., Frattini, A., Bovenzi, J., Alonso, and K. S., Trivedi, “Reliability indices,” in Wiley Encyclopedia of Operations Research and Management Science. JohnWiley & Sons, 2013, pp. 1–21.Google Scholar
[7] R., Sahner, K., Trivedi, and A., Puliafito, Performance and Reliability Analysis of Computer Systems: An Example-based Approach Using the SHARPE Software Package. Kluwer Academic Publishers, 1996.
[8] Y., Bao, X., Sun, and K. S., Trivedi, “A workload-based analysis of software aging, and rejuvenation,IEEE Transactions on Reliability, vol. 54, no. 3, pp. 541–548, 2005.Google Scholar
[9] Y., Huang, C. M. R., Kintala, N., Kolettis, and N. D., Fulton, “Software rejuvenation: Analysis, module and applications,” in Proc. Int. Symp. on Fault-Tolerant Computing (FTCS), 1995, pp. 381–390.Google Scholar
[10] S., Garg, A., Puliafito, M., Telek, and T., Trivedi, “Analysis of preventive maintenance in transactions based software systems,IEEE Transactions on Computers, vol. 47, no. 1, pp. 96–107, Jan. 1998.Google Scholar
[11] J. T., Duane, “Learning curve approach to reliability monitoring,IEEE Transactions on Aerospace, vol. 2, no. 2, pp. 563–566, 1964.Google Scholar
[12] E., Parzen, Stochastic Processes. Holden Day, 1962.
[13] S., Ramani, S. S., Gokhale, and K. S., Trivedi, “SREPT: Software Reliability Estimation and Prediction Tool,Performance Evaluation, vol. 39, no. 1–4, pp. 37–60, 2000.Google Scholar
[14] A., Goel and K., Okumoto, “Time-dependent error-detection rate model for software reliability and other performance measures,IEEE Transactions on Reliability, vol. R-28, no. 3, pp. 206–211, Aug. 1979.Google Scholar
[15] A., Goel, “Software reliability models: Assumptions, limitations, and applicability,” IEEE Transactions on Software Engineering, vol. SE-11, no. 12, pp. 1411–1423, Dec. 1985.Google Scholar
[16] S., Yamada, M., Ohba, and S., Osaki, “S-shaped reliability growth modeling for software error detection,IEEE Transactions on Reliability, vol. R-32, no. 5, pp. 475–484, Dec. 1983.Google Scholar
[17] S., Gokhale, M., Lyu, and K., Trivedi, “Analysis of software fault removal policies using a non-homogeneous continuous timeMarkov chain,Software Quality Journal, vol. 12, no. 3, pp. 211–230, 2004.Google Scholar
[18] J. D., Musa and K., Okumoto, “A logarithmic Poisson execution time model for software reliability measurement,” in Proc. 7th Int. Conf. on Software Engineering. IEEE Press, 1984, pp. 230–238.
[19] J., Musa, Software Reliability Engineering: More Reliable Software Faster and Cheaper, 2nd edn. Print on demand from http://johnmusa.com/book.htm, 2004.
[20] M. M., Grottke and K. S., Trivedi, “On a method for mending time to failure distributions,” in Proc. Int. Conf. on Dependable Systems and Networks, DSN 2005, Jun. 2005, pp. 560–569.Google Scholar
[21] J., Alonso, M., Grottke, A., Nikora, and K. S., Trivedi, “The nature of the times to flight software failure during space missions,” in Proc. IEEE Int. Symp. on Software Reliability Engineering (ISSRE), 2012.
[22] J. D., Lambert, Computational Methods in Ordinary Differential Equations. John Wiley & Sons, 1973.
[23] L. F., Shampine, “Stiffness and nonstiff differential equation solvers, II: Detecting stiffness with Runge–Kutta methods,ACM Transactions in Mathematical Software, vol. 3, no. 1, pp. 44–53, 1977.Google Scholar
[24] E., Hairer, S. P., Nørsett, and G., Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn. Springer, 1993, vol. 8.
[25] A., Reibman and K., Trivedi, “Numerical transient analysis of Markov models,” Computers and Operations Research, vol. 15, pp. 19-36, 1988.Google Scholar
[26] N. M. van, Dijk, “Uniformization for nonhomogeneous Markov chains,” Operations Research Letters 12, pp. 283–291, 1992.Google Scholar
[27] M., Malhotra, J. K., Muppala, and K. S., Trivedi, “Stiffness-tolerant methods for transient analysis of stiff Markov chains,Microelectronics and Reliability, vol. 34, pp. 1825–1841, 1994.Google Scholar
[28] K.W. A., vanMoorsel, “Numerical solution of non-homogeneous Markov processes through uniformization,” in Proc. 12th European Simulation Multiconference on Simulation: Past, Present and Future, 1998, pp. 710–717.Google Scholar
[29] M., Telek, A., Horváth, and G., Horváth, “Analysis of inhomogeneous Markov reward models,Linear Algebra and its Applications, vol. 386, pp. 383–405, 2004.Google Scholar
[30] W., Grassmann, “Transient solution in Markovian queueing systems,Computers and Operations Research, vol. 4, pp. 47–56, 1977.Google Scholar
[31] E., Hairer and G., Wanner, Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems. Springer, 1991.
[32] M., Malhotra, J. K., Muppala, and K. S., Trivedi, “Stiffness-tolerant methods for transient analysis of stiff Markov chains,Journal of Microelectronics and Reliability, vol. 34, pp. 1825–1841, 1994.Google Scholar
[33] R., Geist, M., Smotherman, K. S., Trivedi, and J. B., Dugan, “The reliability of life-critical computer systems,Acta Informatica, vol. 23, no. 6, pp. 621–642, 1986.Google Scholar
[34] R., Geist, M., Smotherman, K. S., Trivedi, and J. B., Dugan, “The use of Weibull fault processes in modeling fault tolerant systems,AIAA Journal of Guidance and Control, vol. 11, no. 1, pp. 91–93, 1988.Google Scholar
[35] M., Smotherman and K., Zemoudeh, “A non-homogeneous Markov model for phased-mission reliability analysis,IEEE Transactions on Reliability, vol. 38, no. 5, pp. 585–590, 1989.Google Scholar
[36] K., Trivedi and R., Geist, A Tutorial on the CARE III Approach to Reliability Modeling, NASA contractor report 3488. National Aeronautics and Space Administration, Scientific and Technical Information Branch, 1981.
[37] K., Trivedi and R., Geist, “Decomposition in reliability analysis of fault-tolerant systems,IEEE Transactions on Reliability, vol. R-32, no. 5, pp. 463–468, Dec. 1983.Google Scholar
[38] Y., Li, E., Zio, and Y., Lin, Methods of Solutions of Inhomogeneous Continuous Time Markov Chains for Degradation Process Modeling. John Wiley & Sons, Ltd, 2013, pp. 3–16.
[39] A., Platis, N., Limnios, and M. L., Du, “Asymptotic availability of systems modeled by cyclic non-homogeneous Markov chains [substation reliability],” in Proc. Ann. Symp. on Reliability and Maintainability, Jan 1997, pp. 293–297.Google Scholar
[40] A., Platis, N., Limnios, and M. L., Du, “Dependability analysis of systems modeled by non-homogeneous Markov chains,Reliability Engineering & System Safety, vol. 61, no. 3, pp. 235–249, 1998.Google Scholar
[41] T., Scholz, V., Lopes, and A., Estanqueiro, “A cyclic time-dependent Markov process to model daily patterns in wind turbine power production,Energy, vol. 67, pp. 557–568, 2014.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×