Skip to main content
SpringerLink
Log in

Using Semi-Markov Chains to Solve Semi-Markov Processes

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

A Correction to this article was published on 07 November 2020

This article has been updated

Abstract

This article provides a novel method to solve continuous-time semi-Markov processes by algorithms from discrete-time case, based on the fact that the Markov renewal function in discrete-time case is a finite series. Bounds of approximate errors due to discretization for the transition function matrix of the continuous-time semi-Markov process are investigated. This method is applied to a reliability problem which refers to the availability analysis of the system subject to sequential cyber-attacks. Two cases where sojourn times follow exponential and Weibull distributions are considered and computed in order to verify and illustrate the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

References

  • Barbu V, Boussemart M, Limnios N (2004) Discrete-time semi-Markov model for reliability and survival analysis. Commun Stat Theory Methods 33 (11):2833–2868

    Article  MathSciNet  Google Scholar 

  • Barbu VS, Limnios N (2009) Semi-Markov chains and hidden semi-Markov models toward applications: their use in reliability and DNA analysis. Springer Science + Business Media

  • Cinlar E (2013) Introduction to stochastic processes. Courier Corporation

  • Corradi G, Janssen J, Manca R (2004) Numerical treatment of homogeneous semi-Markov processes in transient case-a straightforward approach. Methodol Comput Appl Probab 6(2):233–246

    Article  MathSciNet  Google Scholar 

  • DAmico G, Petroni F, Prattico F (2012) Performance analysis of second order semi-Markov chains: an application to wind energy production. Methodol Comput Appl Probab 391(20):4867–4876

    Google Scholar 

  • Hou Y, Limnios N, Schön W (2017) On the existence and uniqueness of solution of MRE and applications. Methodol Comput Appl Probab 19 (4):1241–1250

    Article  MathSciNet  Google Scholar 

  • Isguder HO, Uzunoglu-Kocer U (2014) Analysis of GI/m/n/n queueing system with ordered entry and no waiting line. Appl Math Model 38(3):1024–1032

    Article  MathSciNet  Google Scholar 

  • Karr AF (1975) Weak convergence of a sequence of Markov chains. Z. Wahrscheinlichkeitstheorie verw. Gebiete 33(1):41–48

    Article  MathSciNet  Google Scholar 

  • Levy P (1954) Processus semi-markoviens. In: Proc. Int. Congress. Math. III, Amsterdam

  • Limnios N (1997) Dependability analysis of semi-Markov systems. Reliab Eng Syst Saf 55(3):203–207

    Article  Google Scholar 

  • Limnios N, Oprisan G (2012) Semi-Markov processes and reliability. Springer Science+Business Media.

  • Perman M, Senegacnik A, Tuma M (1997) Semi-markov models with an application to power-plant reliability analysis. IEEE Trans Reliab 46 (4):526–532

    Article  Google Scholar 

  • Liu Q, Xing L, Zhou C (2020) Probabilistic modeling and analysis of sequential cyber-attacks. Eng Rep 1(4):e12065

    Google Scholar 

  • Smith W L (1955) Regenerative stochastic processes. Proc R Soc London A:, Math Phys Sci 232(1188):6–31

    Article  MathSciNet  Google Scholar 

  • Takacs L (1954) Some investigations concerning recurrent stochastic processes of a certain type. Maygyar Tud Akad Mad Kutato int Kozl 3:115–128

    MathSciNet  MATH  Google Scholar 

  • Warr R L (2014) Numerical approximation of probability mass functions via the inverse discrete Fourier transform. Methodol Comput Appl Probab 16 (4):1025–1038

    Article  MathSciNet  Google Scholar 

  • Wu B, Cui L, Fang C (2020) Reliability analysis of semi-Markov systems with restriction on transition times Reliab Eng Syst Saf 106516

  • Wu B, Cui L (2020) Reliability of repairable multi-state two-phase mission systems with finite number of phase switches. Appl Math Model 77:1229–1241

    Article  MathSciNet  Google Scholar 

  • Xia JC, Zeephongsekul P, Packer D (2011) Spatial and temporal modelling of tourist movements using Semi-Markov processes. Tour Manag 32(4):844–851

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China under grant 71631001, which facilitated the travel of the first author to Université de Technologie de Compiègne, France, to collaborate with the other authors on this project.

Author information

Authors and Affiliations

  1. School of Management and Economics, Beijing Institute of Technology, Beijing, 100081, China

    Bei Wu

  2. Université de Technologie de Compiègne, Sorbonne University, Paris, France

    Brenda Ivette Garcia Maya & Nikolaos Limnios

Authors
  1. Bei Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Brenda Ivette Garcia Maya
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nikolaos Limnios
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bei Wu.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The original version of this article was revised: The definition of the norm of A(t) and Propositions 2 and 3, the norm was missing.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, B., Maya, B.I.G. & Limnios, N. Using Semi-Markov Chains to Solve Semi-Markov Processes. Methodol Comput Appl Probab 23, 1419–1431 (2021). https://doi.org/10.1007/s11009-020-09820-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-020-09820-y

Keywords

Search

Navigation