Skip to main content

Immediate Events in Markov Chains

  • Conference paper
Computations with Markov Chains

Abstract

Immediate events are events which happen without delay. There are several contexts in which immediate events occur. In particular, they are important for formulating Markovian systems efficiently, they are an integral part of generalised stochastic Petri nets, and they are useful for analysing processes which have different time scales. These applications are reviewed, and a unified theory for finding equilibrium probabilities for such processes is developed. This theory is based on the GTH algorithm, which is augmented by a special algebra.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berson, S., De Souza e Silva, E., Muntz, R. R. 1991. A Methodology for the Specification and Generation of Markov Models. in: W. J. Stewart (editor), Numerical Solutions of Markov Chains, Marcel Dekker, New York, 11–36.

    Google Scholar 

  2. Ciardo, G., Muppala, J., Trivedi, K. S. 1991. On the Solution of GSPN Reward Models.Performance Evaluation 12, 237–253.

    Article  MATH  Google Scholar 

  3. Coderch, M., Willsky, A. S., Sastry, S. S., Castanon, D. A. 1983. Hierarchical Aggregation of Singularly Perturbed Finite State Markov Processes. Stochastics 8, 259–289.

    Article  MathSciNet  MATH  Google Scholar 

  4. Grassmann, W. K. 1993. Rounding Errors in Certain Algorithms Involving Markov Chains. ACM Transactions on Mathematical Software,19(4), 496–508.

    Article  MATH  Google Scholar 

  5. Grassmann, W. K. 1985. The Factorization of Queueing Equations and Their Interpretation. J.Opl. Res Soc 36, 1041–1051.

    MATH  Google Scholar 

  6. Grassmann, W. K. 1991. Finding Transient Solutions in Markovian Event Systems Through Randomization. In: W. J. Stewart (editor), Numerical Solutions of Markov Chains, Marcel Dekker, New York, 357–371.

    Google Scholar 

  7. W. K. Grassmann, M. I. Taksar, D. P. Heyman. 1985. Regenerative Analysis and Steady State Distributions for Markov Chains. . Op. Res. 33, 1107–1116.

    Article  MathSciNet  MATH  Google Scholar 

  8. Grassmann, W. K. and Heyman, D. P. 1990. Equilibrium Distribution of Block-Structured Markov Chains with Repeating Rows. Jou rn al of Applied Probability 27, 557--576.

    Article  MathSciNet  MATH  Google Scholar 

  9. Gross, D, and D. R. Miller. 1984. The Randomization Technique as a Modelling Tool and Solution Procedure for Transient Markov Processes. Op. Res. 32, 343–361.

    Article  MathSciNet  MATH  Google Scholar 

  10. Marsan, M. A., Conte, G. and Balbo, G. 1984. A Class of Generalized Stochastic Petri Nets for the Performance Evaluation of Multiprocessor Systems. ACM Transactions on Computer Systems, 2 (2), 93–122.

    Article  Google Scholar 

  11. O’Cinneide, C. 1993. Entrywise Perturbation Theory and Error Analysis for Markov Chains. Numer. Math, 65, 109–120.

    Article  MathSciNet  MATH  Google Scholar 

  12. O’Leary, D. and Yuan-Jye, J.W. 1993. A Block GTH Algorithm for Finding the Stationary Vector of a Markov Chain Preprint.

    Google Scholar 

  13. Rohlicek, J. R. 1988. The Reduction of Perturbed Markov Generators: An Algorithm Exposing the Role of Transient States. JA CM 35, 675–696.

    MathSciNet  MATH  Google Scholar 

  14. Stewart, G. W. and Zhang, G. 1990. On a Direct Method for the Solution of Nearly Uncoupled Markov Chains Numer. Math. 59, 1–11, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  15. Stewart, W. J. 1991. MARCA: Markov Chain Analyzer. A Software Package for Markov Modelling. In: W. J. Stewart (editor), Numerical Solutions of Markov Chains, Marcel Dekker, New York, 37–62.

    Google Scholar 

  16. Wallace, V. L. and Rosenberg, R. S. 1966. RQA-1. The Recursive Queue Analyser. Technical Report, System Eng. Lab, University of Michigan, Ann Arbor.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer Science+Business Media New York

About this paper

Cite this paper

Grassmann, W.K., Wang, Y. (1995). Immediate Events in Markov Chains. In: Stewart, W.J. (eds) Computations with Markov Chains. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2241-6_11

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-2241-6_11

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-5943-2

  • Online ISBN: 978-1-4615-2241-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics