Asymptotic Analysis of Reliability for Switching Systems in Light and Heavy Traffic Conditions

  • Vladimir V. Anisimov
Part of the Statistics for Industry and Technology book series (SIT)


An asymptotic analysis of flows of rare events switched by some random environment is provided. An approximation by nonhomogeneous Poisson flows in case of mixing environment is studied. Special notions of S-set and “monotone” structure for finite Markov environment are introduced. An approximation by Poisson flows with Markov switches in case of asymptotically consolidated environment is proved. An analysis of the 1st exit time from a subset is also given. In heavy traffic conditions an averaging principle for trajectories with Poisson approximation for flows of rare events in systems with fast switches is proved. The method of proof is based on limit theorems for processes with semi-Markov switches.

Applications to the reliability analysis of state-dependent Markov and semi-Markov queueing systems in light and heavy traffic conditions are considered

Keywords and phrases

asymptotic analysis reliability switching processes rare events Markov and semi-Markov processes consolidation queueing models light and heavy traffic 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anisimov, V.V. (1970). Limit distributions of functionals of a semi-Markov process given on a fixed set of states up to the time of first exit, Soviet Math. Dokl., 11, No. 4, 1002–1004.zbMATHGoogle Scholar
  2. 2.
    Anisimov, V.V. (1973). Asymptotic consolidation of the states of random processes, Cybernetics, 9, No. 3, 494–504.CrossRefGoogle Scholar
  3. 3.
    Anisimov, V.V. (1974). Limit theorems for sums of random variables in an array of sequences defined on a subset of states of a Markov chain up to the exit time, Theor. Probability and Math. Stat., No. 4, 15–22.Google Scholar
  4. 4.
    Anisimov, V.V. (1978). Applications of limit theorems for switching processes, Cybernetics, 14, No. 6, 917–929.zbMATHCrossRefGoogle Scholar
  5. 5.
    Anisimov, V.V. (1988). Random Processes with Discrete Component Limit Theorems, Publ. Kiev Univ., Kiev (Russian).Google Scholar
  6. 6.
    Anisimov, V.V. (1994). Limit theorems for processes with semi-Markov switches and their applications, Random Oper. & Stoch. Eqv., 2, No. 4, 333–352.Google Scholar
  7. 7.
    Anisimov, V.V. (1995). Switching processes: averaging principle, diffusion approximation and applications, Acta Applicandae Mathematicae, Kluwer, 40, 95–141.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Anisimov, V.V. (1996). Asymptotic analysis of switching queueing systems in conditions of low and heavy loading, Matrix-Analytic Methods in Stochastic Models, Eds. S.R. Chakravarthy and A.S. Alfa, Lect. Notes in Pure and Appl. Mathem. Series, Marcel Dekker, Inc., 183, 241–260Google Scholar
  9. 9.
    Anisimov, V.V. (1998). Asymptotic Analysis of Stochastic Models of Hierarchic Structure and Applications in Queueing Models, Advances in Matrix Analytic Methods for Stochastic Models, Eds. A. S. Alfa, S. R. Chakravarthy, Notable Publ. Inc., USA, 237–259.Google Scholar
  10. 10.
    Anisimov, V.V. (1999). Diffusion approximation for processes with semi-Markov switches and applications in queueing models, Semi-Markov Models and Applications, Eds. J. Janssen and N. Limnios, Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar
  11. 11.
    Anisimov, V.V. and Aliev, A.O. (1990). Limit theorems for recurrent processes of semi-Markov type, Theor. Probab. & Math. Statist., No. 41, 7–1Google Scholar
  12. 12.
    Anisimov, V.V. and Sztrik, Ja. (1989). Asymptotic analysis of some complex renewable systems operating in random environments, European J. of Operational Research, 41, 162–168.Google Scholar
  13. 13.
    Anisimov, V.V., Zakusilo, O.K. and Dontchenko, V.S. (1987). Elements of Queueing Theory and Asymptotic Analysis of Systems, Publ. “Visca Scola”, Kiev (Russian).Google Scholar
  14. 14.
    Bobbio, A. and Trivedi, K.S. (1986). An Aggregation technique for the transient analysis of stiff Markov chains, IEEE Trans. on Computers, C-35, 9, 803–814.Google Scholar
  15. 15.
    Courtois, P.J. (1977). Decomposability: Queueing and Computer Systems Applications, Academic Press, New York.Google Scholar
  16. 16.
    Kovalenko, I.N. (1980). Rare Events Analysis in the Estimation of Systems Efficiency and Reliability, Publ. “Sov. Radio”, Moscow (Russian).Google Scholar
  17. 17.
    Kovalenko, I.N. (1994). Rare events in queueing systems, A survey, Queueing Systems, 16, 1–49.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Korolyuk, V.S. and Turbin A.F. (1978). Mathematical Foundations of Phase Consolidations of Complex Systems, Publ. “Naukova Dumka”, Kiev (Russian).Google Scholar
  19. 19.
    Soloviev A.D. (1970). A redundancy with fast repair, Izv. Acad. Nauk SSSR. Technich. Kibernetika, 1. 56–71 (Russian).Google Scholar
  20. 20.
    Sztrik, Ja. (1992). Asymptotic analysis of a heterogeneous renewable complex system with random environments, Microelectronics and Reliability, 32, 975–986Google Scholar
  21. 21.
    Sztrik, Ja. and Kouvatsos, D. (1991). Asymptotic analysis of a heterogeneous multiprocessor system in a randomly changing environment, IEEE Trans. on Software Engineering, 17, No. 10, 1069–1075.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Vladimir V. Anisimov
    • 1
    • 2
  1. 1.Bilkent UniversityAnkaraTurkey
  2. 2.Turkey & Kiev UniversityKievUkraine

Personalised recommendations