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ISSN 2541-9005 (Online)

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Tananko I. E., Fokina N. P. An Analysis Method of Open Queueing Networks with a Degradable Structure and Instantaneous Repair Times of Systems. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2020, vol. 20, iss. 2, pp. 266-276. DOI: 10.18500/1816-9791-2020-20-2-266-276

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An Analysis Method of Open Queueing Networks with a Degradable Structure and Instantaneous Repair Times of Systems

Tananko Igor' Evstaf'evich, Saratov State University
Fokina Nadezhda Petrovna, Saratov State University

An unreliable open queueing network with Poisson arrivals is considered. For each queueing system the service and failures times are exponentially distributed random variables. The failures of systems lead to changes in the structure of the network and corresponding changes in the performance measures of the queueing network. It is assumed that the times between changes in the network structure are sufficient for the steady-state regime. The main measure of the quality for the network at each structure constancy interval is the average response time. Repairs of all queueing systems occur immediately when the average response time becomes greater than the threshold value. This article presents a method of the network analysis using continuous time Markov chains. It is shown that the steady-state probability distribution of the unreliable queueing network has a product form solution. Expressions for the stationary performance measures of queueing systems and the network including the average of system repair time intervals are obtained. A numerical example to investigate the dependence of the performance measures on some network parameters is demonstrated.

  1. Park K., Kim S. A capacity planning model of unreliable multimedia service systems. Journal of Systems and Software, 2002, vol. 63, iss. 1, pp. 69–76. DOI:
  2. Economides A. А., Silvester J. A. Optimal routing in a network with unreliable links. IEEE INFOCOM’88, 1988, pp. 288–297. DOI:
  3. Thomas N., Thornley D., Zatschler H. Approximate solution of a class of queueing networks with breakdowns. Proc. of 17th European Simulation Multiconference. Nottingham, UK, SCS Publishers, 2003, pp. 251–256.
  4. Chao X. A queueing network model with catastrophes and product form solution. Operations Research Letters, 1995, vol. 18, iss. 2, pp. 75–79. DOI:
  5. Tananko I. E. About closed queueing networks with a variable number of queues. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2005, vol. 5, iss. 1, pp. 138–141 (in Russian).
  6. Tsitsiashvili G. Sh., Osipova M. A. Limiting distributions in queueing networks with unreliable elements. Probl. Inf. Transm., 2008, vol. 44, iss. 4, pp. 385–394. DOI:
  7. Tassiulas L. Scheduling and performance limits of networks with constantly changing topology. IEEE Transactions on Information Theory, 1997, vol. 43, iss. 3, pp. 1067–1073. DOI:
  8. Fokina N. P., Tananko I. E. The Method of Routing in Queueing Networks with Variable Topology. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2013. vol. 13, iss. 2, pt. 2, pp. 82–88 (in Russian). DOI:
  9. Statkevich C. E., Matalytsky M. A. Investigation of queueing network with unreliable systems at transient regime. Tomsk State University Journal of Control and Computer Science, 2012, no. 1 (18), pp. 112–125 (in Russian).
  10. Chakka R., Mitrani I. Approximate solutions for open networks with breakdowns and repairs. In: Stochastic Networks: Theory and Applications (Royal Statistical Society Series) / eds. F. P. Kelly, S. Zachary, I. Ziedins. Oxford, Clarendon Press, 1996. Vol. 4, pp. 267–280.
  11. Vinod B., Altiok T. Approximating Unreliable Queueing Networks Under the Assumption of Exponentiality. J. Opl. Res. Soc., 1986, vol. 37, no. 3, pp. 309–316. DOI:
  12. Thomas N., Bradley J. T., Knottenbelt W. J. Stochastic analysis of scheduling strategies in a Grid-based resource model. IEEE Proceedings – Software, 2004, vol. 151, iss. 5, pp. 232–239. DOI:
  13. Mitrophanov Yu. I. Analiz setei massovogo obsluzhivaniia [Analysis of Queueing Networks]. Saratov, Nauchnaya kniga, 2005. 175 p. (in Russian).
  14. He Q.-M. Fundamentals of matrix-analytic methods. New York, Springer, 2014. 349 p. DOI: