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Rate of Convergence to Stationary Distribution for Unreliable Jackson-Type Queueing Network with Dynamic Routing

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Distributed Computer and Communication Networks (DCCN 2016)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 678))

Abstract

In this paper we consider a Jackson type queueing network with unreliable nodes. The network consists of \( m <\infty \) nodes, each node is a queueing system of M/G/1 type. The input flow is assumed to be the Poisson process with parameter \( \varLambda (t)\). The routing matrix \(\{r_{ij}\}\) is given, \(i, j=0,1,...,m\), \( \sum _{i = 1 } ^ m r_ {0i} \le 1 \). The new request is sent to the node i with the probability \(r_{0i}\), where it is processed with the intensity rate \(\mu _i(t,n_i(t))\). The intensity of service depends on both time t and the number of requests at the node \(n_i(t)\). Nodes in a network may break down and repair with some intensity rates, depending on the number of already broken nodes. Failures and repairs may occur isolated or in groups simultaneously. In this paper we assumed if the node j is unavailable, the request from node i is send to the first available node with minimal distance to j, i.e. the dynamic routing protocol is considered in the case of failure of some nodes. We formulate some results on the bounds of convergence rate for such case.

E.Y. Kalimulina—The work is partially supported by RFBR, research projects No. 14-07-31245 and No. 15-08-08677.

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References

  1. Lakatos, L., Szeidl, L., Telek, M.: Introduction to Queueing Systems with Telecommunication Applications, Springer Science & Business Media, Mathematics, 388 pages. Springer, Heidelberg (2012)

    Google Scholar 

  2. Ghosal, A., Gujaria, S.C., Ghosal, R.: Network Queueing Systems: With Industrial Applications. South Asian Publishers, 1 January 2004. Computer network protocols, 138 pages (2004)

    Google Scholar 

  3. Daigle, J.: Queueing Theory with Applications to Packet Telecommunication. Technology & Engineering, p. 316. Springer Science & Business Media, Berlin (2006)

    Google Scholar 

  4. Alexander, T.: Analysis of fork/join and related queueing systems. ACM Comput. Surv. 47(2), 71 (2014). doi:10.1145/2628913. Article 17

    Google Scholar 

  5. Lakshmi, C., Sivakumar, A.I.: Application of queueing theory in health care: a literature review. Oper. Res. Health Care 2(1–2), 25–39 (2013)

    Google Scholar 

  6. Sterbenz, J.P.G., Çetinkaya, E.K., Hameed, M.A., et al.: Evaluation of network resilience, survivability, and disruption tolerance: analysis, topology generation, simulation, and experimentation. Telecommun. Syst. 52, 705 (2013). doi:10.1007/s11235-011-9573-6

    Google Scholar 

  7. Hou, I.-H., Kumar, P.R.: Queueing systems with hard delay constraints: a framework for real-time communication over unreliable wireless channels. Queueing Syst. 71(1), 1510–1577 (2012). doi:10.1007/s11134-012-9293-y

    MathSciNet  MATH  Google Scholar 

  8. Madhu Jain, G.C., Sharma, R.S.: Unreliable server M/G/1 queue with multi-optional services and multi-optional vacations. Int. J. Math. Oper. Res. 5(2), 145–169 (2013). doi:10.1504/IJMOR.2013.052458

    Article  MathSciNet  Google Scholar 

  9. Ba-Rukab, O.M., Tadj, L., Ke, J.-C.: Binomial schedule for an M/G/1 queueing system with an unreliable server. Int. J. Modell. Oper. Manage. 3(3–4), 206–218 (2013). doi:10.1504/IJMOM.2013.058326

    Google Scholar 

  10. Tadj, L., Choudhury, G., Rekab, K.: A two-phase quorum queueing system with Bernoulli vacation schedule, setup, and N-policy for an unreliable server with delaying repair. Int. J. Serv. Oper. Manage. 12(2), 139–164 (2012). doi:10.1504/IJSOM.2012.047103

    Google Scholar 

  11. Bama, S., Afthab Begum, M.I., Fijy Jose, P.: Unreliable Mx/G/1 queueing system with two types of repair. Int. J. Innov. Res. Dev. 4(10) (2015)

    Google Scholar 

  12. Klimenok, V., Vishnevsky, V.: Unreliable queueing system with cold redundancy. In: Gaj, P., Kwiecień, A., Stera, P. (eds.) CN 2015. CCIS, vol. 522, pp. 336–346. Springer, Heidelberg (2015). doi:10.1007/978-3-319-19419-6_32

    Chapter  Google Scholar 

  13. Chen, H., Yao, D.D.: Fundamentals of Queueing Networks. In: Performance, Asymptotics, and Optimization Springer, Book on Stochastic Modelling and Applied Probability, vol. 46 (2001). doi:10.1007/978-1-4757-5301-1

    Google Scholar 

  14. Vvedenskaya, N.D.: Configuration of overloaded servers with dynamic routing. Probl. Inf. Transm. 47, 289 (2011). doi:10.1134/S0032946011030070

    Article  MathSciNet  MATH  Google Scholar 

  15. Sukhov, Y., Vvedenskaya, N.D.: Fast Jackson networks with dynamic routing. Probl. Inf. Transm. 38, 136 (2002). doi:10.1023/A:1020010710507

    Article  MathSciNet  MATH  Google Scholar 

  16. Kelly, F.P., Laws, C.N.: Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling. Queueing Syst. 13, 47–86 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sauer, C., Daduna, H.: Availability formulas and performance measures for separable degradable networks. Econ. Qual. Control 18(2), 165–194 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Marco, C.: Dynamic Routing in Broadband Networks. Springer, Heidelberg (2012)

    Google Scholar 

  19. Yavuz, F., Zhao, J., Yagan, O., Gligor, V.: Toward k-connectivity of the random graph induced by a pairwise key predistribution scheme with unreliable links. IEEE Trans. Inf. Theor. 61(11), 6251–6271 (2015). doi:10.1109/TIT.2015.2471295

    Article  MathSciNet  Google Scholar 

  20. Lorek, P., Szekli, R.: Computable bounds on the spectral gap for unreliable Jackson networks. Adv. Appl. Probab. 47, 402–424 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lorek, P.: The exact asymptotic for the stationary distribution of some unreliable systems, [math.PR] arXiv:1102.4707 (2011)

  22. Chen, M.F.: Eigenvalues, Inequalities, and Ergodic Theory. Springer, Heidelberg (2005)

    MATH  Google Scholar 

  23. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)

    MATH  Google Scholar 

  24. Saaty, T.L.: Elements of Queueing Theory with Applications. Dover Publications, New York (1983)

    Google Scholar 

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Authors and Affiliations

  1. V. A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya Street 65, Moscow, 11799, Russia

    Elmira Yu. Kalimulina

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  1. Elmira Yu. Kalimulina
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Correspondence to Elmira Yu. Kalimulina .

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Editors and Affiliations

  1. V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    Vladimir M. Vishnevskiy

  2. RUDN University , Moscow, Russia

    Konstantin E. Samouylov

  3. V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    Dmitry V. Kozyrev

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Kalimulina, E.Y. (2016). Rate of Convergence to Stationary Distribution for Unreliable Jackson-Type Queueing Network with Dynamic Routing. In: Vishnevskiy, V., Samouylov, K., Kozyrev, D. (eds) Distributed Computer and Communication Networks. DCCN 2016. Communications in Computer and Information Science, vol 678. Springer, Cham. https://doi.org/10.1007/978-3-319-51917-3_23

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  • DOI: https://doi.org/10.1007/978-3-319-51917-3_23

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  • Online ISBN: 978-3-319-51917-3

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