Advertisement

Modelling Complex Transport Network with Dynamic Routing: A Queueing Networks Approach

  • Elmira Yu. KalimulinaEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 941)

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.

Keywords

Dynamic routing Queueing system Jackson network 

References

  1. 1.
    Lakatos, L., Szeidl, L., Telek, M.: Introduction to Queueing Systems with Telecommunication Applications, 388 p. Springer, Heidelberg (2012)Google Scholar
  2. 2.
    Daigle, J.: Queueing Theory with Applications to Packet Telecommunication, 316 p. Springer, Heidelberg (2006). Technology & EngineeringGoogle Scholar
  3. 3.
    Thomasian, A.: Analysis of fork/join and related queueing systems. ACM Comput. Surv. 47(2), 71 p. (2014).  https://doi.org/10.1145/2628913. Article 17
  4. 4.
    Jain, M., Sharma, G.C., Sharma, R.: Unreliable server M-G-1 queue with multi-optional services and multi-optional vacations. Int. J. Math. Oper. Res. 5(2) (2013).  https://doi.org/10.1504/IJMOR.2013.052458
  5. 5.
    Vvedenskaya, N.D.: Configuration of overloaded servers with dynamic routing. Probl. Inf. Transm. 47, 289 (2011).  https://doi.org/10.1134/S0032946011030070MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Sukhov, Yu.M., Vvedenskaya, N.D.: Fast Jackson networks with dynamic routing. Probl. Inf. Transm. 38, 136 (2002).  https://doi.org/10.1023/A:1020010710507
  7. 7.
    Lorek, P., Szekli, R.: Computable bounds on the spectral gap for unreliable Jackson networks. Adv. Appl. Probab. 47, 402–424 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lorek, P.: The exact asymptotic for the stationary distribution of some unreliable systems. arXiv:1102.4707 [math.PR] (2011)
  9. 9.
    Kalimulina, E.Yu.: Rate of convergence to stationary distribution for unreliable Jackson-type queueing network with dynamic routing. In: Distributed Computer and Communication Networks. Communications in Computer and Information Science, vol. 678, pp. 253–265 (2017).  https://doi.org/10.1007/978-3-319-51917-3_23.
  10. 10.
    Kalimulina, E.Yu.: Queueing system convergence rate. In: Proceedings of the 19th International Conference, Distributed Computer and Communication Networks, DCCN 2016, Moscow, Russia, vol. 3, pp. 203–211. RUDN, Moscow (2016)Google Scholar
  11. 11.
    Kalimulina, E.Yu.: Analysis of system reliability with control, dependent failures, and arbitrary repair times. Int. J. Syst. Assur. Eng. Manag., 1–11 (2016).  https://doi.org/10.1007/s13198-016-0520-5(2016)
  12. 12.
    Kalimulina E. Yu.: Analysis of unreliable Jackson-type queueing networks with dynamic routing, December 2016. SSRN:  https://doi.org/10.2139/ssrn.2881956
  13. 13.
    Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of Networks: From Biological Nets to the Internet and WWW (Physics). Oxford University Press Inc., New York (2003)CrossRefGoogle Scholar
  14. 14.
    Chen, M.F.: Eigenvalues, Inequalities, and Ergodic Theory. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  15. 15.
    Klimenok, V., Vishnevsky, V.: Unreliable queueing system with cold redundancy. In: Gaj, P., Kwiecień, A., Stera, P. (eds.) Computer Networks. Communications in Computer and Information Science, vol. 522, pp. 336–346. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-319-19419-6_32CrossRefGoogle Scholar
  16. 16.
    Chen, H., Yao, D.D.: Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization. Stochastic Modelling and Applied Probability, vol. 46. Springer, Heidelberg (2001).  https://doi.org/10.1007/978-1-4757-5301-1CrossRefzbMATHGoogle Scholar
  17. 17.
    Yavuz, F., Zhao, J., Yaǧan, O., Gligor, V.: Toward k -connectivity of the random graph induced by a pairwise key predistribution scheme with unreliable links. IEEE Trans. Inf. Theory 61(11), 6251–6271 (2015).  https://doi.org/10.1109/TIT.2015.2471295MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sauer, C., Daduna, H.: Availability formulas and performance measures for separable degradable networks. Econ. Qual. Control 18(2), 165–194 (2003)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.V. A. Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia

Personalised recommendations