Two-Server Queueing System with Unreliable Servers and Markovian Arrival Process

  • Valentina KlimenokEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 800)


In this paper, we investigate a queueing system consisting of an infinite buffer and two unreliable heterogeneous servers which fail alternately. If both servers are able to provide the service, they serve a customer in parallel, independently of each other. The service of a customer is completed when his/her service by any of two servers ends. The service times at the servers have PH-type (Phase-type) distributions. The input flow and the flow of breakdowns are described by the MAP (Markovian Arrival Process). An arriving breakdown is directed to the first server with some probability and to the second server with complementary probability. After a breakdown occurrence a server fails and the repair period starts immediately. A customer, whose service is interrupted by the breakdown, goes to another server if it is idle, or enters the queue otherwise We derive a condition for the stable operation of the system, calculate its stationary distribution and base performance measures. Illustrative numerical examples are presented.


Unreliable queueing system Markovian Arrival Process Phase-type service time distribution Stationary distribution Performance measures 


  1. 1.
    Vishnevsky, V., Kozyrev, D., Semenova, O.V.: Redundant queueing system with unreliable servers. In: Proceedings of the 6th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops, Moscow, pp. 383–386 (2014)Google Scholar
  2. 2.
    Arnon, S., Barry, J., Karagiannidis, G., Schober, R., Uysal, M. (eds.): Advanced Optical Wireless Communication Systems. Cambridge University Press, New York (2012)Google Scholar
  3. 3.
    Nadeem, F., Leitgeb, E., Kvicera, V., Grabner, M., Awan, M.S., Kandus, G.: Simulation and analysis of FSO/RF switch over for different armospheric effects. In: Proceedings of the 10th International Conference on Telecommunications, Zagreb, Croatia, pp. 39–43 (2009)Google Scholar
  4. 4.
    Letzepis, N., Nguyen, K.D., Guillen, I., Fabregas, A., Cowley, W.G.: Outage analysis of the hybrid free-space optical and radio-frequency channel. IEEE J. Sel. Areas Commun. 27, 1709–1719 (2009)CrossRefGoogle Scholar
  5. 5.
    Vishnevsky, V.M., Semenova, O.V., Sharov, S.Y.: Modeling and analysis of a hybrid communication channel based on free-space optical and radio-frequency technologies. Autom. Remote Control 72, 345–352 (2013)MathSciNetGoogle Scholar
  6. 6.
    Dudin, A., Klimenok, V., Vishnevsky, V.: Analysis of unreliable single server queueing system with hot back-up server. Commun. Comput. Inf. Sci. 499, 149–161 (2015)Google Scholar
  7. 7.
    Sharov, S.Y., Semenova, O.V.: Simulation model of wireless channel based on FSO and RF technologies. In: Distributed Computer and Communication Networks. Theory and Applications, pp. 368–374. Moscow (2010)Google Scholar
  8. 8.
    Klimenok, V., Vishnevsky, V.M.: Unreliable queueing system with cold redundancy. In: Gaj, P., Kwiecień, A., Stera, P. (eds.) CN 2015. Communications in Computer and Information, vol. 522, pp. 336–346. Springer, Cham (2015). doi: 10.1007/978-3-319-19419-6_32 CrossRefGoogle Scholar
  9. 9.
    Lucantoni, D.M.: New results on the single server queue with a batch Markovian arrival process. Commun. Stat. Stoch. Models 7, 1–46 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Neuts, M.: Matrix-geometric Solutions in Stochastic Models - An Algorithmic Approach. Johns Hopkins University Press, Baltimore (1981)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceBelarusian State UniversityMinskBelarus

Personalised recommendations