Annals of Operations Research

, Volume 247, Issue 1, pp 211–227 | Cite as

A functional approximation for retrial queues with two way communication

  • Sofiane Ouazine
  • Karim Abbas


In this paper we consider retrial queues with two way communication and finite orbit. We will develop a functional approximation of the stationary performances of this queue where the parameter of interest is the outgoing rate. Specifically, we will apply the Taylor series expansion method to propagate parametric uncertainty to performance measures. We provide an analysis of the \(\hbox {M}_{1},\, \hbox {M}_{2}/\hbox {G}_{1},\, \hbox {G}_{2}/1\) retrial queue with two way communication and finite orbit. The sensitivity analysis is carried out by considering several numerical examples. More specifically, this paper proposes a numerical approach based on Taylor series expansion with a statistical aspect for analyzing the stationary performances of the considered queueing model, where we assume that the outgoing rate is not assessed in a perfect manner. Additionally, approximate expressions of the probability density functions, the expectation and the variance of the performance measures are obtained and compared to the corresponding Monte Carlo simulations results.


Retrial queues Numerical approximation Taylor series expansion Parametric uncertainty Monte Carlo simulation 


  1. Abbas, K., Heidergott, B., & Aïssani, D. (2013). A functional approximation for the M/G/1/N queue. Discrete Event Dynamic Systems, 23, 93–104.CrossRefGoogle Scholar
  2. Artalejo, J. R. (1999). Accessible bibliography on retrial queues. Mathematical and Computer Modelling, 30, 1–6.CrossRefGoogle Scholar
  3. Artalejo, J. R. (1999). A classified bibliography of research on retrial queues: Progress in 1990–1999. Top, 7, 187–211.CrossRefGoogle Scholar
  4. Artalejo, J. R. (2010). Accessible bibliography on retrial queues: Progress in 2000–2009. Mathematical and Computer Modelling, 51, 1071–1081.CrossRefGoogle Scholar
  5. Artalejo, J. R., & Gomez-Corral, A. (2008). Retrial queueing systems: A computational approach. Berlin: Springer.CrossRefGoogle Scholar
  6. Artalejo, J. R., Gomez-Corral, A., & Neuts, M. F. (2001). Analysis of multiserver queues with constant retrial rate. European Journal of Operational Research, 135, 569–581.CrossRefGoogle Scholar
  7. Artalejo, J. R., & Phung-Duc, T. (2012). Markovian retrial queues with two way communication. Journal of Industrial and Management Optimization, 8, 781–806.CrossRefGoogle Scholar
  8. Artalejo, J. R., & Phung-Duc, T. (2013). Single server retrial queues with two way communication. Applied Mathematical Modelling, 37, 1811–1822.CrossRefGoogle Scholar
  9. Artalejo, J. R., & Pozo, M. (2002). Numerical calculation of the stationary distribution of the main multiserver retrial queue. Annals of Operations Research, 116, 41–56.CrossRefGoogle Scholar
  10. Artalejo, J. R., & Resing, J. A. C. (2010). Mean value analysis of single server retrial queues. Asia Pacific Journal of Operational Research, 27, 335–345.CrossRefGoogle Scholar
  11. Avrachenkov, K., Dudin, A., & Klimenok, V. (2010). Retrial queueing model MMAP/\(\text{ M }_{2}\)/1 with two orbits. Lecture Note on Computer Science, 6235, 107–118.CrossRefGoogle Scholar
  12. Choi, B. D., Choi, K. B., & Lee, Y. W. (1995). M/G/1 retrial queueing systems with two types of calls and finite capacity. Queueing Systems, 19, 215–229.CrossRefGoogle Scholar
  13. Falin, G. I. (1979). Model of coupled switching in presence of recurrent calls. Engineering Cybernetics Review, 17, 53–59.Google Scholar
  14. Falin, G. I. (1983). Calculation of probability characteristic of a multiline system with repeat calls. Moscow University Computational Mathematics and Cybernetics, 1, 43–49.Google Scholar
  15. Falin, G. I., Martìn Dìaz, M., & Artalejo, J. R. (1994). Information theoretic approximations for the M/G/1 retrial queue. Acta Informatica, 31, 559–571.CrossRefGoogle Scholar
  16. Falin, G. I., & Templeton, J. G. C. (1997). Retrial queues. London: Chapman and Hall.CrossRefGoogle Scholar
  17. Gans, N., Koole, G. M., & Mandelbaum, A. (2003). Telephone call centers: Tutorial, review, and research prospects. Manufacturing & Service Operations Management, 5, 79–141.CrossRefGoogle Scholar
  18. Ganti, R. K., Baccelli, F., & Andrews, J. G. (2012). Series expansion for interference in wireless networks. IEEE Transactions on Information Theory, 58, 2194–2205.CrossRefGoogle Scholar
  19. Heidergott, B., & Hordijk, A. (2003). Taylor expansions for stationary Markov chains. Advances in Applied Probability, 35, 1046–1070.CrossRefGoogle Scholar
  20. Martin, M., & Artalejo, J. R. (1995). Analysis of an M/G/1 queue with two types of impatient units. Advances in Applied Probability, 27, 840–861.CrossRefGoogle Scholar
  21. Mishra, K., & Trivedi, K. S. (2010). A non-obtrusive method for uncertainty propagation in analytic dependability models. In: Proceedings of 4th Asia–Pacific symposium on advanced reliability and maintenance modeling.Google Scholar
  22. Phung-Duc, T., & Rogiest, W. (2012). Two way communication retrial queues with balanced call blending. Proceedings of the 19th international conference on analytic and stochastic modelling techniques and applications (pp. 16–31). ASMTA 12, Grenoble: France.Google Scholar
  23. Sakurai, H., & Phung-Duc, T. (2014). Two-way communication retrial queues with multiple types of outgoing calls. Top. doi: 10.1007/s11750-014-0349-5
  24. Seidel, W., Kocemba, Kv, & Mitreiter, K. (1999). On Taylor series expansions for waiting times in tandem queues: An algorithm for calculating the coefficients and an investigation of the approximation error. Performance Evaluation, 38, 153–173.CrossRefGoogle Scholar
  25. Stark, H., & Woods, J. W. (1994). Probability, random processes, and estimation theory for engineers. Englewood Cliffs: Prentice Hall.Google Scholar
  26. Turck, K. D., Fiems, D., Wittevrongel, S., & Bruneel, H. (2011). A Taylor series expansions approach to queues with train arrivals. In Proceedings of 5th international ICST conference on performance evaluation methodologies and tools.Google Scholar
  27. Yang, Y., Posner, M. J. M., Templeton, J. G. C., & Li, H. (1994). An approximation method for the M/G/1 retrial queue with general retrial times. European Journal of Operational Research, 76, 552–562.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BejaiaBejaiaAlgeria
  2. 2.Research Unit LaMOSUniversity of BejaiaBejaiaAlgeria

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