Annals of Operations Research

, Volume 274, Issue 1–2, pp 267–290 | Cite as

On a 2-class polling model with reneging and \(k_i\)-limited service

  • Kevin Granville
  • Steve DrekicEmail author
Original Research


This paper analyzes a 2-class, single-server polling model operating under a \(k_i\)-limited service discipline with class-dependent switchover times. Arrivals to each class are assumed to follow a Poisson process with phase-type distributed service times. Within each queue, customers are impatient and renege (i.e., abandon the queue) if the time before entry into service exceeds an exponentially distributed patience time. We model the queueing system as a level-dependent quasi-birth-and-death process, and the steady-state joint queue length distribution as well as the per-class waiting time distributions are computed via the use of matrix analytic techniques. The impacts of reneging and choice of service time distribution are investigated through a series of numerical experiments, with a particular focus on the determination of \((k_1,k_2)\) which minimizes a cost function involving the expected time a customer spends waiting in the queue and an additional penalty cost should reneging take place.


Polling model \(k_i\)-limited service discipline Reneging Quasi-birth-and-death process Switchover times Phase-type distribution 



Steve Drekic and Kevin Granville thank the anonymous referee and the editor-in-chief for their supportive comments and helpful suggestions. Steve Drekic and Kevin Granville also acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada through its Discovery Grants program (#RGPIN-2016-03685) and Postgraduate Scholarship-Doctoral program, respectively.

Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest.


  1. Boon, M. A. A. (2011). Polling models: From theory to traffic intersections. Doctoral dissertation, Eindhoven: Technische Universiteit Eindhoven.Google Scholar
  2. Boon, M. A. A. (2012). A polling model with reneging at polling instants. Annals of Operations Research, 198(1), 5–23.CrossRefGoogle Scholar
  3. Boon, M. A. A., van der Mei, R. D., & Winands, E. M. M. (2011). Applications of polling systems. Surveys in Operations Research and Management Science, 16(2), 67–82.CrossRefGoogle Scholar
  4. Boon, M. A. A., van Wijk, A. C. C., Adan, I. J. B. F., & Boxma, O. J. (2010). A polling model with smart customers. Queueing Systems, 66(3), 239–274.CrossRefGoogle Scholar
  5. Boon, M. A. A., & Winands, E. M. M. (2014). Heavy-traffic analysis of \(k\)-limited polling systems. Probability in the Engineering and Informational Sciences, 28(4), 451–471.CrossRefGoogle Scholar
  6. Borst, S. C., Boxma, O. J., & Levy, H. (1995). The use of service limits for efficient operation of multistation single-medium communication systems. IEEE/ACM Transactions on Networking, 3(5), 602–612.CrossRefGoogle Scholar
  7. Bright, L., & Taylor, P. G. (1995). Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes. Stochastic Models, 11(3), 497–525.CrossRefGoogle Scholar
  8. Chang, W., & Down, D. G. (2002). Exact asympototics for \(k_i\)-limited exponential polling models. Queueing Systems, 42(4), 401–419.CrossRefGoogle Scholar
  9. Gaver, D. P., Jacobs, P. A., & Latouche, G. (1984). Finite birth-and-death models in randomly changing environments. Advances in Applied Probability, 16(4), 715–731.CrossRefGoogle Scholar
  10. Graves, S. C. (1982). The application of queueing theory to continuous perishable inventory systems. Management Science, 28(4), 400–406.CrossRefGoogle Scholar
  11. Gromoll, H. C., Robert, P., Zwart, B., & Bakker, R. (2006). The impact of reneging in processor sharing queues. In Proceedings of the joint international conference on measurement and modeling of computer systems (pp. 87–96), Saint Malo, France.Google Scholar
  12. Latouche, G., & Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modeling. Philadelphia, PA: ASA SIAM.CrossRefGoogle Scholar
  13. Levy, H., & Sidi, M. (1990). Polling systems: Applications, modeling and optimization. IEEE Transactions on Communications, COM–38(10), 1750–1760.CrossRefGoogle Scholar
  14. MacPhee, I., Menshikov, M., Petritis, D., & Popov, S. (2007). A Markov chain model of a polling system with parameter regeneration. Annals of Applied Probability, 17(5/6), 1447–1473.CrossRefGoogle Scholar
  15. Mishkoy, G., Krieger, U. R., & Bejenari, D. (2012). Matrix algorithm for polling models with PH distribution. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, 68(1), 70–80.Google Scholar
  16. Perel, E., & Yechiali, U. (2017). Two-queue polling systems with switching policy based on the queue that is not being served. Stochastic Models, 33(3), 430–450.CrossRefGoogle Scholar
  17. Takagi, H. (1988). Queueing analysis of polling models. ACM Computing Surveys, 20(1), 5–28.CrossRefGoogle Scholar
  18. Tijms, H. C. (2003). A first course in stochastic models. Chichester: Wiley.CrossRefGoogle Scholar
  19. van Vuuren, M., & Winands, E. M. M. (2007). Iterative approximation of \(k\)-limited polling systems. Queueing Systems, 55(3), 161–178.CrossRefGoogle Scholar
  20. Vishnevskii, V. M., & Semenova, O. V. (2006). Mathematical methods to study the polling systems. Automation and Remote Control, 67(2), 173–220.CrossRefGoogle Scholar
  21. Vishnevskii, V. M., & Semenova, O. V. (2008). The power-series algorithm for two-queue polling system with impatient customers. In Proceedings of ICT 2008—15th International conference on telecommunications (pp. 1–3), Saint-Petersburg, Russia.Google Scholar
  22. Vishnevskii, V. M., & Semenova, O. V. (2009). The power-series algorithm for \(M/M/1\)-type polling system with impatient customers. In Proceedings of EUROCON 2009—International conference on computer as a tool (pp. 1915–1918), Saint-Petersburg, Russia.Google Scholar
  23. Winands, E. M. M., Adan, I. J. B. F., van Houtum, J., & Down, D. G. (2009). A state-dependent polling model with \(k\)-limited service. Probability in the Engineering and Informational Sciences, 23(2), 385–408.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Statistics and Actuarial ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations