Analysis of a Discrete-Time Queue with Geometrically Distributed Service Capacities

  • Herwig Bruneel
  • Joris Walraevens
  • Dieter Claeys
  • Sabine Wittevrongel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7314)


We consider a discrete-time queueing model whereby the service capacity of the system, i.e., the number of work units that the system can perform per time slot, is variable from slot to slot. Specifically, we study the case where service capacities are independent from slot to slot and geometrically distributed. New customers enter the system according to a general independent arrival process. Service demands of the customers are i.i.d. and arbitrarily distributed. For this (non-classical) queueing model, we obtain explicit expressions for the probability generating functions (pgf’s) of the unfinished work in the system and the queueing delay of an arbitrary customer. In case of geometric service demands, we also obtain the pgf of the number of customers in the system explicitly. By means of some numerical examples, we discuss the impact of the service process of the customers on the system behavior.


Discrete-time queueing model Variable service capacity Analytic study Closed-form results 


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  1. 1.
    Balkhi, Z.T.: On the global optimal solution to an integrated inventory system with general time varying demand, production and deterioration rates. European Journal of Operations Research 114, 29–37 (1999)zbMATHCrossRefGoogle Scholar
  2. 2.
    Bruneel, H.: Buffers with stochastic output interruptions. Electronics Letters 19, 735–737 (1983)CrossRefGoogle Scholar
  3. 3.
    Bruneel, H.: A general model for the behaviour of infinite buffers with periodic service opportunities. European Journal of Operational Research 16(1), 98–106 (1984)zbMATHCrossRefGoogle Scholar
  4. 4.
    Bruneel, H.: Performance of discrete-time queueing systems. Computers and Operations Research 20(3), 303–320 (1993)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bruneel, H., Kim, B.G.: Discrete-time models for communication systems including ATM. Kluwer Academic, Boston (1993)CrossRefGoogle Scholar
  6. 6.
    Chakravarthy, S.R.: Analysis of a multi-server queue with Markovian arrivals and synchronous phase type vacations. Asia-Pacific Journal of Operational Research 26(1), 85–113 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chang, C.S., Thomas, J.A.: Effective bandwidth in high-speed digital networks. IEEE Journal on Selected Areas in Communications 13, 1091–1100 (1995)CrossRefGoogle Scholar
  8. 8.
    Fiems, D., Bruneel, H.: A note on the discretization of Little’s result. Operations Research Letters 30, 17–18 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gao, P., Wittevrongel, S., Bruneel, H.: Discrete-time multiserver queues with geometric service times. Computers and Operations Research 31, 81–99 (2004)zbMATHCrossRefGoogle Scholar
  10. 10.
    Gao, P., Wittevrongel, S., Laevens, K., De Vleeschauwer, D., Bruneel, H.: Distributional Little’s law for queues with heterogeneous server interruptions. Electronics Letters 46, 763–764 (2010)CrossRefGoogle Scholar
  11. 11.
    Georganas, N.D.: Buffer behavior with poisson arrivals and bulk geometric service. IEEE Transactions on Communications 24, 938–940 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Glock, C.H.: Batch sizing with controllable production rates. International Journal of Production Research 48, 5925–5942 (2010)zbMATHCrossRefGoogle Scholar
  13. 13.
    Jin, X., Min, G., Velentzas, S.R.: An analytical queuing model for long range dependent arrivals and variable service capacity. In: Proceedings of IEEE International Conference on Communications (ICC 2008), Beijing, pp. 230–234 (May 2008)Google Scholar
  14. 14.
    Kafetzakis, E., Kontovasilis, K., Stavrakakis, I.: Effective-capacity-based stochastic delay guarantees for systems with time-varying servers, with an application to IEEE 802.11 WLANs. Performance Evaluation 68, 614–628 (2011)CrossRefGoogle Scholar
  15. 15.
    Laevens, K., Bruneel, H.: Delay analysis for discrete-time queueing systems with multiple randomly interrupted servers. European Journal of Operations Research 85, 161–177 (1995)zbMATHCrossRefGoogle Scholar
  16. 16.
    Mitrani, I.: Modelling of Computer and Communication Systems. Cambridge University Press, Cambridge (1987)Google Scholar
  17. 17.
    Takagi, H.: Queueing Analysis, A Foundation of Performance Evaluation. Discrete-time systems, vol. 3. North-Holland, Amsterdam (1993)Google Scholar
  18. 18.
    Yang, H.-L.: A partial backlogging production-inventory lot-size model for deteriorating items with time-varying production and demand rate over a finite time horizon. International Journal of Systems Science 42, 1397–1407 (2011)zbMATHCrossRefGoogle Scholar
  19. 19.
    Yang, X.L., Alfa, A.S.: A class of multi-server queueing system with server failures. Computers & Industrial Engineering 56(1), 33–43 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Herwig Bruneel
    • 1
  • Joris Walraevens
    • 1
  • Dieter Claeys
    • 1
  • Sabine Wittevrongel
    • 1
  1. 1.Department of Telecommunications and Information Processing (TELIN)Ghent UniversityGentBelgium

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