Delay Analysis of a Queue with General Service Demands and Phase-Type Service Capacities

  • Michiel De MuynckEmail author
  • Herwig Bruneel
  • Sabine Wittevrongel
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 383)


We present the analysis of a non-classical discrete-time queueing model where customers demand variable amounts of work from a server that is able to perform this work at a varying rate. The service demands of the customers are integer numbers of work units. They are assumed to be independent and identically distributed (i.i.d.). The service capacities, i.e., the numbers of work units that the server can process in the consecutive slots, are also assumed to be i.i.d. and have a rational probability generating function (pgf). Finally, the numbers of customer arrivals in each slot are i.i.d. as well. We analyze this model analytically using contour integration. Our main result is an expression for the pgf of the customer delay in steady state, from which expressions for the moments of the delay can be derived.


Discrete-time queueing theory Service demands Service capacities Complex contour integration 


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  1. 1.
    Bruneel, H., Wittevrongel, S., Claeys, D., Walraevens, J., Discrete-time queues with variable service capacity: a basic model and its analysis, Annals of operations research (2013) (accepted for publication). doi: 10.1007/s10479-013-1428-yMathSciNetCrossRefGoogle Scholar
  2. 2.
    Walraevens, J., Bruneel, H., Claeys, D., Wittevrongel, S.: The discrete-time queue with geometrically distributed service capacities revisited. In: Dudin, A., De Turck, K. (eds.) ASMTA 2013. LNCS, vol. 7984, pp. 443–456. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  3. 3.
    Bruneel, H., Rogiest, W., Walraevens, J., Wittevrongel, S.: On queues with general service demands and constant service capacity. In: Norman, G., Sanders, W. (eds.) QEST 2014. LNCS, vol. 8657, pp. 210–225. Springer, Heidelberg (2014) Google Scholar
  4. 4.
    De Muynck, M., Wittevrongel, S., Bruneel, H., A discrete-time queue with finite-support service capacities. In: Book of Abstracts of ECQT 2014, Ghent, p. 34, August 20–22, 2014Google Scholar
  5. 5.
    Vinck, B., Bruneel, H.: Analyzing the discrete-time \(G^{(G)}/Geo/1\) queue using complex contour integration. Queueing Systems 18(1–2), 47–67 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Adan, I.J.B.F., van Leeuwaarden, J.S.H., Winands, E.M.M.: On the applicationof Rouchè’s theorem in queueing theory. Oper. Res. Letters, 355–360 (20050MathSciNetCrossRefGoogle Scholar
  7. 7.
    Mitrani, I.: Modelling of Computer and Communication Systems. Cambridge University Press, Cambridge (1987) zbMATHGoogle Scholar
  8. 8.
    Abate, J., Whitt, W.: Numerical inversion of probability generating functions. Oper. Res. Letters 12, 245–251 (1992)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Michiel De Muynck
    • 1
    Email author
  • Herwig Bruneel
    • 1
  • Sabine Wittevrongel
    • 1
  1. 1.Department of Telecommunications and Information Processing, Stochastic Modeling and Analysis of Communication Systems Research GroupGhent UniversityGhentBelgium

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