Exhaustive Vacation Queue with Dependent Arrival and Service Processes

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 383)

Abstract

This paper presents a more general class of MAP/MAP/1 exhaustive vacation queue, in which the Markov modulated arrival and service processes are dependent. This model class requires the evaluation of the busy period of quasi birth death process with arbitrary initial level, which is a new analysis element.

The model is analyzed by applying matrix analytic methods for the underlying quasi birth death process. The main result of the paper is the probability-generating function of the number of jobs in the system. Finally, a numerical example provides an insight into the behavior of the model.

Keywords

Vacation queue MAP Dependent arrival and service process QBD Matrix analytic methods stationary analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alfa, A.S.: A discrete MAP/PH/1 queue with vacations and exhaustive time-limited service. Oper. Res. Lett. 18, 31–40 (1995)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alfa, A.S.: Discrete time analysis of MAP/PH/1 vacation queue with gated time-limited service. Queueing Systems 29(1), 35–54 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Goswami, C., Selvaraju, N.: The discrete-time MAP/PH/1 queue with multiple working vacations. Applied Mathematical Modelling 34, 931–946 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chang, S., Takine, T.: Factorization and stochastic decomposition properties in bulk queues with generalized vacations. Queueing Systems 50(2–3), 165–183 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ke, J.C., Wu, C.H., Zhang, Z.G.: Recent developments in vacation queueing models: a short survey. International Journal of Operations Research 7(4), 3–8 (2010)Google Scholar
  6. 6.
    Latouche, G., Ramaswami, V.: Introduction to matrix analytic methods in stochastic modeling, vol. 5. SIAM (1999)Google Scholar
  7. 7.
    Neuts, M.F.: A versatile markovian point process. Journal of Applied Probability, 764–779 (1979)Google Scholar
  8. 8.
    Saffer, Z., Telek, M.: Analysis of BMAP/G/1 vacation model of non-M/G/1-type. In: Thomas, N., Juiz, C. (eds.) EPEW 2008. LNCS, vol. 5261, pp. 212–226. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  9. 9.
    Saffer, Z., Telek, M.: Closed form results for BMAP/G/1 vacation model with binomial type disciplines. Publ. Math. Debrecen 76(3), 359–378 (2010)MathSciNetGoogle Scholar
  10. 10.
    Tian, N., Zhang, Z.G.: Vacation queueing models: theory and applications, vol. 93. Springer Science & Business Media (2006)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Gábor Horváth
    • 1
    • 2
  • Zsolt Saffer
    • 1
  • Miklós Telek
    • 1
    • 2
  1. 1.Budapest University of Technology and EconomicsBudapestHungary
  2. 2.MTA-BME Information Systems Research GroupBudapestHungary

Personalised recommendations