Advertisement

Queueing Systems

, Volume 89, Issue 3–4, pp 213–241 | Cite as

Heavy traffic limit for a tandem queue with identical service times

  • H. Christian Gromoll
  • Bryce Terwilliger
  • Bert ZwartEmail author
Article
  • 168 Downloads

Abstract

We consider a two-node tandem queueing network in which the upstream queue is M/G/1 and each job reuses its upstream service requirement when moving to the downstream queue. Both servers employ the first-in-first-out policy. We investigate the amount of work in the second queue at certain embedded arrival time points, namely when the upstream queue has just emptied. We focus on the case of infinite-variance service times and obtain a heavy traffic process limit for the embedded Markov chain.

Keywords

Tandem queue Infinite variance Feller process Process limit 

Mathematics Subject Classification

60K25 90B22 

Notes

Acknowledgements

Funding was provided by NWO grant number 639.033.413.

References

  1. 1.
    Boxma, O.: Analysis of models for tandem queues. Ph.D. Thesis, University of Utrecht, Utrecht (1977)Google Scholar
  2. 2.
    Boxma, O.: On a tandem queueing model with identical service times at both counters, I. Adv. Appl. Probab 11, 616–643 (1979)CrossRefGoogle Scholar
  3. 3.
    Karpelevich, F.I., Kreĭnin, A.Y.: Heavy Traffic Limits for Multiphase Queues, vol. 137 of Translations of Mathematical Monographs. American Mathematical Society, Providence (1994). (Translated from the Russian manuscript by Kreĭnin and A. Vainstein)CrossRefGoogle Scholar
  4. 4.
    Karpelevitch, F.I., Kreinin, A.Y.: Asymptotic analysis of queueing systems with identical service. J. Appl. Probab. 33(1), 267–281 (1996)CrossRefGoogle Scholar
  5. 5.
    Boxma, O.: On the longest service time in a busy period of the \(\text{ M }/\text{ G }/1\) queue. Stoch. Process. Appl. 8(1), 93–100 (1978)CrossRefGoogle Scholar
  6. 6.
    Boxma, O.J., Deng, Q.: Asymptotic behaviour of the tandem queueing system with identical service times at both queues. Math. Methods Oper. Res. 52(2), 307–323 (2000)CrossRefGoogle Scholar
  7. 7.
    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence, vol. 282. Wiley, Hoboken (2009)Google Scholar
  8. 8.
    Ballerini, R., Resnick, S.I.: Records in the presence of a linear trend. Adv. Appl. Probab. 19(4), 801–828 (1987)CrossRefGoogle Scholar
  9. 9.
    Billingsley, P.: Convergence of Probability Measures. Tracts on Probability and Statistics. Wiley Series in Probability and Mathematical Statistics. Wiley, Hoboken (1968)Google Scholar
  10. 10.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)CrossRefGoogle Scholar
  11. 11.
    Resnick, S.I.: Heavy-tail Phenomena. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007). (Probabilistic and statistical modeling)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA
  2. 2.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands

Personalised recommendations