Queueing Systems

, Volume 76, Issue 1, pp 21–36 | Cite as

A Lévy input fluid queue with input and workload regulation

  • Zbigniew Palmowski
  • Maria VlasiouEmail author
  • Bert Zwart


We consider a controlled queuing model with Lévy input. The controls take place at random times. They involve the current workload and the input processes and may also depend on whether the workload process has reached certain critical values since the last control epoch. We propose a solution strategy for deriving the steady-state distribution of this model which is based on recent advances in the fluctuation theory of spectrally one-sided Lévy process. We provide illustrative examples involving a clearing model, an inventory model, and a model for the TCP protocol.


Fluctuation theory TCP model Inventory model Clearing model Workload Scale functions 

Mathematics Subject Classification (2000)

Primary: 60K25 Secondary: 60G51 



We are grateful to a referee for two extremely careful reviews of the paper and for providing many constructive comments. The work of Zbigniew Palmowski is partially supported by the Ministry of Science and Higher Education of Poland under the Grant N N201 412239 (2012–2013). The work of Maria Vlasiou is partially supported by an NWO MEERVOUD individual grant through project 632.003.002. Bert Zwart is supported by an NWO VIDI grant and an IBM faculty award, and is also affiliated with Eurandom, VU University Amsterdam, and Georgia Institute of Technology.


  1. 1.
    Asmussen, S.: Applied Probability and Queues. Springer, New York (2003)Google Scholar
  2. 2.
    Bekker, R., Boxma, O.J., Kella, O.: Queues with delays in two-stage startegies and Lévy input. J. Appl. Probab. 45, 314–332 (2008)CrossRefGoogle Scholar
  3. 3.
    Bekker, R., Boxma, O.J., Resing, J.A.C.: Lévy processes with adaptable exponent. Adv. Appl. Probab. 41, 177–205 (2009)CrossRefGoogle Scholar
  4. 4.
    Berman, O., Perry, D., Stadje, W.: A fluid EOQ model with a two-state random environment. Probab. Eng. Inf. Sci. 20, 329–349 (2006)CrossRefGoogle Scholar
  5. 5.
    Bertoin, J.: Lévy Processes. No. 121 in Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1996)Google Scholar
  6. 6.
    Browne, S., Zipkin, P.: Inventory models with continuous stochastic demands. Ann. Appl. Probab. 1, 419–435 (1991)CrossRefGoogle Scholar
  7. 7.
    Dȩbicki, K., Mandjes, M.: Lévy driven queues. Surv. Oper. Res. Manage. Sci. 17, 15–37 (2012)Google Scholar
  8. 8.
    Dshalalow, J.H.: Queueing systems with state dependent parameters. In: Frontiers in queueing. Probability Stochastics Series. pp. 61–116. CRC, Boca Raton, FL (1997)Google Scholar
  9. 9.
    El-Taha, M.: A sample-path condition for the asymptotic uniform distribution of clearing processes. Optimization 51, 965–975 (2002)CrossRefGoogle Scholar
  10. 10.
    Foss, S., Konstantopoulos, T.: An overview of some stochastic stability methods. J. Oper. Res. Soc. Jpn. 47, 275–303 (2004)Google Scholar
  11. 11.
    Ghiani, G., Laporte, G., Musmanno, R.: Introduction to Logistics Systems Planning and Control. Wiley, Chichester (2005)Google Scholar
  12. 12.
    Guillemin, F., Robert, P., Zwart, B.: AIMD algorithms and exponential functionals. Ann. Appl. Probab. 14, 90–117 (2004)CrossRefGoogle Scholar
  13. 13.
    Hubalek, F., Kyprianou, A.E.: Old and new examples of scale functions for spectrally negative Lévy processes. In: Dalang, R., Dozzi, M., Russo, F. (eds.) Seminar on Stochastic Analysis. Random Fields and Applications VI. Progress in Probability, pp. 119–145. Springer, New York (2011)Google Scholar
  14. 14.
    Kella, O., Perry, D., Stadje, W.: A stochastic clearing model with a Brownian and a compound Poisson component. Probab. Eng. Inf. Sci. 17, 1–22 (2003)CrossRefGoogle Scholar
  15. 15.
    Kyprianou, A.E.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext, Springer, Berlin (2006)Google Scholar
  16. 16.
    Kyprianou, A.E., Palmowski, Z.: A martingale review of some fluctuation theory for spectrally negative Lévy processes. In: Séminaire de Probabilités XXXVIII. vol. 1857 of Lecture Notes in Math, pp. 16–29. Springer, Berlin (2005)Google Scholar
  17. 17.
    Kyprianou, A., Rivero, V.: Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Probab. 13, 1672–1701 (2008)Google Scholar
  18. 18.
    Lambert, A.: Completely asymmetric Lévy processes confined in a finite interval. Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 36, 251–274 (2000)CrossRefGoogle Scholar
  19. 19.
    Maulik, K., Zwart, B.: An extension of the square root law of TCP. Ann. Oper. Res. 170, 217–232 (2009)CrossRefGoogle Scholar
  20. 20.
    Palmowski, Z., Vlasiou, M.: A Lévy input model with additional state-dependent services. Stoch. Process. Appl. 121, 1546–1564 (2011)CrossRefGoogle Scholar
  21. 21.
    Pistorius, M.: Exit problems of Lévy processes with applications in finance. PhD thesis. Utrecht University, The Netherlands (2003)Google Scholar
  22. 22.
    Serfozo, R., Stidham, S.: Semi-stationary clearing processes. Stoch. Process. Appl. 6, 165–178 (1978)CrossRefGoogle Scholar
  23. 23.
    Song, J., Zipkin, P: Inventory control with information about supply conditions. Manage. Sci. 42, 1409–1419 (1993)Google Scholar
  24. 24.
    Suprun, V.N.: The ruin problem and the resolvent of a killed independent increment process. Akademiya Nauk Ukrainskoĭ SSR. Institut Matematiki. Ukrainskiĭ Matematicheskiĭ Zhurnal 28, 53–61, 142 (1976)Google Scholar
  25. 25.
    Whitt, W.: The stationary distribution of a stochastic clearing process. Oper. Res. 29, 294–308 (1981)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.University of WrocławWrocławPoland
  2. 2.Eindhoven University of TechnologyEindhovenThe Netherlands
  3. 3.CWIAmsterdamThe Netherlands

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