Abstract
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.
Similar content being viewed by others
References
Asmussen, S.: Applied Probability and Queues. Springer, New York (2003)
Bekker, R., Boxma, O.J., Kella, O.: Queues with delays in two-stage startegies and Lévy input. J. Appl. Probab. 45, 314–332 (2008)
Bekker, R., Boxma, O.J., Resing, J.A.C.: Lévy processes with adaptable exponent. Adv. Appl. Probab. 41, 177–205 (2009)
Berman, O., Perry, D., Stadje, W.: A fluid EOQ model with a two-state random environment. Probab. Eng. Inf. Sci. 20, 329–349 (2006)
Bertoin, J.: Lévy Processes. No. 121 in Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1996)
Browne, S., Zipkin, P.: Inventory models with continuous stochastic demands. Ann. Appl. Probab. 1, 419–435 (1991)
Dȩbicki, K., Mandjes, M.: Lévy driven queues. Surv. Oper. Res. Manage. Sci. 17, 15–37 (2012)
Dshalalow, J.H.: Queueing systems with state dependent parameters. In: Frontiers in queueing. Probability Stochastics Series. pp. 61–116. CRC, Boca Raton, FL (1997)
El-Taha, M.: A sample-path condition for the asymptotic uniform distribution of clearing processes. Optimization 51, 965–975 (2002)
Foss, S., Konstantopoulos, T.: An overview of some stochastic stability methods. J. Oper. Res. Soc. Jpn. 47, 275–303 (2004)
Ghiani, G., Laporte, G., Musmanno, R.: Introduction to Logistics Systems Planning and Control. Wiley, Chichester (2005)
Guillemin, F., Robert, P., Zwart, B.: AIMD algorithms and exponential functionals. Ann. Appl. Probab. 14, 90–117 (2004)
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)
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)
Kyprianou, A.E.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext, Springer, Berlin (2006)
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)
Kyprianou, A., Rivero, V.: Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Probab. 13, 1672–1701 (2008)
Lambert, A.: Completely asymmetric Lévy processes confined in a finite interval. Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 36, 251–274 (2000)
Maulik, K., Zwart, B.: An extension of the square root law of TCP. Ann. Oper. Res. 170, 217–232 (2009)
Palmowski, Z., Vlasiou, M.: A Lévy input model with additional state-dependent services. Stoch. Process. Appl. 121, 1546–1564 (2011)
Pistorius, M.: Exit problems of Lévy processes with applications in finance. PhD thesis. Utrecht University, The Netherlands (2003)
Serfozo, R., Stidham, S.: Semi-stationary clearing processes. Stoch. Process. Appl. 6, 165–178 (1978)
Song, J., Zipkin, P: Inventory control with information about supply conditions. Manage. Sci. 42, 1409–1419 (1993)
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)
Whitt, W.: The stationary distribution of a stochastic clearing process. Oper. Res. 29, 294–308 (1981)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Palmowski, Z., Vlasiou, M. & Zwart, B. A Lévy input fluid queue with input and workload regulation. Queueing Syst 76, 21–36 (2014). https://doi.org/10.1007/s11134-013-9358-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-013-9358-6