Abstract
This chapter addresses the limiting regime in which the drift of the driving Lévy process is just ‘slightly negative’, commonly referred to as ‘heavy traffic’. Resorting to the steady-state and transient results that were derived in the previous chapters, it appears that we observe an interesting dichotomy, in that one should distinguish between two scenarios that show intrinsically different behavior. In the case that the underlying Lévy process has a finite variance, the appropriately scaled workload process tends to a Brownian motion reflected at 0 (i.e. a Lévy-driven queue with Brownian input). If the variance is infinite, on the contrary, we establish convergence to a Lévy-driven queue fed by an α-stable Lévy motion.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1999)
Boxma, O., Cohen, J.: Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions. Queueing Syst. 33, 177–204 (1999)
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events. Springer, New York (1997)
Furrer, H.: Risk Theory and Heavy-Tailed Lévy Processes. Ph.D. thesis, Eidgenössische Technische Hochschule, Zürich (1997). http://e-collection.ethbib.ethz.ch/eserv/eth:22556/eth-22556-02.pdf.
Glynn, P.: Diffusion approximations. In: Heyman, D., Sobel, M. (eds.) Handbooks on Operations Research & Management Science, vol. 2, pp. 145–198. Elsevier, Amsterdam (1990)
Kingman, J.: The single server queue in heavy traffic. Proc. Camb. Philos. Soc. 57, 902–904 (1961)
Kingman, J.: The heavy traffic approximation in the theory of queues. In: Smith, W.L., Wilkinson, W.E. (eds.) Proceedings of Symposium on Congestion Theory, Chapel Hill, pp. 137–159. University of North Carolina Press, Chapel Hill (1965)
Kosiński, K., Boxma, O., Zwart, B.: Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime. Queueing Syst. 67, 295–304 (2011)
Prohorov, V.: Transition phenomena in queueing processes, I. Litovsk. Mat. Sb. 3, 199–205 (1963)
Pruitt, E.W. The growth of random walks and Lévy processes. Ann. Probab. 9, 948–956 (1981)
Robbins, H., Siegmund, D.: Boundary crossing probabilities for the Wiener process and sample sums. Ann. Math. Stat. 41, 1410–1429 (1970)
Robert, P.: Stochastic Networks and Queues. Springer, Berlin (2003)
Shneer, S., Wachtel, V.: Heavy-traffic analysis of the maximum of an asymptotically stable random walk. Teor. Verojatn. i Primenen. 55, 335–344 (2010). (English translation in Theor. Probab. Appl. 55, 332–341, 2011)
Whitt, W.: Stochastic-Process Limits. Springer, New York (2002)
Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Dębicki, K., Mandjes, M. (2015). Heavy Traffic. In: Queues and Lévy Fluctuation Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-20693-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-20693-6_5
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-20692-9
Online ISBN: 978-3-319-20693-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)