Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-02T21:20:52.905Z Has data issue: false hasContentIssue false
  • English
  • Français

Functional limit theorems for the number of busy servers in a G/G/∞ queue

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  28 March 2018

Alexander Iksanov ,
Wissem Jedidi  and
Fethi Bouzeffour
Alexander Iksanov*
Affiliation:
Taras Shevchenko National University of Kyiv and University of Wrocław
Wissem Jedidi*
Affiliation:
King Saud University and Université de Tunis El Manar
Fethi Bouzeffour*
Affiliation:
King Saud University
*
* Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine. Email address: [email protected]
** Postal address: Department of Statistics & OR, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia. Email address: [email protected]
*** Postal address: Department of Mathematics, College of Sciences, King Saud University, Riyadh, 11451, Saudi Arabia. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss weak convergence of the number of busy servers in a G/G/∞ queue in the J1-topology on the Skorokhod space. We prove two functional limit theorems with random and nonrandom centering, thereby solving two open problems stated in Mikosch and Resnick (2006). A new integral representation for the limit Gaussian process is given.

Keywords

Functional limit theoremG/G/∞ queueperturbed random walktightness

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 15 - 29
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alsmeyer, G., Iksanov, A. and Marynych, A. (2017). Functional limit theorems for the number of occupied boxes in the Bernoulli sieve. Stoch. Process. Appl. 127, 995–1017. Google Scholar
[2]Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York. Google Scholar
[3]Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press. Google Scholar
[4]Chow, Y. S. and Teicher, H. (1997). Probability Theory: Independence, Interchangeability, Martingales, 3rd edn. Springer, New York. Google Scholar
[5]Csörgő, M., Horváth, L. and Steinebach, J. (1987). Invariance principles for renewal processes. Ann. Prob. 15, 1441–1460. Google Scholar
[6]Gut, A. (2009). Stopped Random Walks: Limit Theorems and Applications, 2nd edn. Springer, New York. Google Scholar
[7]Iksanov, A. (2016). Renewal Theory for Perturbed Random Walks and Similar Processes. Birkhäuser, Cham. Google Scholar
[8]Iksanov, A. and Meiners, M. (2010). Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks. J. Appl. Prob. 47, 513–525. Google Scholar
[9]Iksanov, A., Marynych, A. and Meiners, M. (2017). Asymptotics of random processes with immigration I: Scaling limits. Bernoulli 23, 1233–1278. Google Scholar
[10]Itô, K. (1951). Multiple Wiener integral. J. Math. Soc. Japan 3, 157–169. Google Scholar
[11]Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin. Google Scholar
[12]Kaplan, N. (1975). Limit theorems for a GI/G/∞ queue. Ann. Prob. 3, 780–789. Google Scholar
[13]Konstantopoulos, T. and Lin, S.-J. (1998). Macroscopic models for long-range dependent network traffic. Queueing Systems Theory Appl. 28, 215–243. Google Scholar
[14]Krichagina, E. V. and Puhalskii, A. A. (1997). A heavy-traffic analysis of a closed queueing system with a GI/∞ service center. Queueing Systems Theory Appl. 25, 235–280. Google Scholar
[15]Lindvall, T. (1973). Weak convergence of probability measures and random functions in the function space D(0, ∞). J. Appl. Prob. 10, 109–121. Google Scholar
[16]Marynych, A. V. (2015). A note on convergence to stationarity of random processes with immigration. Theory Stoch. Process. 20, 84–100. Google Scholar
[17]Marynych, A. and Verovkin, G. (2017). A functional limit theorem for random processes with immigration in the case of heavy tails. Modern Stoch. Theory Appl. 4, 93–108. Google Scholar
[18]Mikosch, T. and Resnick, S. (2006). Activity rates with very heavy tails. Stoch. Process. Appl. 116, 131–155. Google Scholar
[19]Resnick, S. and Rootzén, H. (2000). Self-similar communication models and very heavy tails. Ann. Appl. Prob. 10, 753–778. Google Scholar
[20]Walsh, J. B. (1986). Martingales with a multidimensional parameter and stochastic integrals in the plane. In Lectures in Probability and Statistics (Lecture Notes Math. 1215), Springer, Berlin, pp. 329–491. Google Scholar