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

Part of: Limit theorems Special processes

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]
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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 

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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