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Shot noise processes with randomly delayed cluster arrivals and dependent noises in the large-intensity regime

Part of: Stochastic processes Mathematical economics Special processes Operations research and management science

Published online by Cambridge University Press:  22 November 2021

Bo Li  and
Guodong Pang
Bo Li*
Affiliation:
Nankai University
Guodong Pang*
Affiliation:
Pennsylvania State University
*
*Postal address: School of Mathematics and LPMC, Nankai University, Tianjin, 300071 China. Email address: libo@nankai.edu.cn
**Postal address: The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, College of Engineering, Pennsylvania State University, University Park, PA 16802, USA. Email address: gup3@psu.edu
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Abstract

We study shot noise processes with cluster arrivals, in which entities in each cluster may experience random delays (possibly correlated), and noises within each cluster may be correlated. We prove functional limit theorems for the process in the large-intensity asymptotic regime, where the arrival rate gets large while the shot shape function, cluster sizes, delays, and noises are unscaled. In the functional central limit theorem, the limit process is a continuous Gaussian process (assuming the arrival process satisfies a functional central limit theorem with a Brownian motion limit). We discuss the impact of the dependence among the random delays and among the noises within each cluster using several examples of dependent structures. We also study infinite-server queues with cluster/batch arrivals where customers in each batch may experience random delays before receiving service, with similar dependence structures.

Keywords

Shot noise processinfinite-server queuescluster arrival(dependent) random delays of arrivalsdependent noiseslarge-intensity asymptotic regimeGaussian limit processes

MSC classification

Primary: 60G15: Gaussian processes 60G20: Generalized stochastic processes 60G17: Sample path properties
Secondary: 60K25: Queueing theory 90B22: Queues and service 91B30: Risk theory, insurance 60G55: Point processes
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 4 , December 2021 , pp. 1190 - 1221
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Copyright
© The Author(s) 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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