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Approximate waiting times for queuing systems with variable cross-correlated arrival rates

Author

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  • Bogachev, Mikhail I.
  • Pyko, Nikita S.
  • Tymchenko, Nikita
  • Pyko, Svetlana A.
  • Markelov, Oleg A.

Abstract

Modern information and telecommunication, transportation and logistic, economic and financial systems are represented by complex networks exhibiting traffic flows with spatio-temporal long-term persistence. Conventional queuing theory relies largely upon stationary models where traffic flows are assumed independent and are typically characterized by the first two moments of inter-arrival and service time distributions, leading to drastic underestimations of traffic flow delays. Here we extend a recent superstatistical approach focusing on traffic models with variable arrival rates by accounting for interdependent activity patterns on multiple network nodes. We suggest an analytical correction to the conventional stationary queue model given by the Kingman’s formula based on the calculation of aggregated inter-arrival times variability from the variabilities of arrival rates at individual nodes and cross-correlations between them. We confirm our analytical approximations by comparing with computer simulation results and large-batch empirical traffic analysis from the backbone of a major academic network. We believe that our results, in combination with recent data on the effects of long-term temporal persistence in network traffic flow, are applicable to various complex networks not limited to information and telecommunication, transportation, and logistics but also to economics and finance, rainfall and river flow dynamics, water accumulation in reservoirs, and many other research domains exhibiting spatio-temporal interdependence patterns.

Suggested Citation

  • Bogachev, Mikhail I. & Pyko, Nikita S. & Tymchenko, Nikita & Pyko, Svetlana A. & Markelov, Oleg A., 2024. "Approximate waiting times for queuing systems with variable cross-correlated arrival rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 654(C).
  • Handle: RePEc:eee:phsmap:v:654:y:2024:i:c:s0378437124006617
    DOI: 10.1016/j.physa.2024.130152
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    References listed on IDEAS

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    1. Joel E. Cohen, 2019. "Sum of a Random Number of Correlated Random Variables that Depend on the Number of Summands," The American Statistician, Taylor & Francis Journals, vol. 73(1), pages 56-60, January.
    2. Bogachev, Mikhail I. & Kuzmenko, Alexander V. & Markelov, Oleg A. & Pyko, Nikita S. & Pyko, Svetlana A., 2023. "Approximate waiting times for queuing systems with variable long-term correlated arrival rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 614(C).
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    6. Markelov, Oleg & Nguyen Duc, Viet & Bogachev, Mikhail, 2017. "Statistical modeling of the Internet traffic dynamics: To which extent do we need long-term correlations?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 485(C), pages 48-60.
    7. Tsallis, Constantino, 2016. "Inter-occurrence times and universal laws in finance, earthquakes and genomes," Chaos, Solitons & Fractals, Elsevier, vol. 88(C), pages 254-266.
    8. Xi-Yuan Qian & Ya-Min Liu & Zhi-Qiang Jiang & Boris Podobnik & Wei-Xing Zhou & H. Eugene Stanley, 2015. "Detrended partial cross-correlation analysis of two nonstationary time series influenced by common external forces," Papers 1504.02435, arXiv.org, revised Apr 2015.
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