Annals of Operations Research

, Volume 264, Issue 1–2, pp 157–191 | Cite as

Functional law of the iterated logarithm for multi-server queues with batch arrivals and customer feedback

  • Yongjiang Guo
  • Yunan Liu
  • Renhu Pei
Original Paper


A functional law of the iterated logarithm (FLIL) and its corresponding law of the iterated logarithm (LIL) are established for a multi-server queue with batch arrivals and customer feedback. The FLIL and LIL, which quantify the magnitude of asymptotic fluctuations of the stochastic processes around their mean values, are developed in three cases: underloaded, critically loaded and overloaded, for five performance measures: queue length, workload, busy time, idle time and departure process. Both FLIL and LIL are proved using an approach based on strong approximations.


Functional law of the iterated logarithm Multi-server queue Batch arrival Customer feedback Nonexponential service times Strong approximation 



The first author acknowledge support from NSFC Grant 11471053. The second author also acknowledges support from NSF Grant CMMI 1362310.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina
  2. 2.Industrial and Systems EngineeringNorth Carolina State UniversityRaleighUSA

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