# Many-server Gaussian limits for overloaded non-Markovian queues with customer abandonment

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## Abstract

Extending Ward Whitt’s pioneering work “Fluid Models for Multiserver Queues with Abandonments, Operations Research, 54(1) 37–54, 2006,” this paper establishes a many-server heavy-traffic functional central limit theorem for the overloaded \(G{/}GI{/}n+GI\) queue with stationary arrivals, nonexponential service times, *n* identical servers, and nonexponential patience times. Process-level convergence to non-Markovian Gaussian limits is established as the number of servers goes to infinity for key performance processes such as the waiting times, queue length, abandonment and departure processes. Analytic formulas are developed to characterize the distributions of these Gaussian limits.

## Keywords

Many-server queues Many-server heavy-traffic limits Nonexponential service times Efficiency-driven regime Customer abandonment Gaussian approximation Functional central limit theorem## Mathematics Subject Classification

60B12 60B05 60F17 60K25 60M20 90B22## Notes

### Acknowledgements

We thank editors Amy Ward and Guodong Pang for inviting us to contribute to this special issue and anonymous referees for providing constructive comments. The third author would like to thank Ward Whitt for his support and guidance through the years, and for being a tremendous source of inspiration. The first and third authors acknowledge supports from NSF Grant CMMI 1362310.

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