Queueing Systems

, Volume 78, Issue 1, pp 1–29 | Cite as

Validity of heavy-traffic steady-state approximations in many-server queues with abandonment

  • J. G. Dai
  • A. B. Dieker
  • Xuefeng Gao


We consider \(GI/Ph/n+M\) parallel-server systems with a renewal arrival process, a phase-type service time distribution, \(n\) homogenous servers, and an exponential patience time distribution with positive rate. We show that in the Halfin–Whitt regime, the sequence of stationary distributions corresponding to the normalized state processes is tight. As a consequence, we establish an interchange of heavy-traffic and steady-state limits for \(GI/Ph/n+M\) queues.


Diffusion approximations Stationary distribution Geometric Lyapunov function Weak convergence Multi-server queues Customer abandonment Halfin–Whitt regime Phase-type distribution Piecewise OU processes 

Mathematics Subject Classification

60K25 93E15 60J70 



JGD is supported in part by NSF Grants CMMI-0727400, CMMI-0825840, CMMI-1030589, and CNS-1248117. ABD is supported in part by NSF Grant CMMI-1252878, and he also gratefully acknowledges the hospitality of the Korteweg-de Vries Institute.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of ORIECornell UniversityIthacaUSA
  2. 2.H. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  3. 3.Department of Systems Engineering and Engineering ManagementThe Chinese University of Hong KongShatinHong Kong

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