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

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## Abstract

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.

## Keywords

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## Notes

### Acknowledgments

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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