A Diffusion Approximation for a GI/GI/1 Queue with Balking or Reneging

Authors:
Amy R. Ward;Peter W. Glynn
Affiliations:
School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, USA 30332-0205;Department of Management Science & Engineering, Stanford University, Stanford, USA 94305
Venue:
Queueing Systems: Theory and Applications
Year:
2005

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Abstract

Consider a single-server queue with a renewal arrival process and generally distributed processing times in which each customer independently reneges if service has not begun within a generally distributed amount of time. We establish that both the workload and queue-length processes in this system can be approximated by a regulated Ornstein-Uhlenbeck (ROU) process when the arrival rate is close to the processing rate and reneging times are large. We further show that a ROU process also approximates the queue-length process, under the same parameter assumptions, in a balking model. Our balking model assumes the queue-length is observable to arriving customers, and that each customer balks if his or her conditional expected waiting time is too large.