A heavy traffic limit theorem for a class of open queueing networks with finite buffers

Authors:
J. G. Dai;W. Dai
Affiliations:
School of Industrial and Systems Engineering, and School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA E-mail: dai@isye.gatech.edu;School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA
Venue:
Queueing Systems: Theory and Applications
Year:
1999

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Abstract

We consider a queueing network of d single server stations. Each station has a finite capacity waiting buffer, and all customers served at a station are homogeneous in terms of service requirements and routing. The routing is assumed to be deterministic and hence feedforward. A server stops working when the downstream buffer is full. We show that a properly normalized d-dimensional queue length process converges in distribution to a d-dimensional semimartingale reflecting Brownian motion (RBM) in a d-dimensional box under a heavy traffic condition. The conventional continuous mapping approach does not apply here because the solution to our Skorohod problem may not be unique. Our proof relies heavily on a uniform oscillation result for solutions to a family of Skorohod problems. The oscillation result is proved in a general form that may be of independent interest. It has the potential to be used as an important ingredient in establishing heavy traffic limit theorems for general finite buffer networks.