Journal of Applied Mathematics and Stochastic Analysis
Volume 7 (1994), Issue 3, Pages 337-356

asymptotics for open-loop window flow control

Arthur W. Berger1 and Ward Whitt2

1AT&T Bell Laboratories, Room HO-3H-601, Holmdel 07730-3030, NJ, USA
2AT&T Bell Laboratories, Room MH-2C-178, Murray Hill 07974-0636, NJ, USA

Received 1 May 1993; Revised 1 October 1993

Copyright © 1994 Arthur W. Berger and Ward Whitt. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


An open-loop window flow-control scheme regulates the flow into a system by allowing at most a specified window size W of flow in any interval of length L. The sliding window considers all subintervals of length L, while the jumping window considers consecutive disjoint intervals of length L. To better understand how these window control schemes perform for stationary sources, we describe for a large class of stochastic input processes the asymptotic behavior of the maximum flow in such window intervals over a time interval [0,T] as T and Lget large, with T substantially bigger than L. We use strong approximations to show that when TLlogT an invariance principle holds, so that the asymptotic behavior depends on the stochastic input process only via its rate and asymptotic variability parameters. In considerable generality, the sliding and jumping windows are asymptotically equivalent. We also develop an approximate relation between the two maximum window sizes. We apply the asymptotic results to develop approximations for the means and standard deviations of the two maximum window contents. We apply computer simulation to evaluate and refine these approximations.