Journal of Applied Mathematics and Stochastic Analysis
Volume 12 (1999), Issue 1, Pages 35-62

A queueing system with queue length dependent service times, with applications to cell discarding in ATM networks

Doo Il Choi, Charles Knessl, and Charles Tier

University of Illinois at Chicago, 851 South Morgan Street, Chicago 60607, IL, USA

Received 1 September 1997; Revised 1 August 1998

Copyright © 1999 Doo Il Choi et al. 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.


A queueing system (M/G1,G2/1/K) is considered in which the service time of a customer entering service depends on whether the queue length, N(t), is above or below a threshold L. The arrival process is Poisson, and the general service times S1 and S2 depend on whether the queue length at the time service is initiated is <L or L, respectively. Balance equations are given for the stationary probabilities of the Markov process (N(t),X(t)), where X(t) is the remaining service time of the customer currently in service. Exact solutions for the stationary probabilities are constructed for both infinite and finite capacity systems. Asymptotic approximations of the solutions are given, which yield simple formulas for performance measures such as loss rates and tail probabilities. The numerical accuracy of the asymptotic results is tested.