A queueing system with queue length dependent service times, with applications to cell discarding in ATM networks (Q1305821)

State-dependent queues have received wide attention due to their applications in communication systems. An excellent survey of the work on state-dependent queues has been given by \textit{J. H. Dshalalow} [in: Frontiers in queueing: models and applications in science and engineering, 61-116 (1997; Zbl 0871.60076)]. In this paper, the authors consider a state-dependent single server queueing system wherein customers arrive at the queue by a Poisson process. If the queue length at a customer service initiation is less than a threshold \(L\), the service time of the customer follows a distribution with probability density function (pdf) \(b_1(\cdot)\); otherwise, the service time has pdf \(b_2(\cdot)\). The authors analyse this system using the supplementary variable method. Balance equations are given for the stationary probabilities of the process. Exact solutions are constructed for both infinite and finite capacity systems. Asymptotic approximations of the solutions are given, which yield simple formulas for loss rates and tail probabilities. The numerical accuracy of the asymptotic results is tested.