Processor-Shared Buffers With Reneging

Motivated by buffer design questions in data networks, we consider a processor shared queue with finite buffer capacity and reneging. Reneging models events such as the timing-out of a communication path due to excessive delays, or decisions to preempt a path in order to re-route it according to some congestion control mechanism. The focus of the paper is on the asymptotic behavior of this system in a variety of regimes that correspond to large service rates (bandwidths). Congestion is measured by the blocking probability P, the mean number in system Q, and a cost function L giving the sum of the blocking and reneging rates normalized by the arrival rate. Convenient parameters of the analysis are the respective ratios rho and eta of the arrival rate to the service rate and the service rate to the reneging rate. Fixing the relative rates at which eta --> infinity and the buffer capacity N --> infinity in an exponential model, asymptotic estimates of P, Q, and L are derived as functions of eta with rho a parameter. Similar results are obtained under a suitable scaling leading to heavy-traffic diffusion limits. These latter results are extended to more general distributional assumptions, and used in the solution of a problem in which bandwidth and reneging parameters must be chosen so as to minimize L.