Fluid queues and regular variation

This paper considers a fluid queueing system, fed by $N$ independent sources that alternate between silence and activity periods. We assume that the distribution of the activity periods of one or more sources is a regularly varying function of index $zeta$. We show that its fat tail gives rise to an even fatter tail of the buffer content distribution, viz., one that is regularly varying of index $zeta +1$. In the special case that $zeta in (-2,-1)$, which implies long-range dependence of the input process, the buffer content does not even have a finite first moment. As a queueing-theoretic by-product of the analysis of the case of $N$ identical sources, with $N rightarrow infty$, we show that the busy period of an M/G/$infty$ queue is regularly varying of index $zeta$ iff the service time distribution is regularly varying of index $zeta$

Workloads and waiting times in single-server systems with multiple customer classes

One of the most fundamental properties that single-server multi-class service systems may possess is the property of work conservation. Under certain restrictions, the work conservation property gives rise to a conservation law for mean waiting times, i.e., a linear relation between the mean waiting times of the various classes of customers. This paper is devoted to single-server multi-class service systems in which work conservation is violated in the sense that the server's activities may be interrupted although work is still present. For a large class of such systems with interruptions, a decomposition of the amount of work into two independent components is obtained; one of these components is the amount of work in the corresponding systemwithout interruptions. The work decomposition gives rise to a (pseudo)conservation law for mean waiting times, just as work conservation did for the system without interruptions

Analysis of an M/G/1 queue with customer impatience and an adaptive arrival process

We study an M/G/1 queue with impatience and an adaptive arrival process. The rate of the arrival process changes according to whether an incoming customer is accepted or rejected. We analyse two different models for impatience : (i) based on workload, and (ii) based on queue length. For the workload-based model, we obtain the Laplace-Stieltjes Transform of the joint stationary workload and arrival rate process, and that of the waiting time. For the queue-length based model we obtain the analogous z-transform. These queueing models also capture the interaction between congestion control algorithms and queue management schemes in the Internet