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

Waiting-time approximations in multiqueue systems with cyclic service

This study is devoted to mean waiting-time approximations in a single-server multi-queue model with cyclic service and zero switching times of the server between consecutive queues. Two different service disciplines are considered: exhaustive service and (ordinary cyclic) nonexhaustive service. For both disciplines it is shown how estimates of the mean waiting times at the various queues can be obtained when no explicit information on arrival intensities and service-time distributions is available, while only the utilizations at the queues and the lengths of the busy periods of the system can be measured. In the exhaustive case, a known mean waiting-time approximation is shown to be suitable for our purposes; in the nonexhaustive case, a new approximation has been derived which is simple and yet more accurate than existing approximations. Extensive simulation validates the approximation methods