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 $zeta$

On the Large Deviations of a Class of Stationary On/Off Sources which Exhibit Long Range Dependence

We present a class of stationary two-state sources which exhibit long range dependence: We relate the large deviations of their sojourn times to the large deviations of the sources themselves. We calculate the rate-function, on a non-linear scale, for a two-state source whose sojourn times are distributed by a semi-exponential distribution, and we calculate the rate-function for the multiplex of a finite collection of such sources

Generalized processor sharing with long-tailed traffic sources

We analyze the queueing behavior of longtailed traffic sources under the Generalized Processor Sharing (GPS) discipline. GPS-based scheduling algorithms, such as Weighted Fair Queueing, have emerged as important mechanisms for accommodating heterogeneous quality-of-service requirements in integrated-services networks. Under mild stability conditions, we show that the tail behavior of the buffer content of an individual source with long-tailed traffic characteristics is equivalent to the tail behavior when that source is served in isolation at a constant rate which is equal to the link rate minus the aggregate average rate of all other sources. Thus, asymptotically, the buffer content of the source is only affected by the traffic characteristics of the other sources through their aggregate average rate. In particular, the source is essentially immune from excessive activity of sources with 'heavier'-tailed traffic characteristics. This suggests that GPS-based scheduling algorithms provide an effective mechanism for extracting high multiplexing gains, while protecting individual connections

Exact asymptotics for fluid queues fed by multiple heavy-tailed on-off flows

We consider a fluid queue fed by multiple On-Off flows with heavy-tailed (regularly varying) On periods. Under fairly mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a reduced system. The reduced system consists of a ``dominant'' subset of the flows, with the original service rate subtracted by the mean rate of the other flows. We describe how a dominant set may be determined from a simple knapsack formulation. The dominant set consists of a ``minimally critical'' set of On-Off flows with regularly varying On periods. In case the dominant set contains just a single On-Off flow, the exact asymptotics for the reduced system follow from known results. For the case of several On-Off flows, we exploit a powerful intuitive argument to obtain the exact asymptotics. Combined with the reduced-load equivalence, the results for the reduced system provide a characterization of the tail of the workload distribution for a wide range of traffic scenarios

Performance analysis of a fluid queue with random service rate in discrete-time

We consider a fluid queue in discrete time with random service rate. Such a queue has been used in several recent studies on wireless networks where the packets can be arbitrarily fragmented. We provide conditions on finiteness of moments of stationary delay, its Laplace-Stieltjes transform and various approximations under heavy traffic. Results are extended to the case where the wireless link can transmit in only a few slots during a frame

On a Random Sum Formula for the Busy Period of the M|G|Infinity Queue with Applications

A random sum formula is derived for the forward recurrence time associated with the busy period length of the M|G|infinity queue. This result is then used to (1) provide a necessary and sufficient condition for the subexponentiality of this forward recurrence time, and (2) establish a stochastic comparison in the convex increasing (variability) ordering betweenthe busy periods in two M|G|infinity queues with service times comparablein the convex increasing ordering

Heavy Loads and Heavy Tails

The present paper is concerned with the stationary workload of queues with heavy-tailed (regularly varying) characteristics. We adopt a transform perspective to illuminate a close connection between the tail asymptotics and heavy-traffic limit in infinite-variance scenarios. This serves as a tribute to some of the pioneering results of J.W. Cohen in this domain. We specifically demonstrate that reduced-load equivalence properties established for the tail asymptotics of the workload naturally extend to the heavy-traffic limit