Queue Length Asymptotics for Generalized Max-Weight Scheduling in the presence of Heavy-Tailed Traffic

We investigate the asymptotic behavior of the steady-state queue length distribution under generalized max-weight scheduling in the presence of heavy-tailed traffic. We consider a system consisting of two parallel queues, served by a single server. One of the queues receives heavy-tailed traffic, and the other receives light-tailed traffic. We study the class of throughput optimal max-weight-alpha scheduling policies, and derive an exact asymptotic characterization of the steady-state queue length distributions. In particular, we show that the tail of the light queue distribution is heavier than a power-law curve, whose tail coefficient we obtain explicitly. Our asymptotic characterization also contains an intuitively surprising result - the celebrated max-weight scheduling policy leads to the worst possible tail of the light queue distribution, among all non-idling policies. Motivated by the above negative result regarding the max-weight-alpha policy, we analyze a log-max-weight (LMW) scheduling policy. We show that the LMW policy guarantees an exponentially decaying light queue tail, while still being throughput optimal

A queueing theory description of fat-tailed price returns in imperfect financial markets

In a financial market, for agents with long investment horizons or at times of severe market stress, it is often changes in the asset price that act as the trigger for transactions or shifts in investment position. This suggests the use of price thresholds to simulate agent behavior over much longer timescales than are currently used in models of order-books. We show that many phenomena, routinely ignored in efficient market theory, can be systematically introduced into an otherwise efficient market, resulting in models that robustly replicate the most important stylized facts. We then demonstrate a close link between such threshold models and queueing theory, with large price changes corresponding to the busy periods of a single-server queue. The distribution of the busy periods is known to have excess kurtosis and non-exponential decay under various assumptions on the queue parameters. Such an approach may prove useful in the development of mathematical models for rapid deleveraging and panics in financial markets, and the stress-testing of financial institutions

Transient bayesian inference for short and long-tailed GI/G/1 queueing systems

In this paper, we describe how to make Bayesian inference for the transient behaviour and busy period in a single server system with general and unknown distribution for the service and interarrival time. The dense family of Coxian distributions is used for the service and arrival process to the system. This distribution model is reparametrized such that it is possible to define a non-informative prior which allows for the approximation of heavytailed distributions. Reversible jump Markov chain Monte Carlo methods are used to estimate the predictive distribution of the interarrival and service time. Our procedure for estimating the system measures is based in recent results for known parameters which are frequently implemented by using symbolical packages. Alternatively, we propose a simple numerical technique that can be performed for every MCMC iteration so that we can estimate interesting measures, such as the transient queue length distribution. We illustrate our approach with simulated and real queues

Inference for double Pareto lognormal queues with applications

In this article we describe a method for carrying out Bayesian inference for the double Pareto lognormal (dPlN) distribution which has recently been proposed as a model for heavy-tailed phenomena. We apply our approach to inference for the dPlN/M/1 and M/dPlN/1 queueing systems. These systems cannot be analyzed using standard techniques due to the fact that the dPlN distribution does not posses a Laplace transform in closed form. This difficulty is overcome using some recent approximations for the Laplace transform for the Pareto/M/1 system. Our procedure is illustrated with applications in internet traffic analysis and risk theory