Heavytailed queues in the HalfinWhitt regime
Abstract
We consider the FCFS G/G/n queue in the HalfinWhitt regime, in the presence of heavytailed distributions (i.e. infinite variance). We prove that under minimal assumptions, i.e. only that processing times have finite 1 + epsilon moment and interarrival times have finite second moment, the sequence of stationary queue length distributions, normalized by $n^{\frac{1}{2}}$, is tight. All previous tightness results for the stationary queue length required that processing times have finite 2 + epsilon moment. Furthermore, we develop simple and explicit bounds on the stationary queue length in that setting. When processing times have an asymptotically Pareto tail with index alpha in (1,2), we bound the large deviations behavior of the limiting process, and derive a matching lower bound when interarrival times are Markovian. Interestingly, we find that the large deviations behavior of the limit has a subexponential decay, differing fundamentally from the exponentially decaying tails known to hold in the lighttailed setting, and answering an open question of Gamarnik and Goldberg. For the setting where instead the interarrival times have an asymptotically Pareto tail with index alpha in (1,2), we extend recent results of Hurvich and Reed (who analyzed the case of deterministic processing times) by proving that for general processing time distributions, the sequence of stationary queue length distributions, normalized by $n^{\frac{1}{\alpha}}$, is tight (here we use the scaling of Hurvich and Reed, i.e. HalfinWhittReed regime). We are again able to bound the largedeviations behavior of the limit, and find that our derived bounds do not depend on the particular processing time distribution, and are in fact tight even for the case of deterministic processing times.
 Publication:

arXiv eprints
 Pub Date:
 July 2017
 arXiv:
 arXiv:1707.07775
 Bibcode:
 2017arXiv170707775G
 Keywords:

 Mathematics  Probability;
 60K25