Manyserver asymptotics for JointheShortest Queue in the SuperHalfinWhitt Scaling Window
Abstract
The JointheShortest Queue (JSQ) policy is a classical benchmark for the performance of manyserver queueing systems due to its strong optimality properties. While the exact analysis of the JSQ policy is an open question to date, even under Markovian assumption on the service requirements, recently, there has been a significant progress in understanding its manyserver asymptotic behavior since the work of Eschenfeldt and Gamarnik (Math.~Oper.~Res.~43 (2018) 867886). In this paper, we analyze the manyserver limits of the JSQ policy in the \emph{superHalfinWhitt} scaling window when load per server $\lambda_N$ scales with the system size $N$ as $\lim_{N\rightarrow\infty}N^{\alpha}(1\lambda_N)=\beta$ for $\alpha\in (1/2, 1)$ and $\beta>0$. We establish that the centered and scaled total queue length process converges to a certain Bessel process with negative drift and the associated centered and scaled steadystate total queue length, indexed by $N$, converges to a $\mathrm{Gamma}(2,\beta)$ distribution. Both the transient and steadystate limit laws are universal in the sense that they do not depend on the value of the scaling parameter $\alpha$, and exhibit fundamentally different qualitative behavior from both the HalfinWhitt regime ($\alpha = 1/2$) and the Nondegenerate Slowdown (NDS) regime ($\alpha=1$).
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 arXiv:
 arXiv:2106.00121
 Bibcode:
 2021arXiv210600121Z
 Keywords:

 Mathematics  Probability;
 Primary: 60K25;
 60J60;
 Secondary: 60K05;
 60H20
 EPrint:
 69 pages