Mean-field fluctuations at diffusion scale in threshold-based randomized routing for processor sharing systems and applications

Abstract

In this article, we study the fluctuations of the empirical occupation measures of servers around their mean-field limit in a large system of heterogeneous processor sharing servers. It is assumed that there are M different servers grouped by their speeds and that the total number of servers is N. In particular, we study the sensitivity of the fluctuations to arrival rate parameters at the diffusion scale. The job arrival process is assumed to be Poisson with rate $N(\lambda -\frac{\beta }{\sqrt{N}})$ and the job lengths are assumed to be exponentially distributed with unit mean. On arrival, a finite number of servers from each group are selected and the destination server depends on the server occupancy normalized to their speeds and pre-defined thresholds, referred to as the Join-Below-Threshold scheme. We derive Functional Central Limit Theorems (FCLTs) for the fluctuations that enable us to estimate the error of the mean-field approximations to the empirical measures associated with a system with N servers. We then use these results to show mean response time for finite systems can be approximated by the response time given by the mean-field limit and the error is $O\left(\frac{1}{\sqrt{N}}\right)$ for which the constants can be precisely calculated.

KEYWORDS:

• FCLT
• fluctuations
• join below threshold
• mean-field
• processor sharing

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported in part by a grant from Orange (France) through the CROP programme.