Sub-additive service curves in the Network Calculus analysis

TL;DR

This paper revisits Network Calculus, demonstrating that traditional assumptions suffice for analysis, correcting previous claims about negative-valued functions and highlighting issues with feedback control system analysis.

Contribution

It shows conventional non-negative functions are sufficient for analysis, corrects prior claims about negative functions, and identifies stability issues in feedback control system analysis.

Findings

Conventional analysis is valid for all cases discussed.

Negative-valued functions are unnecessary for accurate analysis.

Previous feedback control analysis has stability issues.

Abstract

Network Calculus is a theoretical model that aims at providing upper bounds of worst-case performance (such as delay or buffer occupancy). This is a mathematical framework that handles both network modeling and network analysis. As such it has requirements regarding the space of functions needed for a safe analysis. Namely, the functions need to be non-negative, as they model a quantity of data. This results in some pitfall for the analysis, where hypothesis matter. A recent paper by Hamscher et al. states that allowing functions with negative values can also lead to a valid analysis, in cases that would be untractable with the non-negative assumption results, especially when feedback control is present in the system. In this paper, we show that, on the contrary, a more conventional analysis is possible in all the mentioned cases. The key is a detailed analysis of sub-additive functions. Second, we show that the analysis of complex feedback control systems, presented by Hamscher et al. in a second paper that uses functions with negative values, is unsound and has stability issues. We give a corrected analysis, when possible, with conventional hypotheses.