Extending the Network Calculus Algorithmic Toolbox for Ultimately Pseudo-Periodic Functions: Pseudo-Inverse and Composition

TL;DR

This paper extends the Network Calculus algorithmic toolbox by developing algorithms for the pseudo-inverse and composition operations on ultimately pseudo-periodic functions, enabling more comprehensive performance analysis tools.

Contribution

It introduces and proves the correctness of algorithms for pseudo-inverse and composition operations on UPP curves within Network Calculus, expanding computational capabilities.

Findings

Algorithms for pseudo-inverse operations on UPP curves are now available.

The composition operation for UPP curves has been formalized and validated.

The extended toolbox improves the computational handling of complex network performance guarantees.

Abstract

Network Calculus (NC) is an algebraic theory that represents traffic and service guarantees as curves in a Cartesian plane, in order to compute performance guarantees for flows traversing a network. NC uses transformation operations, e.g., min-plus convolution of two curves, to model how the traffic profile changes with the traversal of network nodes. Such operations, while mathematically well-defined, can quickly become unmanageable to compute using simple pen and paper for any non-trivial case, hence the need for algorithmic descriptions. Previous work identified the class of piecewise affine functions which are ultimately pseudo-periodic (UPP) as being closed under the main NC operations and able to be described finitely. Algorithms that embody NC operations taking as operands UPP curves have been defined and proved correct, thus enabling software implementations of these operations. However, recent advancements in NC make use of operations, namely the lower pseudo-inverse, upper pseudo-inverse, and composition, that are well defined from an algebraic standpoint, but whose algorithmic aspects have not been addressed yet. In this paper, we introduce algorithms for the above operations when operands are UPP curves, thus extending the available algorithmic toolbox for NC. We discuss the algorithmic properties of these operations, providing formal proofs of correctness.