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

TL;DR

The paper extends the Network Calculus algorithmic toolbox to include lower and upper pseudo-inverses and composition for ultimately pseudo-periodic (UPP) curves, enabling automated, correct computation of these operations. It introduces by-curve and by-sequence frameworks, proves that pseudo-inverses and composition preserve the UPP class, and provides linear-time algorithms in the number of segments, with specialized UA-based optimizations for efficiency. A proof-of-concept demonstrates replication of a recent NC result (IWRR) and showcases significant runtime improvements via the optimized UA cases, all implemented in the open-source Nancy library. This work enhances practical NC analyses for complex schedulers and cross-traffic scenarios, facilitating scalable end-to-end 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.

Figures (8)

• Figure 1: Example of leaky-bucket shaper, taken from andreozzi2020heterogeneous. The traffic process $A(t)$ is always below the arrival curve $\alpha(t)$ and its translations along $A(t)$.
• Figure 2: Graphical interpretation of the convolution operation.
• Figure 3: Graphical example of a delay bound.
• Figure 4: Example of ultimately pseudo-periodic piecewise affine function $f$ and its representation $R_f$.
• Figure 5: Example of lower pseudo-inverse of a sequence $S$. Since $S$ is left-continuous, $S = \left(S^{-1}_{\downarrow}\right)^{-1}_{\downarrow}$

Theorems & Definitions (32)

• Definition 1: Piecewise Affine Function bouillard2008algorithmic
• Definition 2: Ultimately Pseudo-Periodic Function bouillard2008algorithmic
• Definition 3: Point
• Definition 4: Segment
• Definition 5: Sequence
• Definition 6: Ultimately Affine Function
• Definition 7: Weakly Ultimately Infinite Function
• Definition 8: Lower and Upper Pseudo-Inverse Liebeherr2017
• Theorem 9
• proof