An Algorithmic Toolbox for Network Calculus

TLDR: This paper presents a class containing the piecewise affine functions which are ultimately pseudo-periodic and can be finitely described, which enables us to propose some algorithms for each of the network calculus operations.

Abstract: Network calculus offers powerful tools to analyze the performances in communication networks, in particular to obtain deterministic bounds. This theory is based on a strong mathematical ground, notably by the use of (min,+) algebra. However, the algorithmic aspects of this theory have not been much addressed yet. This paper is an attempt to provide some efficient algorithms implementing network calculus operations for some classical functions. Some functions which are often used are the piecewise affine functions which ultimately have a constant growth. As a first step towards algorithmic design, we present a class containing these functions and closed under the main network calculus operations (min, max, +, -, convolution, subadditive closure, deconvolution): the piecewise affine functions which are ultimately pseudo-periodic. They can be finitely described, which enables us to propose some algorithms for each of the network calculus operations. We finally analyze their computational complexity.

Figures

Figure 12: A simple data structure to store ultimately pseudo-periodic functions.

Figure 4: Issues for some interpolations of functions.

Figure 10: Subadditive closure of the two types of segments.

Figure 6: f is ultimately affine but f∗ is not.

Figure 2: A piecewise affine function with respect to Definition 2.

Figure 8: Convolution of two segments.

Full text

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An Algorithmic Toolbox for Network Calculus
Anne Bouillard, Eric Thierry
To cite this version:
Anne Bouillard, Eric Thierry. An Algorithmic Toolbox for Network Calculus. [Research Report]
RR-6094, INRIA. 2007, pp.44. �inria-00123643v2�

apport 

de recherche
ISSN 0249-6399 ISRN INRIA/RR--6094--FR+ENG
Thème COM
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
An Algorithmic Toolbox for Network Calculus
Anne Bouillard and Éric Thierry
N° 6094
Janvier 2007

Unité de recherche INRIA Rennes
IRISA, Campus universitaire de Beaulieu, 35042 Rennes Cedex (France)
Téléphone : +33 2 99 84 71 00 — Télécopie : +33 2 99 84 71 71
An Algorithmic Toolbox for Network Calculus
Anne Bouillard
∗
and
´
Eric Thierry
†
Th`eme COM — Syst`emes communicants
Projet Distribcom
Rapport de recherche n

6094 — Janvier 2007 — 44 pages
Abstract: Network Calculus offers powerful tools to analyze the performances in communication networks,
in particular to obtain deterministic bounds. This theory is based on a strong mathematical ground, notably
by the use of (min,+) algebra. However the algorithmic aspects of this theory have not been much addressed
yet. This paper is an attempt to provide some efficient algorithms implementing Network Calculus operations
for some classical functions.
Some functions which are often used are the piecewise affine functions which ultimately have a constant
growth. As a first step towards algorithmic design, we present a class containing these functions and closed
under the Network Calculus operations: the piecewise affine functions which are ultimately pseudo-periodic.
They can be finitely described which enables us to propose some algorithms for each of the Network Calculus
operations. We finally analyze their computational complexity.
Key-words: Network Calculus, functional (min,+) algebra, algorithmics, computational complexity.
∗
Anne.Bouillard@bretagne.ens-cachan.fr
†
Eric.Thierry@ens-lyon.fr

Stabilit´e et ´etude algorithmique des op´erations du Network Calculus
R´esum´e : Le Network Calculus est un outil puissant pour analyser les performances des r´eseaux de commu-
nication, en particulier pour obtenir des bornes d´eterministes. Cette th´eorie s’appuie sur l’alg`ebre (min, +) et
ses aspects th´eoriques ont fait l’objet de nombreuses ´etudes. Malgr´e cela, les aspects algorithmiques de cette
th´eorie ont tr`es peu ´et´e regard´es. Dans ce rapport, nous fournissons des algorithmes efficaces pour impl´ementer
les op´erations du Network Calculus.
Les fonctions les plus utilis´ees sont les fonctions affines qui ont ultimement un taux de croissance constant.
Une premi`ere ´etape de notre travail est de pr´esenter une classe stable de fonctions sous les op´erations du Network
Calculus contenant ces fonctions (ce sont les fonctions affines par morceaux ultimement pseudo-p´eriodiques).
Comme ces fonctions sont finiment repr´esentables, on peut trouver des algorithmes pour les op´erations ´etudi´ees.
Nous nous int´eressons aussi `a leur complexit´e.
Mots-cl´es : Network Calculus, alg`ebre des fonctions (min,+), algorithmique, complexit´e

Frequently asked questions

Q1. How do you get the hardness result for non-decreasing functions?

To get the hardness result for non-decreasing functions, in the construction, “lift” the function by considering f(t)+ t instead of f(t). 

Q2. What is the main question raised by all these algorithms?

An important question raised by all these algorithms is whether the computed output is a compressed representation of the output. 

Q3. What is the way to organize the pairwise sums of functions?

Another way to organize the pairwise sums of functions is to use the binary tree constructed with Huffman algorithm (where weights are the number of jump points, i.e. tuples in the data structure), it is proven that the overall complexity is better than the balanced binary tree. 

Q4. What is the main issue with the transform?

Its use seemed promising (e.g. convolution becomes addition for the transformed functions), however one important issue is that this transform is not injective for non-convex functions. 

Q5. What is the smallest rank of pseudo-periodicity of f?

It is an ultimately affine function, and its smallest rank of pseudo-periodicity is exactly Frob(a1, . . . , an)−1. Thus the computation of the smallest rank of pseudo-periodicity of f ∗ is NP-hard. 

Q6. What is the proof for the subadditive closure of ultimately plain functions?

The subadditive closure of ultimately plain function is not necessarily ultimately plain: let f ∈ D (or F) such that f(t) = 0 if t = 2 and = +∞ otherwise, it is ultimately plain but f∗(t) = 0 if t is an even integer and = +∞ otherwise, is not. 

Q7. What is the ultimate pseudo-periodicity of f?

For instance f1(t) = 1 1−(t−btc) and f2(t) = t are both plain and pseudo-periodic but f1 is not locally bounded and min(f1, f2) is not ultimately pseudo-periodic. 

Q8. What is the way to ensure that min(f1) is locally bounded?

As a complement, note that if c′1 ≤ c′2, another way to ensure that min(f1, f2) is locally bounded, ultimately plain and pseudo-periodic, is to keep locally bounded and pseudo-periodicity assumptions on inputs but suppose that only f1 is ultimately plain with ∀t ≥ T1, f1(t) ∈ R and that f2 is not necessarily ultimately plain (it may have +∞ values from T2) but ∀t ≥ T2, f2(t) 6= −∞. 

Q9. What is the definition of a function closed under a set of operations?

A class of functions is closed under some set of operations if combining members of the class with any of these operations outputs (if defined) a function which remains in the class. 

Q10. What is the hardness result for pseudo-periodic functions?

Note that the hardness result is also true when the authors deal with ultimately pseudo-periodic functions in F [Q+, Q], even if they are continuous, non-decreasing and ultimately affine (see [8]).