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.

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.

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.

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= −∞.

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.

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]).

Open AccessJournal Article10.1007/S10626-007-0028-X

An Algorithmic Toolbox for Network Calculus

Anne Bouillard, +1 more

- 01 Mar 2008

- Discrete Event Dynamic Systems

- Vol. 18, Iss: 1, pp 3-49

114

TL;DR: 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.

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Most frequently asked questions

1. 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).

2. 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.

3. 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.

4. 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.

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.

Citations

Convex Analysisの二,三の進展について

徹丸山

- 01 Feb 1977

5.9K

Journal Article•10.1109/SURV.2010.020110.00019

Survey of deterministic and stochastic service curve models in the network calculus

Markus Fidler

- 01 Jan 2010

- IEEE Communications Surveys and Tutorial...

TL;DR: In this paper, the authors survey the state-of-the-art of the deterministic and the recent probabilistic network calculus and discuss the use of service curves, its use in the network calculus, and the relation to systems theory under the min-plus algebra.

...read moreread less

213

Proceedings Article•10.1109/INFOCOM.2008.228

Delay Bounds under Arbitrary Multiplexing: When Network Calculus Leaves You in the Lurch...

Jens B. Schmitt, +2 more

- 13 Apr 2008

TL;DR: This paper addresses the problem of bounding the delay of individual traffic flows in feed-forward networks under arbitrary multiplexing and finds that direct application of network calculus results in loose bounds even in seemingly simple scenarios.

...read moreread less

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•Journal Article•10.1007/S10626-007-0028-X

An Algorithmic Toolbox for Network Calculus

Anne Bouillard, +1 more

- 01 Mar 2008

- Discrete Event Dynamic Systems

...read moreread less

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•Proceedings Article•10.1109/INFCOM.2010.5461912

Tight Performance Bounds in the Worst-Case Analysis of Feed-Forward Networks

Anne Bouillard, +2 more

- 14 Mar 2010

TL;DR: The first algorithm which computes the maximum end-to-end delay for a given flow, as well as the maximum backlog at a server, for any feed-forward network under blind multiplexing, with concave arrival curves and convex service curves is described.

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...

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References

•Book

Introduction to Algorithms

Thomas H. Cormen, +2 more

- 01 Jan 1990

TL;DR: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures and presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers.

...read moreread less

24.8K

Convex Analysisの二,三の進展について

徹丸山

- 01 Feb 1977

5.9K

Introduction to Algorithms, 2nd edition.

TH Cormen, +3 more

- 01 Jan 2001

•Book

Network Calculus: A Theory of Deterministic Queuing Systems for the Internet

Jean-Yves Le Boudec, +1 more

- 06 Jul 2001

TL;DR: The application of Network Calculus to the Internet and basic Min-plus and Max-plus Calculus and Optimal Multimedia Smoothing and Adaptive and Packet Scale Rate Guarantees are studied.

...read moreread less

1.8K

•Book

Davenport-Schinzel sequences and their geometric applications

Micha Sharir, +1 more

- 01 Jan 1995

TL;DR: A close to linear bound on the maximum length of Davenport--Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.

...read moreread less

1.1K

...

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An Algorithmic Toolbox for Network Calculus

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AI Agents for this Paper

Most frequently asked questions

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

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

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

4. What is the main issue with the transform?

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

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

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

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

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

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

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Citations

Convex Analysisの二,三の進展について

Survey of deterministic and stochastic service curve models in the network calculus

Delay Bounds under Arbitrary Multiplexing: When Network Calculus Leaves You in the Lurch...

An Algorithmic Toolbox for Network Calculus

Tight Performance Bounds in the Worst-Case Analysis of Feed-Forward Networks

References

Introduction to Algorithms

Convex Analysisの二,三の進展について

Introduction to Algorithms, 2nd edition.

Network Calculus: A Theory of Deterministic Queuing Systems for the Internet

Davenport-Schinzel sequences and their geometric applications

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