Deterministic Traffic Specification via Projections under the Min-plus Algebra

In this paper, we address the parameterization problem for traffic envelopes needed for deterministic traffic regulation and service guarantees. A parameterized function is a good "substitute" for a traffic envelope if (i) the substitute is not smaller than the envelope and (ii) no other functions not smaller than the envelope are smaller than the substitute. Analogous to the least square approximation problem in a vector space, we use projections under the (min; +)-algebra to find a substitute for a traffic envelope. To facilitate the computation of operations under the (min; +)- algebra, we develop the concept of ordered orthogonal bases. A substitute for a traffic envelope can be represented by a coordinate vector with respect to an ordered orthogonal basis. Operations under the (min; +)-algebra, including pointwise minimum, convolution, subadditive closure, and pointwise maximum, can then be computed on the domain of coordinate vectors. A substitute and its coordinate vector forms a transform pair, called C-transform in the paper. The C-transform is related to the Legendre (or convex) transform and has many properties such as Parseval's formula

An Algorithmic Toolbox for Network Calculus

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

An Algorithmic Toolbox for Network Calculus

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