Abstract
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 point-wise 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.