A dual approach to network calculus applying the legendre transform

Abstract

Network calculus has successfully been applied to derive service guarantees for per-flow Integrated Services networks. Recent extensions also allow providing performance bounds for aggregate-based Differentiated Services domains, but with a significant increase in complexity. Further on, a number of issues still remain unsolved or are not well understood yet. Founded on convolution and de-convolution, network calculus obeys a strong analogy to system theory. However, system theory has been extended beyond the time domain, applying the Fourier transform and allowing for an efficient analysis in the frequency domain. A corresponding dual domain for network calculus has not been elaborated, so far. In this paper we show that in analogy to system theory such a dual domain for network calculus is given by the Legendre transform. We provide solutions for dual operations and show that min-plus convolution and de-convolution become simple addition and subtraction in Legendre space. Finally we address an aggregate scheduling example, where the Legendre transform allows deriving a new, explicit, and clear solution. © Springer-Verlag Berlin Heidelberg 2005.