Conjugate network calculus: A dual approach applying the Legendre transform

Network calculus is a theory of deterministic queuing systems that has successfully been applied to derive performance bounds for communication networks. Founded on min–plus convolution and de-convolution, network calculus obeys a strong analogy to system theory. Yet, system theory has been extended beyond the time domain applying the Fourier transform thereby 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 convex/concave conjugates referred to also as the Legendre transform. We provide solutions for dual operations and show that min–plus convolution and de-convolution become simple addition and subtraction in the Legendre domain. Additionally, we derive expressions for the Legendre domain to determine upper bounds on backlog and delay at a service element and provide representative examples for the application of conjugate network calculus.