Network Calculus Made Easy

We define general concepts that extend, and simplify, the network calculus developed by Cruz [1,2, 3] for guaranteed quality of service networks. We introduce a definition of service curve, the extended service curve, which is the same for isolated queues and for networks of queues, and makes it possible to model nodes that otherwise do not fit in the calculus of Cruz. We introduce two key technical tools, the $\oplus$ and $\ominus$ operators, on arrival and service curves. We show how their systematic use simplifies the derivation of many fundamental results. We obtain a simple characterization of the output flow from a shaper. We show that, if the shaping and arrival curves are concave (which is the case in practice), then the output of a shaper is still constrained by the same arrival curve as the input, in addition to being constrained by the shaper curve. Lastly, we define a general form of deterministic effective bandwidth that is particularly simple and efficient.