Network Calculus (NC) is an algebraic theory that represents traffic and service guarantees as curves in a Cartesian plane, in order to compute performance guarantees for flows traversing a network. NC uses transformation operations, e.g., min-plus convolution of two curves, to model how the traffic profile changes with the traversal of network nodes. Such operations, while mathematically well-defined, can quickly become unmanageable to compute using simple pen and paper for any non-trivial case, hence the need for algorithmic descriptions. Previous work identified the class of piecewise affine functions which are ultimately pseudo-periodic (UPP) as being closed under the main NC operations and able to be described finitely. Algorithms that embody NC operations taking as operands UPP curves have been defined and proved correct, thus enabling software implementations of these operations. However, recent advancements in NC make use of operations, namely the lower pseudo-inverse, upper pseudo-inverse, and composition, that are well defined from an algebraic standpoint, but whose algorithmic aspects have not been addressed yet. In this paper, we introduce algorithms for the above operations when operands are UPP curves, thus extending the available algorithmic toolbox for NC. We discuss the algorithmic properties of these operations, providing formal proofs of correctness.

翻译：网络演算是一种代数理论，它将流量和服务保证表示为笛卡尔平面中的曲线，以计算流经网络的流的性能保证。网络演算使用变换操作（例如两条曲线的极小加卷积）来建模流量剖面如何随着网络节点的穿越而变化。此类操作虽然在数学上定义明确，但对于任何非平凡情况，使用简单的纸笔计算可能迅速变得难以处理，因此需要算法化描述。先前的研究确定了最终伪周期的分段仿射函数类，该类在主网络演算操作下封闭且能够被有限描述。已有研究定义并证明了以最终伪周期曲线为操作数的网络演算操作算法的正确性，从而实现了这些操作的软件实现。然而，网络演算的最新进展利用了从代数角度定义明确但算法层面尚未得到解决的操作，即下伪逆、上伪逆和复合。本文针对操作数为最终伪周期曲线的情况，为上述操作引入算法，从而扩展了可用的网络演算算法工具箱。我们讨论了这些操作的算法性质，并提供了正确性的形式化证明。