Networks with hop-by-hop flow control occur in several contexts, from data centers to systems architectures (e.g., wormhole-routing networks on chip). A worst-case end-to-end delay in such networks can be computed using Network Calculus (NC), an algebraic theory where traffic and service guarantees are represented as curves in a Cartesian plane. NC uses transformation operations, e.g., the min-plus convolution, to model how the traffic profile changes with the traversal of network nodes. NC allows one to model flow-controlled systems, hence one can compute the end-to-end service curve describing the minimum service guaranteed to a flow traversing a tandem of flow-controlled nodes. However, while the algebraic expression of such an end-to-end service curve is quite compact, its computation is often intractable from an algorithmic standpoint: data structures tend to grow quickly to unfeasibly large sizes, making operations intractable, even with as few as three hops. In this paper, we propose computational and algebraic techniques to mitigate the above problem. We show that existing techniques (such as reduction to compact domains) cannot be used in this case, and propose an arsenal of solutions, which include methods to mitigate the data representation space explosion as well as computationally efficient algorithms for the min-plus convolution operation. We show that our solutions allow a significant speedup, enable analysis of previously unfeasible case studies, and -- since they do not rely on any approximation -- still provide exact results.

翻译：采用逐跳流控制的网络广泛存在于数据中心和系统架构（如片上网络中的虫洞路由）等多种场景中。此类网络的最坏情况端到端时延可通过网络演算（一种将流量与服务保障表示为笛卡尔平面曲线的代数理论）计算。网络演算通过最小加卷积等变换操作，模拟流量特征随网络节点遍历的变化过程。该理论可对流控系统建模，从而计算描述流经串联流控节点的流量所获最低服务保障的端到端服务曲线。然而，尽管此类端到端服务曲线的代数表达式较为简洁，其计算在算法层面往往难以处理：数据结构会迅速膨胀至不可行的规模，即便仅有三跳网络，相关操作也可能变得不可解。本文提出计算与代数技术以缓解上述问题。研究表明，现有方法（如压缩域规约技术）在此场景下无法适用，我们提出了一套解决方案组合，包括缓解数据表示空间爆炸的方法，以及针对最小加卷积操作的高效计算算法。实验证明，所提方案能显著加速计算过程，支持此前无法处理的案例研究，且因不依赖任何近似，仍能提供精确结果。