Minimal Per-Flow Backlog Bounds at an Aggregate FIFO Server under Piecewise-Linear Arrival Curves

Abstract

Network Calculus (NC) is a versatile methodology based on min-plus algebra to derive worst-case per-flow performance bounds in networked systems with many concurrent flows. In particular, NC can analyze many scheduling disciplines; yet, somewhat surprisingly, an aggregate FIFO server is a notoriously hard case due to its min-plus non-linearity. A resort is to represent the FIFO residual service by a family of functions with a free parameter instead of just a single curve. For simple token-bucket arrival curves, literature provides optimal choices for that free parameter to minimize delay and backlog bounds. In this paper, we tackle the challenge of more general arrival curves than just token buckets. In particular, we derive residual service curves resulting in minimal backlog bounds for general piecewise-linear arrival curves. To that end, we first show that a backlog bound can always be calculated at a breakpoint of either the arrival curve of the flow of interest or its residual service curve. Further, we define a set of curves that characterize the backlog for a fixed breakpoint, depending on the free parameter of the residual service curve. We show that the backlog-minimizing residual service curve family parameter corresponds to the largest intersection of those curves with the arrival curve. In more complex scenarios finding this largest intersection can become inefficient as the search space grows in the number of flows. Therefore, we present an efficient heuristic that finds, in many cases, the optimal parameter or at least a close conservative approximation. This heuristic is evaluated in terms of accuracy and execution time. Finally, we utilize these backlog-minimizing residual service curves to enhance the DiscoDNC tool and observe considerable reductions in the corresponding backlog bounds.