Time-sensitive networks require timely and accurate monitoring of the status of the network. To achieve this, many devices send packets periodically, which are then aggregated and forwarded to the controller. Bounding the aggregate burstiness of the traffic is then crucial for effective resource management. In this paper, we are interested in bounding this aggregate burstiness for independent and periodic flows. A deterministic bound is tight only when flows are perfectly synchronized, which is highly unlikely in practice and would be overly pessimistic. We compute the probability that the aggregate burstiness exceeds some value. When all flows have the same period and packet size, we obtain a closed-form bound using the Dvoretzky-Kiefer-Wolfowitz inequality. In the heterogeneous case, we group flows and combine the bounds obtained for each group using the convolution bound. Our bounds are numerically close to simulations and thus fairly tight. The resulting aggregate burstiness estimated for a non-zero violation probability is considerably smaller than the deterministic one: it grows in $\sqrt{n\log{n}}$, instead of $n$, where $n$ is the number of flows.

翻译：时间敏感网络需要及时准确地监控网络状态。为此，许多设备周期性发送数据包，这些数据包被聚合后转发至控制器。限制聚合流量的突发性对于有效资源管理至关重要。本文针对独立的周期性流，研究其聚合突发性的上界问题。确定性上界仅在流量完美同步时成立，而现实中这几乎不可能发生，因此过于悲观。我们计算了聚合突发性超过某个值的概率。当所有流具有相同周期和数据包大小时，利用Dvoretzky-Kiefer-Wolfowitz不等式得到了闭式上界。在异构情况下，我们将流分组，并结合卷积界对每组得到的上界进行组合。我们的上界在数值上接近仿真结果，因此相当紧致。对于非零违反概率，估计出的聚合突发性显著小于确定性上界：当流量数为n时，其增长阶为$\sqrt{n\log{n}}$而非$n$。