Abstract
Networks for safety-critical operation must guarantee deterministic bounds on the end-to-end delay of data transmission despite the usually many data flows that all share the available data forwarding resources. Queueing is inevitable and the queueing delay becomes the important impact factor for communication delays. Network Calculus can calculate verifiable delay bounds in networks of such queues and the tighter the bounds are, the less over-provisioning is required when they are used for the design of safety-critical networked systems.
Tightening delay bounds is an important objective of Network Calculus research. In this paper, we focus on the improvement of the overall analysis algorithm bounding delays in feedforward networks. FIFO queueing is widespread in practice, yet, considering it to model the fraction any data flow gets of the forwarding resource turned out to be complex with Network Calculus. The currently only analysis with practically usable performance was developed for tandem topologies. On the other hand, there are sophisticated algorithms for the feedforward analysis without considering the FIFO property. Here, big gains in tightness were achieved by properly extending the algorithms for tandem topologies. We aim at bringing these gains to the FIFO analysis – and the FIFO analysis to feedforward networks. We provide a thorough integration of both – theoretically and with novel tool support. Our new analysis shows a considerable tightness improvement over the feedforward analysis without FIFO considerations as well as a straightforward extension of the FIFO analysis.
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Notes
- 1.
Available at http://dnc.networkcalculus.org.
- 2.
Available at http://cng1.iet.unipi.it/wiki/index.php/Deborah.
- 3.
A tandem has nested interference iff for every pair of flows either both flows do not have common servers or the path of one flow is included in the other flow’s path.
- 4.
The DEBORAH tool is licensed under GPL while the NCorg DNC is licensed under LGPL. Thus, we cannot redistribute DEBORAH and we opted for a new implementation in the NCorg DNC.
- 5.
DEBORAH works with the pseudoaffine curve framework. Although it outputs variables \(s_i\) that relate to the variables \(\theta _i\) from the FIFO residual service curve in Theorem 2, it is not trivial to infer \(\theta _i\) from \(s_i\): for a crossflow indexed with i, we can compute \(\theta _i = hdev(\alpha _i, \beta _i) + s_i\) from its residual service curve \(\beta _i\) and the \(s_i\). However, this requires to additionally create a tree-structure that captures the relative paths of crossflows to the foi on the analyzed tandem, the so-called nesting tree, first.
- 6.
For the resulting output bound \(\gamma _{\sigma ^{\prime }, \rho ^{\prime }} = \min \limits _{\theta _1, ... ,\theta _{|F_x|}} \left( \gamma _{\sigma , \rho } \oslash \beta _{\text {foi}}^{\text {l.o.}} (\theta _1, ..., \theta _{|F_x|})\right) \) holds.
- 7.
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A SFA-FIFO
A SFA-FIFO
SFA-FIFO is the simple hop-by-hop analysis where we set every occurring FIFO parameter given some (output) arrival curve \(\alpha _f\) and (residual) service curve \(\beta _f^{\text {l.o.}}\) to \(\underline{\theta }( \beta _f^{\text {l.o.}},\alpha _f)\) (see Definition 3). Algorithm 3 below shows the details of the approach where 1, ..., n represents the server indices on the path p and \(\text {Flows}(i)\) with \(i \in \{1,...,n\}\) denotes all flows that cross server i.
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Scheffler, A., Bondorf, S. (2021). Network Calculus for Bounding Delays in Feedforward Networks of FIFO Queueing Systems. In: Abate, A., Marin, A. (eds) Quantitative Evaluation of Systems. QEST 2021. Lecture Notes in Computer Science(), vol 12846. Springer, Cham. https://doi.org/10.1007/978-3-030-85172-9_8
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