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On the Stochastic End-to-End Delay Analysis in Sink Trees Under Independent and Dependent Arrivals

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Measurement, Modelling and Evaluation of Computing Systems (MMB 2020)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12040))

Abstract

Sink trees are a frequent topology in many networked systems; typical examples are multipoint-to-point label switched paths in Multiprotocol Label Switching networks or wireless sensor networks with sensor nodes reporting to a base station. In this paper, we compute end-to-end delay bounds using a stochastic network calculus approach for a flow traversing a sink tree.

For n servers with one flow of interest and n cross-flows, we derive solutions for a general class of arrivals with moment-generating function bounds. Comparing algorithms known from the literature, our results show that, e.g., pay multiplexing only once has to consider less stochastic dependencies in the analysis.

In numerical experiments, we observe that the reduced dependencies to consider, and therefore less applications of Hölder’s inequality, lead to a significant improvement of delay bounds with fractional Brownian motion as a traffic model. Finally, we also consider a sink tree with dependent cross-flows and evaluate the impact on the delay bounds.

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Authors and Affiliations

  1. Distributed Computer Systems (DISCO) Lab, TU Kaiserslautern, Kaiserslautern, Germany

    Paul Nikolaus & Jens Schmitt

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  1. Paul Nikolaus
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  2. Jens Schmitt
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Correspondence to Jens Schmitt .

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  1. Saarland University, Saarbrücken, Germany

    Holger Hermanns

A Appendix

A Appendix

1.1 A.1 Proof of Proposition 3

Proof

We prove the theorem via induction. The base case \( n = 2 \) is already treated in Subsect. 3.1.

Assume now that the induction hypothesis (IH) is true for some \( n \in \mathbb {N}. \) We denote the end-to-end service of tandems of length n by \( S_{\mathrm {e2e}}^n \). Observe that extending the sink tree basically means that we prolong all flows and add one flow that only traverses the last hop. Therefore, we apply the induction hypothesis on the last server n servers \( S_2, \dots , S_{n+1} \) and receive \( S_{\mathrm {e2e}}^n \). Afterwards, we basically apply the base case, as the network is reduced to the network consisting of \( S_1\) and \(S_{\mathrm {e2e}}^n \). This gives

$$\begin{aligned} S_{\mathrm {e2e}}^{n+1} =&\left[ \left( S_{\mathrm {e2e}}^n \otimes S_{1}\right) - A_2\right] ^+\\ \overset{(\text {IH})}{=}&\left[ \left( \left[ \left( \left[ \left( \left[ S_{n+1}-A_{n+2}\right] ^+ \otimes S_{n}\right) -A_{n+1}\right] ^+\otimes \cdots \otimes S_{2}\right) -A_{3}\right] ^+ \otimes S_{1}\right) - A_2\right] ^+. \end{aligned}$$

For the delay bound, it follows that

This finishes the proof.

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Nikolaus, P., Schmitt, J. (2020). On the Stochastic End-to-End Delay Analysis in Sink Trees Under Independent and Dependent Arrivals. In: Hermanns, H. (eds) Measurement, Modelling and Evaluation of Computing Systems. MMB 2020. Lecture Notes in Computer Science(), vol 12040. Springer, Cham. https://doi.org/10.1007/978-3-030-43024-5_9

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  • DOI: https://doi.org/10.1007/978-3-030-43024-5_9

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