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

  • Paul Nikolaus
  • Jens SchmittEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12040)


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.


Network calculus Sink trees Moment-generating functions Hölder’s inequality Fractional Brownian motion 


  1. 1.
    Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.P.: Synchronization and Linearity: An Algebra for Discrete Event Systems. Wiley, Hoboken (1992)zbMATHGoogle Scholar
  2. 2.
    Beck, M.: Towards the analysis of transient phases with stochastic network calculus. In: IEEE 17th International Network Strategy and Planning Symposium (Networks 2016) (2016)Google Scholar
  3. 3.
    Beck, M.A.: Advances in theory and applicability of stochastic network calculus. Ph.D. thesis, TU Kaiserslautern (2016)Google Scholar
  4. 4.
    Becker, N., Fidler, M.: A non-stationary service curve model for performance analysis of transient phases. In: 2015 27th International Teletraffic Congress, pp. 116–124. IEEE (2015)Google Scholar
  5. 5.
    Bondorf, S., Nikolaus, P., Schmitt, J.B.: Quality and cost of deterministic network calculus - design and evaluation of an accurate and fast analysis. Proc. ACM Meas. Anal. Comput. Syst. (POMACS) 1(1), 34 (2017)Google Scholar
  6. 6.
    Bouillard, A., Comte, C., de Panafieu, É., Mathieu, F.: Of kernels and queues: when network calculus meets analytic combinatorics. In: 2018 30th International Teletraffic Congress (ITC 30), vol. 2, pp. 49–54. IEEE (2018)Google Scholar
  7. 7.
    Bouillard, A., Thierry, É.: Tight performance bounds in the worst-case analysis of feed-forward networks. Discrete Event Dyn. Syst. 26(3), 383–411 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Boyes, H., Hallaq, B., Cunningham, J., Watson, T.: The industrial internet of things (IIOT): an analysis framework. Comput. Ind. 101, 1–12 (2018)CrossRefGoogle Scholar
  9. 9.
    Burchard, A., Liebeherr, J., Ciucu, F.: On superlinear scaling of network delays. IEEE/ACM Trans. Netw. 19(4), 1043–1056 (2010)CrossRefGoogle Scholar
  10. 10.
    Champati, J.P., Al-Zubaidy, H., Gross, J.: Transient delay bounds for multi-hop wireless networks. CoRR (2018)Google Scholar
  11. 11.
    Chang, C.S.: Stability, queue length, and delay of deterministic and stochastic queueing networks. IEEE Trans. Autom. Control 39(5), 913–931 (1994)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chang, C.S.: Performance Guarantees in Communication Networks. Springer, London (2000). Scholar
  13. 13.
    Ciucu, F., Burchard, A., Liebeherr, J.: Scaling properties of statistical end-to-end bounds in the network calculus. IEEE Trans. Inf. Theory 52(6), 2300–2312 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ciucu, F., Schmitt, J.: Perspectives on network calculus - no free lunch, but still good value. In: Proceedings of the ACM Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications (SIGCOMM 2012), New York, NY, USA, pp. 311–322 (2012)Google Scholar
  15. 15.
    Cruz, R.L.: Quality of service management in integrated services networks. In: Proceedings of the Semi-Annual Research Review, CWC, UCSD (1996)Google Scholar
  16. 16.
    Dong, F., Wu, K., Srinivasan, V.: Copula analysis for statistical network calculus. In: Proceedings of the IEEE INFOCOM 2015, pp. 1535–1543 (2015)Google Scholar
  17. 17.
    Fettweis, G.P.: The tactile internet: applications and challenges. IEEE Veh. Technol. Mag. 9(1), 64–70 (2014)CrossRefGoogle Scholar
  18. 18.
    Fidler, M.: An end-to-end probabilistic network calculus with moment generating functions. In: Proceedings of the IEEE IWQoS 2006, pp. 261–270 (2006)Google Scholar
  19. 19.
    Fidler, M., Rizk, A.: A guide to the stochastic network calculus. IEEE Commun. Surv. Tutor. 17(1), 92–105 (2015)CrossRefGoogle Scholar
  20. 20.
    Fonseca, N.L., Mayor, G.S., Neto, C.A.: On the equivalent bandwidth of self-similar sources. ACM Trans. Model. Comput. Simul. (TOMACS) 10(2), 104–124 (2000)CrossRefGoogle Scholar
  21. 21.
    Jafari, F., Lu, Z., Jantsch, A., Yaghmaee, M.H.: Buffer optimization in network-on-chip through flow regulation. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 29(12), 1973–1986 (2010)CrossRefGoogle Scholar
  22. 22.
    Jasperneite, J., Neumann, P., Theis, M., Watson, K.: Deterministic real-time communication with switched ethernet. In: 4th IEEE International Workshop on Factory Communication Systems, pp. 11–18. IEEE (2002)Google Scholar
  23. 23.
    Jiang, Y., Liu, Y.: Stochastic Network Calculus, vol. 1. Springer, London (2008). Scholar
  24. 24.
    Kelly, F.P.: Notes on effective bandwidths. In: Kelly, F.P., Zachary, S., Ziedins, I. (eds.) Stochastic Networks: Theory and Applications. Royal Statistical Society Lecture Notes Series, vol. 4, pp. 141–168. Oxford University Press, Oxford (1996)Google Scholar
  25. 25.
    Koubaa, A., Alves, M., Tovar, E.: Modeling and worst-case dimensioning of cluster-tree wireless sensor networks. In: 2006 27th IEEE International Real-Time Systems Symposium (RTSS 2006), pp. 412–421. IEEE (2006)Google Scholar
  26. 26.
    Leiserson, C.E.: Fat-trees: universal networks for hardware-efficient supercomputing. IEEE Trans. Comput. 100(10), 892–901 (1985)CrossRefGoogle Scholar
  27. 27.
    Li, C., Burchard, A., Liebeherr, J.: A network calculus with effective bandwidth. IEEE/ACM Trans. Netw. 15(6), 1442–1453 (2007)CrossRefGoogle Scholar
  28. 28.
    Liebeherr, J., Burchard, A., Ciucu, F.: Delay bounds in communication networks with heavy-tailed and self-similar traffic. IEEE Trans. Inf. Theory 58(2), 1010–1024 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Mellia, M., Stoica, I., Zhang, H.: Tcp model for short lived flows. IEEE Commun. Lett. 6(2), 85–87 (2002)CrossRefGoogle Scholar
  30. 30.
    Mitrinovic, D.S., Vasic, P.M.: Analytic Inequalities, vol. 1. Springer, Heidelberg (1970)CrossRefGoogle Scholar
  31. 31.
    Nelson, R.: Probability, Stochastic Processes, and Queueing Theory: The Mathematics of Computer Performance Modeling. Springer, New York (1995). Scholar
  32. 32.
    Nikolaus, P., Schmitt, J.: On per-flow delay bounds in tandem queues under (in)dependent arrivals. In: Proceedings of 16th IFIP Networking 2017 Conference (NETWORKING 2017). IEEE (2017)Google Scholar
  33. 33.
    Nikolaus, P., Schmitt, J., Ciucu, F.: Dealing with dependence in stochastic network calculus - using independence as a bound. Technical report, 394/19, TU Kaiserslautern, Department of Computer Science (2019).
  34. 34.
    Nikolaus, P., Schmitt, J., Schütze, M.: h-Mitigators: improving your stochastic network calculus output bounds. Comput. Commun. 144, 188–197 (2019)CrossRefGoogle Scholar
  35. 35.
    Norros, I.: On the use of fractional Brownian motion in the theory of connectionless networks. IEEE J. Sel. Areas Commun. 13(6), 953–962 (1995)CrossRefGoogle Scholar
  36. 36.
    Qian, Z., Bogdan, P., Tsui, C.Y., Marculescu, R.: Performance evaluation of NoC-based multicore systems: From traffic analysis to noc latency modeling. ACM Trans. Des. Autom. Electron. Syst. (TODAES) 21(3), 52 (2016)Google Scholar
  37. 37.
    Rizk, A., Fidler, M.: Leveraging statistical multiplexing gains in single-and multi-hop networks. In: Proceedings of the IEEE Nineteenth IEEE International Workshop on Quality of Service (IWQoS 2011), pp. 1–9 (2011)Google Scholar
  38. 38.
    Rizk, A., Fidler, M.: Non-asymptotic end-to-end performance bounds for networks with long range dependent fbm cross traffic. Comput. Netw. 56(1), 127–141 (2012)CrossRefGoogle Scholar
  39. 39.
    Rosen, E., Viswanathan, A., Callon, R.: Multiprotocol label switching architecture. RFC 3031, RFC Editor (2001)Google Scholar
  40. 40.
    Schmitt, J., Bondorf, S., Poe, W.Y.: The sensor network calculus as key to the design of wireless sensor networks with predictable performance. J. Sensor Actuator Netw. 6(3), 21 (2017)CrossRefGoogle Scholar
  41. 41.
    Schmitt, J., Zdarsky, F.A., Fidler, M.: Delay bounds under arbitrary multiplexing: When network calculus leaves you in the lurch ... In: Proceedings of the IEEE International Conference on Computer Communications (INFOCOM 2008), Phoenix, AZ, USA, pp. 1669–1677 (2008)Google Scholar
  42. 42.
    Schmitt, J., Zdarsky, F.A., Martinovic, I.: Improving performance bounds in feed-forward networks by paying multiplexing only once. In: Proceedings of the GI/ITG Conference on Measurement, Modeling, and Evaluation of Computer and Communication Systems (MMB 2008), pp. 1–15 (2008)Google Scholar
  43. 43.
    Singla, A., Chandrasekaran, B., Godfrey, P.B., Maggs, B.: The internet at the speed of light. In: Proceedings of the ACM Workshop on Hot Topics in Networks 2014, pp. 1–7. HotNets-XIII (2014)Google Scholar
  44. 44.
    Wang, H., Shen, H., Wieder, P., Yahyapour, R.: A data center interconnects calculus. In: 2018 IEEE/ACM 26th International Symposium on Quality of Service (IWQoS), pp. 1–10. IEEE (2018)Google Scholar
  45. 45.
    Yaron, O., Sidi, M.: Performance and stability of communication networks via robust exponential bounds. IEEE/ACM Trans. Netw. 1(3), 372–385 (1993)CrossRefGoogle Scholar
  46. 46.
    Zhu, T., Berger, D.S., Harchol-Balter, M.: SNC-meister: admitting more tenants with tail latency SLOs. In: Proceedings of the ACM Symposium on Cloud Computing (SoCC 2016), pp. 374–387 (2016)Google Scholar
  47. 47.
    Zografos, K.G., Androutsopoulos, K.N., Vasilakis, G.M.: A real-time decision support system for roadway network incident response logistics. Transp. Res. Part C: Emerg. Technol. 10(1), 1–18 (2002)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Distributed Computer Systems (DISCO) LabTU KaiserslauternKaiserslauternGermany

Personalised recommendations