Discrete Event Dynamic Systems

, Volume 26, Issue 3, pp 383–411 | Cite as

Tight performance bounds in the worst-case analysis of feed-forward networks

  • Anne BouillardEmail author
  • Éric Thierry


Network Calculus theory aims at evaluating worst-case performances in communication networks. It provides methods to analyze models where the traffic and the services are constrained by some minimum and/or maximum envelopes (arrival/service curves). While new applications come forward, a challenging and inescapable issue remains open: achieving tight analyzes of networks with aggregate multiplexing. The theory offers efficient methods to bound maximum end-to-end delays or local backlogs. However as shown in a recent breakthrough paper (Schmitt et al. 2008), those bounds can be arbitrarily far from the exact worst-case values, even in seemingly simple feed-forward networks (two flows and two servers), under blind multiplexing (i.e. no information about the scheduling policies, except FIFO per flow). For now, only a network with three flows and three servers, as well as a tandem network called sink tree, have been analyzed tightly.We describe the first algorithm which computes the maximum end-to-end delay for a given flow, as well as the maximum backlog at a server, for any feed-forward network under blind multiplexing, with piecewise affine concave arrival curves and piecewise affine convex service curves. Its computational complexity may look expensive (possibly super-exponential), but we show that the problem is intrinsically difficult (NP-hard). Fortunately we show that in some cases, like tandem networks with cross-traffic interfering along intervals of servers, the complexity becomes polynomial. We also compare ourselves to the previous approaches and discuss the problems left open.


Network calculus Feed-forward networks Blind multiplexing Linear programming 


  1. Andrews M (2007) Instability of FIFO in the permanent sessions model at arbitrarily small network loads. In: Proceeding of SODA’07Google Scholar
  2. Baccelli F, Cohen G, Olsder GY, Quadrat JP (1992) Synchronization and linearity. WileyGoogle Scholar
  3. Bisti L, Lenzini L, Mingozzi E, Stea G (2008) Estimating the worst-case delay in FIFO tandems using network calculus. In: Proceeding of valuetools’2008Google Scholar
  4. Borodin A, Kleinberg JM, Raghavan P, Sudan M, Williamson DP (2001) Adversarial queuing theory. J ACM 48(1):13–38MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bouillard A, Gaujal B, Lagrange S, Thierry E (2008) Optimal routing for end-to-end guarantees using network calculus. Perform Eval 65(11-12):883–906CrossRefGoogle Scholar
  6. Bouillard A, Jouhet L, Thierry E (2010) Tight performance bounds in the worst-case analysis of feed-forward networks. In: Proceedings of infocom’2010Google Scholar
  7. Bouillard A, Thierry E (2008) An algorithmic toolbox for network calculus. Discret Event Dyn Syst 18(1):3–49MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bouillard A, Junier A (2011) Worst-case delay bounds with fixed priorities using network calculus. In: Proceedings of valuetools’11Google Scholar
  9. Boyer M, Fraboul C (2008) Tightening end to end delay upper bound for AFDX network calculus with rate latency FCFS servers using network calculus. In: Proceeding of WFCS’2008Google Scholar
  10. Chakraborty S, Künzli S, Thiele L, Herkersdorf A, Sagmeister P (2003) Performance evaluation of network processor architectures: Combining simulation with analytical estimation. Comput Netw 41(5):641–665CrossRefzbMATHGoogle Scholar
  11. Chang C-S (1997) A filtering theory for deterministic traffic regulation. In: Proceeding of INFOCOM’97, pp 436–443Google Scholar
  12. Chang CS (2000) performance guarantees in communication networks, TNCS. SpringerGoogle Scholar
  13. Cruz RL (1991) A calculus for network delay, part i: Network elements in isolation. IEEE Trans Inf Theory 37(1):114–131MathSciNetCrossRefzbMATHGoogle Scholar
  14. Cruz RL (1991) A calculus for network delay, part ii: Network analysis. IEEE Trans Inf Theory 37(1):132–141MathSciNetCrossRefzbMATHGoogle Scholar
  15. Fidler M (2006) A network calculus approach to probabilistic quality of service analysis of fading channels. In: Proceeding of GLOBECOM’2006Google Scholar
  16. Firoiu V, Le Boudec J-Y, Towsley D, Zhang Z-L (2002) Theories and models for internet quality of service. Proc IEEE 90(9):1565–1591CrossRefGoogle Scholar
  17. Hendriks M, Verhoef M (2006) Timed automata based analysis of embedded system architectures. In: Proceeding of IPDPSGoogle Scholar
  18. Jiang Y, Liu Y (2008) Stochastic network calculus. SpringerGoogle Scholar
  19. Le Boudec J-Y, Thiran P (2004) Network calculus: A theory of deterministic queuing systems for the internet, volume LNCS 2050. revised version 4Google Scholar
  20. Lenzini L, Mingozzi E, Stea G (2008) A methodology for computing end-to-end delay bounds in FIFO-multiplexing tandems. Perform Eval 65(11-12):922–943CrossRefGoogle Scholar
  21. Martin S, Minet P, George L (2005) End-to-end response time with fixed priority scheduling: trajectory approach versus holistic approach. Int J Communication Systems 18(1):37–56CrossRefGoogle Scholar
  22. Perathoner S, Wandeler E, Thiele L, Hamann A, Schliecker S, Henia R, Racu R, Ernst R, Harbour M G (2007) Influence of different system abstractions on the performance analysis of distributed real-time systems. In: Proceeding of EMSOFT’2007Google Scholar
  23. Pruesse G, Ruskey F (1994) Generating linear extensions fast. SIAM J Comput 23(2):373–386MathSciNetCrossRefzbMATHGoogle Scholar
  24. Rizzo G, Le Boudec J-Y (2008) Stability and delay bounds in heterogeneous networks of aggregate schedulers. In: Proceeding of INFOCOM’2008Google Scholar
  25. Rizzo G, Le Boudec J-Y (2005) ”pay bursts only once” does not hold for non-FIFO guaranteed rate nodes. Perform Eval 62(1-4):366–381CrossRefGoogle Scholar
  26. Schmitt JB, Zdarsky FA (2006) The disco network calculator: A toolbox for worst case analysis. In: Proceeding of Valuetools’2006Google Scholar
  27. Schmitt JB, Zdarsky FA, Fidler M (2007) Delay bounds under arbitrary multiplexing. University of Kaiserslautern, Technical reportGoogle Scholar
  28. Schmitt JB, Zdarsky FA, Fidler M (2008) Delay bounds under arbitrary multiplexing: When network calculus leaves you in the lurch... In: Proceeding of INFOCOM’2008Google Scholar
  29. Skeie T, Johannessen S, Holmeide O (2006) Timeliness of real-time IP communication in switched industrial ethernet networks. IEEE Trans Ind Inform 2:25–39CrossRefGoogle Scholar
  30. Tassiulas L, Georgiadis L (1996) Any work-conserving policy stabilizes the ring with spatial re-use. IEEE/ACM Trans Networking 4(2):205–208CrossRefGoogle Scholar
  31. Thiele L, Chakraborty S, Gries M, Maxiaguine A, Greutert J (2001) Embedded software in network processors models and algorithms. In: Proceeding of EMSOFT’2001Google Scholar
  32. Thiele L, Chakraborty S, Naedele M (2000) Real-time calculus for scheduling hard real-time systems. In: Proceeding of ISCAS’2000Google Scholar
  33. Tindell K, Clark J (1994) Holistic schedulability analysis for distributed hard real-time systems. Microprocess Microprogram 40(2-3):117–134CrossRefGoogle Scholar
  34. Vanderbei RJ (2000) Linear programming, foundations and extensions. Kluwer Academic PublisherGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.ENS/INRIAParisFrance
  2. 2.ENS LyonLyonFrance

Personalised recommendations