# Numerical analysis of worst-case end-to-end delay bounds in FIFO tandem networks

## Abstract

This paper addresses the problem of computing end-to-end delay bounds for a traffic flow traversing a tandem of FIFO multiplexing network nodes using Network Calculus. Numerical solution methods are required, as closed-form delay bound expressions are unknown except for few specific cases. For the methodology called the Least Upper Delay Bound, the most accurate among those based on Network Calculus, exact and approximate solution algorithms are presented, and their accuracy and computation cost are discussed. The algorithms are inherently exponential, yet affordable for tandems of up to few tens of nodes, and amenable to online execution in cases of practical significance. This complexity is, however, required to compute accurate bounds. As the LUDB may actually be larger than the worst-case delay, we assess how close the former is to the latter by computing lower bounds on the worst-case delay and measuring the gap between the lower and upper bound.

## Keywords

Network Calculus Delay bounds FIFO Numerical analysis## References

- Agrawal R, Cruz RL, Okino C, Rajan R (1999) Performance bounds for flow control protocols. IEEE/ACM Trans Netw 7(3):310–323 CrossRefGoogle Scholar
- Bauer H, Scharbarg J-L, Fraboul C (2010) Improving the worst-case delay analysis of an AFDX network using an optimized trajectory approach. IEEE Trans Ind Inform 6(4):521–533 CrossRefGoogle Scholar
- Bisti L, Lenzini L, Mingozzi E, Stea G (2010) DEBORAH: a tool for worst-case analysis of FIFO tandems. In: ISoLA 2010, Crete, Oct 18–20, 2010 Google Scholar
- Bisti L, Lenzini L, Mingozzi E, Stea G (2008) Estimating the worst-case delay in FIFO tandems using network calculus. In: Proc of VALUETOOLS 2008, Athens, Greece, 21–23 October, 2008 Google Scholar
- Blake S, Black D, Carlson M, Davies E, Wang Z, Weiss W (1998) An architecture for differentiated services. IETF RFC 2475 Google Scholar
- Bouillard A, Thierry E (2008) An algorithmic toolbox for Network Calculus. J Discrete Event Dyn Syst 18(1):3–49 MathSciNetzbMATHCrossRefGoogle Scholar
- Bouillard A, Jouhet L, Thierry E (2010) Tight performance bounds in the worst-case analysis of feed-forward networks. In: Proc of INFOCOM 2010, S. Diego, CA, 14–19 March, 2010 Google Scholar
- Boyer M, Navet N, Olive X, Thierry E (2010) The Pegase project: precise and scalable analysis for aerospace communication systems with Network Calculus. In: Proc of ISoLA 2010, Crete, October 18–20, 2010 Google Scholar
- Braden R, Clark D, Shenker S (1994) Integrated services in the Internet architecture: an overview. IETF RFC 1633 Google Scholar
- Chakraborty S, Künzli S, Thiele L, Herkersdorf A, Sagmeister P (2003) Performance evaluation of network processor architectures: combining simulation with analytical estimation. Comput Netw 42(5):641–665 CrossRefGoogle Scholar
- Chang CS (2000) Performance guarantees in communication networks. Springer, New York zbMATHCrossRefGoogle Scholar
- Cruz RL (1991) A calculus for network delay. IEEE Trans Inf Theory 37(1):114–141 MathSciNetzbMATHCrossRefGoogle Scholar
- Fourer R (2010) A simplex algorithm for piecewise-linear programming I: derivation and proof. Math Program 33(2):204–233. doi: 10.1007/BF01582246 MathSciNetCrossRefGoogle Scholar
- Kiefer A, Gollan N, Schmitt JB (2010) Searching for tight performance bounds in feed-forward networks. In: Proc of MMB/DFT, pp 227–241 Google Scholar
- Koubaa A, Alves M, Tovar E (2006) Modeling and worst-case dimensioning of cluster-tree wireless sensor networks. In: Proc of IEEE RTSS’06, pp 412–421 Google Scholar
- Le Boudec J-Y, Thiran P (2001) Network Calculus. LNCS, vol 2050. Springer, Berlin zbMATHCrossRefGoogle Scholar
- Lenzini L, Martorini L, Mingozzi E, Stea G (2006) Tight end-to-end per-flow delay bounds in FIFO multiplexing sink-tree networks. Perform Eval 63:956–987 CrossRefGoogle Scholar
- Lenzini L, Mingozzi E, Stea G (2008) A methodology for computing end-to-end delay bounds in FIFO-multiplexing tandems. Perform Eval 65:922–943 CrossRefGoogle Scholar
- Lenzini L, Mingozzi E, Stea G (2005) Delay bounds for FIFO aggregates: a case study. Comput Commun 28(3):287–299 CrossRefGoogle Scholar
- Martin S, Minet P (2006) Schedulability analysis of flows scheduled with FIFO: application to the expedited forwarding class. In: Proc of IPDPS 2006, Rhodes Island, 25–29 April, 2006 Google Scholar
- Martin S, Minet P, George L (2004) Deterministic end-to-end guarantees for real-time applications in a DiffServ-MPLS domain. In: Proc of SERA 2003. LNCS, vol 3026. Springer, Berlin, pp 51–73 Google Scholar
- Schmitt JB, Roedig U (2005) Sensor Network Calculus—a framework for worst case analysis. In: DCOSS’05, pp 141–154, June 2005 Google Scholar
- Schmitt JB, Zdarsky FA, Fidler M (2008) Delay bounds under arbitrary multiplexing: when Network Calculus leaves you in the lurch. In: Proc of INFOCOM 2008, pp 1669–1677 Google Scholar
- Skeie T, Johannessen S, Holmeide O (2006) Timeliness of real-time IP communication in switched industrial Ethernet networks. IEEE Trans Ind Inform 2:25–39 CrossRefGoogle Scholar
- Urvoy-Keller G, Hèbuterne G, Dallery Y (2002) Traffic engineering in a multipoint-to-point network. IEEE J Sel Areas Commun 20(4):834–849 CrossRefGoogle Scholar