Duality of the Max-Plus and Min-Plus Network Calculus

The network calculus is a framework for the analysis of communication net-works, which exploits that many computer network models become tractablefor analysis if they are expressed in a min-plus or max-plus algebra. In amin-plus algebra, the network calculus characterizes amounts of traffic andavailable service as functions of time. In a max-plus algebra, the network cal-culus works with functions that express the arrival and departure times or therequired service time for a given amount of traffic. While the min-plus net-work calculus is more convenient for capacity provisioning in a network, themax-plus network calculus is more compatible with traffic control algorithmsthat involve the computation of timestamps. Many similarities and relation-ships between the two versions of the network calculus are known, yet theyare largely viewed as distinct analytical approaches with different capabili-ties and limitations. We show that there exists a one-to-one correspondencebetween the min-plus and max-plus network calculus, as long as traffic andservice are described by functions with real-valued domains and ranges. Con-sequently, results from one version of the network calculus can be readilyapplied for computations in the other version. The ability to switch betweenmin-plus and max-plus analysis without any loss of accuracy provides addi-tional flexibility for characterizing and analyzing traffic control algorithms.This flexibility is exploited for gaining new insights into link scheduling al-gorithms that offer rate and delay guarantees to traffic flows. J. Liebeherr. Duality of the Max-Plus and Min-Plus Network Calculus. Foundations andTrends R© in Networking, vol. 11, no. 3-4, pp. 139–282, 2016.DOI: 10.1561/1300000059.Full text available at: http://dx.doi.org/10.1561/1300000059