Network Calculus theory aims at evaluating worstcase 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 (service/arrival 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 endto-end delays or local backlogs. However as shown recently, 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 concave arrival curves and convex service curves. Its computational complexity may look expensive (possibly superexponential), 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.

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Network Calculus is an attractive theory to derive deterministic bounds on end-to-end performance measures. Nevertheless bounding tightly and quickly the worst-case delay or backlog of a flow over a path with cross-traffic remains a challenging problem. This paper carries on with the study of configurations where a main flow encounters some cross-traffic flows which interfere over connected sub-paths. We also assume that no information is available about scheduling policies at the nodes (blind multiplexing). Such configurations were first analyzed in [25,27] where a "Pay Multiplexing Only Once" (PMOO) phenomenon was identified, and then in [6,7] where a (min, +) multi-dimensional operator was introduced to compute a minimum service curve for the whole path. Under usual assumptions (concave arrival curves and convex service curves), we prove some properties of this new operator and we show how to use it to derive bounds on delays and backlogs in polynomial time.We also discuss the simpler case when there is no crosstraffic. Then the analysis is known to boil down to the (min, +) convolution of all the service curves over the path. For convex and piecewise affine service curves, a specific theorem enables to compute efficiently the convolution. This theorem has been used by several authors [6,8,17,21,22,25,27], but they all refer to a proof which is unfortunately incomplete [5]. To set definitely this theorem, we provide three different proofs. We also investigate the complexity of computing performances bounds in this case.

Le calcul réseau manipule de nombreuses notions propres à la fois à son domaine de travail (flux, serveur, délai...), et à la modélisation (courbe d'arrivée, de service...). Il utilise le dioïde (min, +) comme support mathématique. Cet article se propose de définir un ensemble de notations associé à ces notions. L'objectif des notations est double: non seulement de simplifier la manipulation (modélisation, preuves) et l'appropriation (diffusion, enseignement) du calcul réseau, mais aussi de redéfinir formellement certaines notions, et ce faisant, de les clarifier. ABSTRACT. Network calculus involves many notions that are both due to the applications concerned (flow, server, delay...) and to the model (arrival curves, service curves...). The mathematical framework is based on the (min,plus) algebra. This article attempts to provide some notations associated to these notions. The aim is twofold: on the one hand, simplification of the manipulation (model, proofs) and appropriation (diffusion, teaching) of network calculus and in the second hand, formalization and clarification of some notions (notion of server for example).