Section: New Results
Fundamentals and algorithms: timed models
Participants : Claude Jard, Anne Bouillard, Bartosz Grabiec.
Our work on timed models was focused on the study and use of two different techniques: unfoldings of network of timed automata (and Time Petri nets) and the network calculus. The goals are supervision with time and performance evaluation.
In the context of the PhD of B. Grabiec, we are trying to extend the model of networks of timed automata by modifying the time semantics to take into account more realistic time models, allowing for instance some drift between clocks of different components. This has some consequences on the way of representing generic time zones and unfoldings. The focus this year was to introduce parameters in models in order to facilitate the modelling phase. The interest is to handle symbolically both the timings and the parameters. Unfoldings are an efficient method to determine the causal relations between the events in a system. Given a partial observation, as a list of actions, we propose to use our method to determine the causal relation between events in the model that explain the observation [24] . We can also synthesise parametric constraints associated with these explanations. In collaboration with the IRCCyN's group in the DOTS project, this method was implemented in the Romeo tool.
Network Calculus is a quite recent theory developed to compute deterministic worstcase bounds in queuing networks. Computing such bounds is necessary when dealing with realtime and critical systems (that can be found for example in embedded systems of airplanes or cars). The Network Calculus is based on the (min,plus) algebra. It models constraints on arrival and output processes by means of arrival and service curves. Our work has focused on three main aspects:

We first studied the algorithmic aspects of the Network Calculus operators, namely the (min,plus) convolution, the (min,plus)deconvolution and the subadditive closure. We have exhibited a stable class of functions regarding those operators and have given efficient algorithms to compute them [38] . A small software COINC has been written to implement those algorithms and a first version is now available (http://www.istia.univangers.fr/ lagrange/spip.php?article21 ).

We then studied the composition of network elements in presence of crosstraffic. Our contribution concerns two kinds of scenarios:

One flow in a network in presence of independent crosstraffic, that has to be transmitted from a given source to a given destination. What is the best path concerning the worstcase delay/backlog) for that flow?

One flow on a fixed path interfering with dependent crosstraffic. Can we compute a service curve for the effective traffic for the flow?
To answer the first question, we derived some polynomialtime algorithms that can be seen as shortestpath algorithms with functional weights on the arcs, instead of constant weights as in the classical case ([36] ). To answer the second question, we introduce a new operator, the multidimensional convolution. It appears that we are not able to compute the multidimensional convolution (hence the effective service curve) in polynomial time, but, thanks to linear programming, we are able to derive bounds on the delay and backlog in polynomial time([37] ).


Finally some work has been initiated thanks to the associated team CASDS. It concerns the study of multimode Network Calculus: in classical Network calculus, constraints are static. Here, they evolve with time, and this evolution is modeled with a finitestate automaton. A first study concerns the blockwriting server (a server either serves packets with a given constraint on the guaranteed service or serves nothing). Corresponding paper [21] has been accepted at RTAS'09.