## Section: Research Program

### Network Calculus

Network calculus [53] is a theory for obtaining deterministic upper bounds in networks that has been developed by R. Cruz [41], [42]. From the modelling point of view, it is an algebra for computing and propagating constraints given in terms of envelopes. A flow is represented by its cumulative function $R\left(t\right)$ (that is, the amount of data sent by the flow up to time $t$). A constraint on a flow is expressed by an arrival curve $\alpha \left(t\right)$ that gives an upper bound for the amount of data that can be sent during any interval of length $t$. Flows cross service elements that offer guarantees on the service. A constraint on a service is a service curve $\beta \left(t\right)$ that is used to compute the amount of data that can be served during an interval of length t. It is also possible to define in the same way minimal arrival curves and maximum service curves. Then such constraints envelop the processes and the services. Network calculus enables the following operations:

$\u2022$ computing the exact output cumulative function or at least bounding functions;

$\u2022$ computing output constraints for a flow (like an output arrival curve);

$\u2022$ computing the remaining service curve (that is, the service that of not used by the flows crossing a server);

$\u2022$ composing several servers in tandem;

$\u2022$ giving upper bounds on the worst-case delay and backlog (bounds are tight for a single server or a single flow).

The operations used for this are an adaptation of filtering theory to $(min,+)$: $(min,+)$ convolution and deconvolution, sub-additive closure.

We investigate the complexity of computing exact worst-case performance bounds in network calculus and to develop algorithms that present a good trade off between algorithmic efficiency and accuracy of the bounds.