Extending Network Calculus to Deal with Min-Plus Service Curves in Multiple Flow Scenarios

Network Calculus (NC) is a versatile analytical methodology to efficiently compute performance bounds in networked real-time systems. The arrival and service curve abstractions allow to model diverse and heterogeneous distributed real-time systems. The operations to compute residual service curves and to concatenate sequences of systems enable an efficient and accurate calculation of per-flow timing guarantees. Yet, in some scenarios involving multiple concurrent flows at a system, the central notion of so-called min-plus service curves is too weak to still be able to compute a meaningful residual service curve. In these cases, one usually resorts to so-called strict service curves that enable the computation of per-flow bounds. However, strict service curves are restrictive: (1) there are service elements for which only min-plus service curves can be provided but not strict ones and (2) strict service curves generally have no concatenation property, i.e., a sequence of two strict systems does not yield a strict service curve. In this paper, we extend NC to deal with systems only offering aggregate min-plus service curves to multiple flows. The key to this extension is the exploitation of minimal arrival curves, i.e., lower bounds on the arrival process. Technically speaking, we provide basic performance bounds (backlog and delay) for the case of negative service curves. We also discuss their accuracy and show them to be tight in many cases and approximately tight up to a constant in the others. In order to illustrate their usefulness we also present patterns of application of these new results for: (1) heterogeneous systems involving computation and communication resources and (2) finite buffers that are shared between multiple flows.