Arrival curves for real-time calculus: the causality problem and its solutions

The Real-Time Calculus (RTC) [19] is a framework to analyze heterogeneous real-time systems that process event streams of data. The streams are characterized by pairs of curves, called arrival curves, that express upper and lower bounds on the number of events that may arrive over any specified time interval. System properties may then be computed using algebraic techniques in a compositional way. A well-known limitation of RTC is that it cannot model systems with states and recent works [10, 9, 2, 16, 14] studied how to interface RTC curves with state-based models. Doing so, while trying, for example to generate a stream of events that satisfies some given pair of curves, we faced a causality problem [17]: it can be the case that, once having generated a finite prefix of an event stream, the generator deadlocks, since no extension of the prefix can satisfy the curves afterwards. When trying to express the property of the curves with state-based models, one may face the same problem. This paper formally defines the problem on arrival curves, and gives algebraic ways to characterize causal pairs of curves, i.e. curves for which the problem cannot occur. Then, we provide algorithms t