Causality problem in real-time calculus

Real-Time Calculus (RTC) [37] 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. A well-known limitation of RTC is that it cannot model systems with states and several works [1, 4, 5, 11, 18, 19, 22, 23, 30, 24, 31, 38] 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 [33]: it can be the case that, after generating a finite prefix of an event stream, the generator deadlocks, since no extension of the prefix can satisfy the curves afterwards. This paper formally defines the problem; it states and proves algebraic results that characterize causal pairs of curves, i.e. curves for which the problem cannot occur. We consider the general case of infinite curve models, either discrete or continuous time and events. The paper provides an analysis on how causality issues appear when using arrival curves and how they could be handled. It also provides an overview of algorithms to compute causal curves in several models. These algorithms compute a canonical representation of a pair of curves, which is the best pair of curves among the curves equivalent to the ones they take as input.