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The Real-Time Calculus (RTC) framework proposed in [Chakraborty et al., DATE 2003] and subsequently extended in [Wandeler et al., Real-Time Systems 29(2-3), 2005] and a number of other papers is geared towards the analysis of real-time systems that process various types of streaming data. The main strength of RTC is a count-based abstraction, where arrival patterns of event streams are specified as constraints on the number of events that may arrive over any specified time interval. In this framework, algebraic techniques can be used to compute system properties in a compositional way. However, the main drawback of RTC is that it cannot model state information in a natural way. For example, when a scheduling policy depends on the fill-level of a certain buffer or there is a shift from one type of data stream into another. In this paper, we extend RTC in a manner that enables state information to be easily captured while limiting the state-space explosion caused by fine grained state-based models such as timed automata. Our model, called "multi-mode RTC", specifies event streams as finite automata whose states are annotated with functions that specify constraints on the arrival patterns of event streams or the service available to process them. Our new framework combines the expressiveness of state-based models with the algebraic and compositional features of the RTC formalism. In particular, system properties within a single mode can be analyzed using the RTC-based algebraic techniques and state-space exploration can be used to piece together the results obtained algebraically for the individual modes. We show how to determine typical system properties with the focus on efficient approximate techniques and illustrate the advantages of multi-mode RTC using two case studies.