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Streaming computations on massive data sets are an attractive candidate for parallelization, particularly when they exhibit independence (and hence data parallelism) between items in the stream. However, some streaming computations are stateful, which disrupts independence and can limit parallelism. In this work, we consider how to extract data parallelism from streaming computations with a common, limited form of statefulness. The stream is assumed to be divided into variably-sized regions, and items in the same region are processed in a common context of state. In general, the computation to be performed on a stream is also irregular, with each item potentially undergoing different, data-dependent processing. This work describes mechanisms to implement such computations efficiently on a SIMD-parallel architecture such as a GPU. We first develop a low-level protocol by which a data stream can be augmented with control signals that are delivered to each stage of a computation at precise points in the stream. We then describe an abstraction, enumeration and aggregation, by which an application developer can specify the behavior of a streaming application with region-based state. Finally, we study an implementation of our ideas as part of the MERCATOR system for irregular streaming computations on GPUs, investigating how the frequency of region boundaries in a stream impacts SIMD occupancy and hence application performance.