Determining Asynchronous Pipeline Execution

Asynchronous pipelining is a form of parallelism in which processors execute diierent loop tasks (loop statements) as opposed to diierent loop iterations. An asynchronous pipeline schedule for a loop is an assignment of loop tasks to processors, plus an order on instances of tasks assigned to the same processor. This variant of pipelining is particularly relevant in distributed memory systems (since pipeline control may be distributed across processors), but may also be used in shared memory systems. Accurate estimation of the execution time of a pipeline schedule is needed to determine if pipelining is appropriate for a loop, and to compare alternative schedules. Pipeline execution of n iterations of a loop requires time at most a + bn, for some constants a and b. The coeecient b is the iteration interval of the pipeline schedule, and is the primary measure of the performance of a schedule. The startup time a is a secondary performance measure. We generalize previous work on determining if a pipeline schedule will deadlock, and generalize Reiter's well-known formula 21] for determining the iteration interval b of a deadlock-free schedule, to account for nonzero communication times (easy) and the assignment of multiple tasks to processors (nontrivial). Two key components of our generalization are the use of pipeline scheduling edges, and the notion of negative data dependence distances (in a single unnested loop). We also discuss implementation of an asynchronous pipeline schedule at runtime; show how to eeciently simulate pipeline execution on a sequential processor; derive bounds on the startup time a; and discuss evaluation of the iteration interval formula, including development of a new algorithm.