Computing the throughput of probabilistic streaming applications with replication

In this paper, we investigate how to compute the throughput of probabilistic streaming applications. Given a streaming application whose dependence graph is a linear chain, a fully heterogeneous target platform, and a one-to-one mapping of the application onto the platform (a processor is assigned at most one application stage), how can we compute the throughput of the application, i.e., the rate at which data sets can be processed? The problem is easy when workflow stages are not replicated, i.e., assigned to a single processor: in that case the throughput is dictated by the critical hardware resource. However, when stages are replicated, i.e., assigned to several processors, the problem becomes surprisingly complicated. Even in the case when execution and communication times are deterministic, there are examples where the optimal period (i.e., the inverse of the throughput) is larger than the largest cycle-time of any resource. We model the problem as a timed Petri net to compute the optimal throughput in the general case, and we show how the throughput is impacted when execution and communication times are no longer deterministic, but follow some random variable laws. Finally, we prove that the problem of finding a one-to-one mapping with replication which maximizes the throughput is NP-complete, even with no communication costs.