Computing the throughput of probabilistic and replicated streaming applications

In this paper, we investigate how to compute the throughput of probabilistic and replicated streaming applications. We are given (i) a streaming application whose dependence graph is a linear chain; (ii) a one-to-many mapping of the application onto a fully heterogeneous target platform, where a processor is assigned at most one application stage, but where a stage can be replicated onto a set of processors; and (iii) a set of random variables modeling the computation and communication times in the mapping. We show how to compute the throughput of the application, i.e., the rate at which data sets can be processed, under two execution models, the Strict model where the actions of each processor are sequentialized, and the Overlap model where a processor can compute and communicate in parallel. The problem is easy when application 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 deterministic case, the optimal throughput may be lower than the smallest internal resource throughput. The first contribution of the paper is to provide a general method to compute the throughput when mapping parameters are constant or follow I.I.D. exponential laws. The second contribution is to provide bounds for the throughput when stage parameters (computation and communication times) form associated random sequences, and are N.B.U.E. (New Better than Used in Expectation) variables: the throughput is bounded from below by the exponential case and bounded from above by the deterministic case. An extensive set of simulation allows us to assess the quality of the model, and to observe the actual behavior of several distributions

Computing the throughput of probabilistic and replicated streaming applications

International audienceIn this paper, we investigate how to compute the throughput of probabilistic and replicated streaming applications. We are given (i) a streaming application whose dependence graph is a linear chain; (ii) a one-to-many mapping of the application onto a fully heterogeneous target, where a processor is assigned at most one application stage, but where a stage can be replicated onto a set of processors; and (iii) a set of IID (Independent and Identically-Distributed) variables to model each computation and communication time in the mapping. How can we compute the throughput of the application, i.e., the rate at which data sets can be processed? We consider two execution models, the STRICT model where the actions of each processor are sequentialized, and the OVERLAP model where a processor can compute and communicate in parallel. The problem is easy when application 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 deterministic case, the optimal throughput may be lower than the smallest internal resource throughput. To the best of our knowledge, the problem has never been considered in the probabilistic case. The first main contribution of the paper is to provide a general method (although of exponential cost) to compute the throughput when mapping parameters follow IID exponential laws. This general method is based upon the analysis of timed Petri nets deduced from the application mapping; it turns out that these Petri nets exhibit a regular structure in the OVERLAP model, thereby enabling to reduce the cost and provide a polynomial algorithm. The second main contribution of the paper is to provide bounds for the throughput when stage parameters are arbitrary IID and NBUE (New Better than Used in Expectation) variables: the throughput is bounded from below by the exponential case and bounded from above by the deterministic case

Computing the throughput of probabilistic and replicated streaming applications

In this paper, we investigate how to compute the throughput of probabilistic and replicated streaming applications. We are given (i) a streaming application whose dependence graph is a linear chain; (ii) a one-to-many mapping of the application onto a fully heterogeneous target, where a processor is assigned at most one application stage, but where a stage can be replicated onto a set of processors; and (iii) a set of I.I.D. (Independent and Identically-Distributed) variables to model each computation and communication time in the mapping. How can we compute the throughput of the application, i.e., the rate at which data sets can be processed? We consider two execution models, the STRICT model where the actions of each processor are sequentialized, and the OVERLAP model where a processor can compute and communicate in parallel. The problem is easy when application 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 deterministic case, the optimal throughput may be lower than the smallest internal resource throughput. To the best of our knowledge, the problem has never been considered in the probabilistic case. The first main contribution of the paper is to provide a general method to compute the throughput when mapping parameters are constant or follow I.I.D. exponential laws. The second main contribution is to provide bounds for the throughput when stage parameters are arbitrary I.I.D. and N.B.U.E. (New Better than Used in Expectation) variables: the throughput is bounded from below by the exponential case and bounded from above by the deterministic case