Computing the Throughput of Replicated Workflows on Heterogeneous Platforms

Abstract

In this paper, we focus on computing the throughput of replicated workflows. Given a streaming application whose dependence graph is a linear chain, and a mapping of this application onto a fully heterogeneous platform, how can we compute the optimal throughput, or equivalently the minimal period? The problem is easy when workflow stages are not replicated, i.e., assigned to a single processor: in that case the period is dictated by the critical hardware resource. But when stages are replicated, i.e., assigned to several processors, the problem gets surprisingly complicated, and we provide examples where the optimal period is larger than the largest cycle-time of any resource. We then show how to model the problem as a timed Petri net to compute the optimal period in the general case, and we provide a polynomial algorithm for the one-port communication model with overlap. Finally, we report comprehensive simulation results on the gap between the optimal period and the largest resource cycle-time.Keywords: Scheduling, workflows, heterogeneous platforms, period, critical resource, timed Petri nets. RésuméDans ce papier, nous étudions le débit de graphes de tâches répliqués. Étant donnée une application de streaming dont le graphe de dépendance est une chaîne, et un placement de cette application sur une plate-forme hétérogène, comment pouvons-nous calculer le débit optimal, ou, de façon équivalente, la période minimale ? Ce problème est simple quand les différentes tâches ne sont traitées que par un seul processeur : dans ce cas, la période est donnée par le débit de la ou des ressources critiques. Cependant, quand les tâches sont répliquées, c'est-à-dire placées sur plusieurs processeurs, le problème devient étonnamment compliqué, et nous présentons des exemples d'instances sans aucune ressource critique, c'est-à-dire que chacune des ressources connaît des moments d'inactivité lors de l'exécution du système. Nous montrons comment calculer la période du système en utilisant les réseaux de Petri temporisés, et nous donnons un algorithme polynomial pour la calculer pour le modèle de communication avec overlap. Nous exposons également les résultats de nombreuses simulations montrant l'écart entre la période réelle entre le système et le maximum des temps d'occupation des ressources.

Computing the throughput of probabilistic and replicated streaming applications

“…However, when stages are replicated, the previous result is no longer true, and more sophisticated techniques are required. Here are the main results that we established previously in [3]:…”

“…However, when stages are replicated, the problem becomes surprisingly complicated: even in the deterministic case, the optimal throughput may be lower than the smallest internal resource throughput. This result was shown in [3], using a representation of the mapping based on timed Petri nets. In this paper we build upon the latter construction to tackle the probabilistic case.…”

“…As in [3], we model the system formed by a given mapping (of an application onto a platform) by a timed Petri net. Here we briefly recall design principles:…”

“…As said before, the transformation of the initial system into a timed Petri net is fully described in [3] and we do not detail it entirely here. Recall from Section 3.1 that it consists in R = lcm 1≤i≤N (Ri) rows and 2N − 1 columns, and examples for both models are depicted in Figure 2.…”

“…First, let us give the overall structure of the proof: 1. split the timed Petri net into columns Ci, with 1 ≤ i ≤ 2N − 1; 2. separately consider each column Ci; 3. separately consider each connected component Dj of Ci; 4. note that each component Dj is made of many copies of the same pattern Pj, of size uj × vj; 5. transform Pj into a Markov chain Mj; 6. determine a stationary measure of Mj, using a combinatorial trick based on Young diagrams [13]; 7. compute the throughput of Pj in isolation (called hereinafter inner throughput of component Dj in the following); 8. combine the inner throughputs of all components to get the global throughput of the system. To decrease the overall complexity, we use the same idea as in [3]: thanks to the regularity of the global timed Petri net, we split it into a polynomial number of columns, and we compute the throughput of each column independently.…”