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. The problem is easy when application stages are not replicated, i.e., each application stage is assigned to a single processor: in that case the throughput is dictated by the critical hardware resource. However, when stages are replicated, i.e., each application stage may be 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 computation and communication times, also called stage parameters, are constant or follow I.I.D. exponential laws. The second contribution is to provide bounds for the throughput when stage parameters form associated random sequences (correlation between communication and processing times of a given data set on the different application stages, i.e., a data set that takes a long time on the first stage is likely to be large, and to take a long time on the next stages), and are N.B.U.E. (New Better than Used in Expectation) variables (if an operation has already been processed for some duration, the remaining time is smaller than the processing time of a fresh operation): 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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16


  1. 1.

    The product ⊗ is defined as: (VM) k =max i (V i +M ik ).


  1. 1.

    Baccelli, F., Cohen, G., Gaujal, B.: Evolution equations of timed Petri nets. In: Proceedings of the 30th IEEE Conference on Decision and Control, 1991, vol. 2, pp. 1139–1144 (1991). doi:10.1109/CDC.1991.261523

    Google Scholar 

  2. 2.

    Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.-P.: Synchronization and Linearity. Wiley, New York (1992)

    Google Scholar 

  3. 3.

    Benoit, A., Robert, Y.: Mapping pipeline skeletons onto heterogeneous platforms. J. Parallel Distrib. Comput. 68(6), 790–808 (2008)

    Article  MATH  Google Scholar 

  4. 4.

    Benoit, A., Gallet, M., Gaujal, B., Robert, Y.: Computing the throughput of replicated workflows on heterogeneous platforms. In: Proceedings of ICPP’2009, the 38th International Conference on Parallel Processing (2009)

    Google Scholar 

  5. 5.

    Benoit, A., Dufossé, F., Gallet, M., Gaujal, B., Robert, Y.: Computing the throughput of probabilistic and replicated streaming applications. In: Proceedings of SPAA 2010, the 22nd ACM Symposium on Parallelism in Algorithms and Architectures. ACM, New York (2010)

    Google Scholar 

  6. 6.

    Beynon, M.D., Kurc, T., Sussman, A., Saltz, J.: Optimizing execution of component-based applications using group instances. Future Gener. Comput. Syst. 18(4), 435–448 (2002)

    Article  MATH  Google Scholar 

  7. 7.

    Casanova, H., Legrand, A., Quinson, M.: SimGrid: a generic framework for large-scale distributed experiments. In: Proceedings of UKSim, the 10th EUROS/UKSim International Conference on Computer Modelling and Simulation, pp. 126–131 (2008)

    Google Scholar 

  8. 8.

    Chiola, G., Franceschinis, G., Gaeta, R., Ribaudo, M.: GreatSPN: graphical editor and analyzer for timed and stochastic Petri nets. Perform. Eval. 24(1–2), 47–68 (1995)

    Article  MATH  Google Scholar 

  9. 9.

    Cole, M.: Bringing skeletons out of the closet: a pragmatic manifesto for skeletal parallel programming. Parallel Comput. 30(3), 389–406 (2004)

    Article  Google Scholar 

  10. 10.

    DataCutter Project. Middleware for filtering large archival scientific datasets in a grid environment.

  11. 11.

    Esary, J.D., Proschan, F., Walkup, D.W.: Association of random variables, with applications. Ann. Math. Stat. 38(5), 1466–1474 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Gaujal, B., Vincent, J.-M.: Comparisons of stochastic task-resource systems. In: Introduction to Scheduling. CRC Press, Boca Raton (2009)

    Google Scholar 

  13. 13.

    Häggström, O.: Finite Markov Chains and Algorithmic Applications. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  14. 14.

    Hillion, H., Proth, J.-M.: Performance evaluation of job shop systems using timed event graphs. IEEE Trans. Autom. Control 34(1), 3–9 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Jean-Marie, A.: ERS: a tool set for performance evaluation of discrete event systems.

  16. 16.

    Kamburowski, J.: Bounding the distribution of project duration in pert networks. Oper. Res. Lett. 12, 17–22 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

    Knuth, D.E.: The Art of Computer Programming, vol. 3, 2nd edn. Addison-Wesley, Reading (1998)

    Google Scholar 

  18. 18.

    Kumazawa, Y.: Tests for new better than used in expectation with randomly censored data. Seq. Anal. 5(1), 85–92 (1986)

    Article  MathSciNet  Google Scholar 

  19. 19.

    Spencer, M., Ferreira, R., Beynon, M., Kurc, T., Catalyurek, U., Sussman, A., Saltz, J.: Executing multiple pipelined data analysis operations in the grid. In: Proceedings of Supercomputing’02, the 2002 ACM/IEEE Conference on Supercomputing, pp. 1–18. IEEE Comput. Soc., Los Alamitos (2002)

    Google Scholar 

  20. 20.

    Subhlok, J., Vondran, G.: Optimal mapping of sequences of data parallel tasks. In: Proceedings of PPoPP’95, the ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming, pp. 134–143. ACM, New York (1995)

    Google Scholar 

  21. 21.

    Subhlok, J., Vondran, G.: Optimal latency-throughput tradeoffs for data parallel pipelines. In: Proceedings of SPAA’96, the 8th ACM Symposium on Parallel Algorithms and Architectures, pp. 62–71. ACM, New York (1996)

    Google Scholar 

  22. 22.

    Taura, K., Chien, A.: A heuristic algorithm for mapping communicating tasks on heterogeneous resources. In: Proceedings of HCW’00, the 9th Heterogeneous Computing Workshop, pp. 102–115. IEEE Comput. Soc., Los Alamitos (2000)

    Google Scholar 

  23. 23.

    Velho, P., Legrand, A.: Accuracy study and improvement of network simulation in the SimGrid framework. In: Proceedings of Simutools’09, the 2nd International Conference on Simulation Tools and Techniques, ICST, pp. 1–10 (2009)

    Google Scholar 

  24. 24.

    Vydyanathan, N., Çatalyurek, Ü.V., Kurc, T., Saddayappan, P., Saltz, J.: Toward optimizing latency under throughput constraints for application workflows on clusters. In: Proceedings of Euro-Par’07. LNCS, vol. 4641, pp. 173–183. Springer, Berlin (2007)

    Google Scholar 

  25. 25.

    Vydyanathan, N., Çatalyurek, Ü.V., Kurc, T., Saddayappan, P., Saltz, J.: A duplication based algorithm for optimizing latency under throughput constraints for streaming workflows. In: Proceedings of ICPP’2008, the 37th International Conference on Parallel Processing, pp. 254–261. IEEE Comput. Soc., Los Alamitos (2008)

    Google Scholar 

  26. 26.

    Wu, Q., Gu, Y.: Supporting distributed application workflows in heterogeneous computing environments. In: Proceedings of ICPADS’08, the 14th IEEE International Conference on Parallel and Distributed Systems, pp. 3–10. IEEE Comput. Soc., Los Alamitos (2008)

    Google Scholar 

Download references


The authors thank the reviewers for their numerous comments and suggestions, which greatly improved the final version of the paper.

Author information



Corresponding author

Correspondence to Anne Benoit.

Additional information

Part of this work has appeared in ICPP’09 and SPAA’10. Anne Benoit and Yves Robert are with the Institut Universitaire de France. This work was supported in part by the ANR StochaGrid and RESCUE projects, and by the Inria ALEAE project.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Benoit, A., Gallet, M., Gaujal, B. et al. Computing the Throughput of Probabilistic and Replicated Streaming Applications. Algorithmica 69, 925–957 (2014).

Download citation


  • Communication Time
  • Deterministic Case
  • Critical Resource
  • Overlap Model
  • Streaming Application