Dynamic Control of the Join-Queue Lengths in Saturated Fork-Join Stations

Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9826)

Abstract

The analysis of fork-join queueing systems has played an important role for the performance evaluation of distributed systems where parallel computations associated with the same job are carried out and a job is considered served only when all the parallel tasks it consists of are served and then joined. The fork-join nodes that we consider consist of \(K\ge 2\) parallel servers each of which is equipped with two FCFS queues, namely the service-queue and the join-queue. The former stores the tasks waiting for being served while the latter stores the served tasks waiting for being joined. When the queueing station is saturated, i.e., the service-queues are never empty, we observe that the join-queue sizes tend to grow infinitely even if the expected service times at the servers are the same. In fact, this is due to the variance of the service time distribution. To tackle this problem, we propose a simple service-rate control mechanism, and show that under the exponential assumption on the service times, we can analytically study a set of relevant performance indices. We show that by selectively reducing the speed of some servers, significant energy saving can be achieved.

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References

  1. 1.
    Baccelli, F., Makowski, M.A.: Queueing models for systems with synchronization constraints. Proc. IEEE 77(1), 138–161 (1989)CrossRefGoogle Scholar
  2. 2.
    Baccelli, F., Liu, Z.: On the execution of parallel programs on multiprocessor systems: a queuing theory approach. J. ACM 37(2), 373–414 (1990)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Baccelli, F., Massey, W.A., Towsley, D.: Acyclic fork-join queuing networks. J. ACM 36(3), 615–642 (1989)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bruell, S.C., Balbo, G., Afshari, P.V.: Mean value analysis of mixed, multiple class BCMP networks with load dependent service stations. Perform. Eval. 4, 241–260 (1984)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Buzen, J.P.: Computational algorithms for closed queueing networks with exponential servers. Commun. ACM 16(9), 527–531 (1973)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Casale, G.: A generalized method of moments for closed queueing networks. Perform. Eval. 68(2), 180–200 (2011)CrossRefGoogle Scholar
  7. 7.
    Fiorini, P.M., Lipsky, L.: Exact analysis of some split-merge queues. SIGMETRICS Perform. Eval. Rev. 43(2), 51–53 (2015)CrossRefGoogle Scholar
  8. 8.
    Flatto, L., Hahn, S.: Two parallel queues created by arrivals with two demands. SIAM J. Appl. Math. 44(5), 1041–1053 (1984)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Heidelberger, P., Trivedi, K.: Analytic queueing models for programs with internal concurrency. IEEE Trans. Comput. C–32, 73–82 (1983)CrossRefMATHGoogle Scholar
  10. 10.
    Hoekstra, G.J., van der Mei, R.D., Bhulai, S.: Optimal job splitting in parallel processor sharing queues. Stoch. Models 28, 144–166 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kelly, F.: Reversibility and Stochastic Networks. Wiley, New York (1979)MATHGoogle Scholar
  12. 12.
    Lebrecht, A.S., Dingle, N.J., Knottenbelt, W.J.: Modelling zoned RAID systems using fork-join queueing simulation. In: Bradley, J.T. (ed.) EPEW 2009. LNCS, vol. 5652, pp. 16–29. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  13. 13.
    Lu, H., Pang, G.: Gaussian limits for a fork-join network with nonexchangeable synchronization in heavy traffic. Math. Oper. Res. 41(2), 560–595 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Marin, A., Rossi, S.: On discrete time reversibility modulo state renaming and its applications. In: 8th International Conference on Performance Evaluation Methodologies and Tools, VALUETOOLS, pp. 1–8 (2014)Google Scholar
  15. 15.
    Marin, A., Rossi, S.: On the relations between lumpability and reversibility. In: Proceedings of the IEEE 22nd International Symposium on Modeling, Analysis and Simulation of Computer and Telecommunication Systems (MASCOTS 2014), pp. 427–432 (2014)Google Scholar
  16. 16.
    Marin, A., Rossi, S.: On the relations between Markov chain lumpability and reversibility. Acta Inform. 1–39 (2016)Google Scholar
  17. 17.
    Nelson, R., Tantawi, A.N.: Approximate analysis of fork/join synchronization in parallel queues. IEEE Trans. Comput. 37(6), 739–743 (1986)CrossRefGoogle Scholar
  18. 18.
    Nguyen, V.: Processing networks with parallel and sequential tasks: heavy traffic analysis and Browinian limits. Ann. Appl. Probab. 3(1), 28–55 (1993)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Olver, F.W., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions, 1st edn. Cambridge University Press, New York (2010)MATHGoogle Scholar
  20. 20.
    Rauber, T., Rünger, G.: Energy-aware execution of fork-join-based task parallelism. In: Proceedings of the 20th IEEE International Symposium on Modeling, Analysis and Simulation of Computer and Telecommunication Systems, (MASCOTS), pp. 231–240 (2012)Google Scholar
  21. 21.
    Towsley, D., Romel, G., Astankovic, J.: Analysis of fork-join program response times on multiprocessors. IEEE Trans. Parallel Distrib. Syst. 1(3), 286–303 (1990)CrossRefGoogle Scholar
  22. 22.
    Tsimashenka, I., Knottenbelt, W., Harrison, P.G.: Controlling variability in split-merge systems. In: Al-Begain, K., Fiems, D., Vincent, J.-M. (eds.) ASMTA 2012. LNCS, vol. 7314, pp. 165–177. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  23. 23.
    Tsimashenka, I., Knottenbelt, W.J., Harrison, P.G.: Controlling variability in split-merge systems and its impact on performance. Ann. Oper. Res. 239(2), 569–588 (2016)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Welch, P.D.: On the problem of the initial transient in steady-state simulations. Technical report, IBM Watson Research Center, Yorktown Heights, NY (1981)Google Scholar
  25. 25.
    Wright, P.E.: Two parallel processors with coupled inputs. Adv. Appl. Probab. 24, 986–1007 (1992)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.DAIS - Università Ca’ FoscariVeneziaItaly

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