Stochastic Bounds for Markov Chains with the Use of GPU

  • Jarosław Bylina
  • Jean-Michel Fourneau
  • Marek Karwacki
  • Nihal Pekergin
  • Franck Quessette
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 522)


The authors present a new approach to find stochastic bounds for a Markov chain – namely with the use of the GPU for computing the bounds. A known algorithm [1, 2] is used and it is rewritten to suit the GPU architecture with the cooperation of the CPU. The authors do some experiments with matrices from various models as well as some random matrices. The tests are analyzed and some future considerations are given.


Markov chains GPU Stochastic bounds Sparse matrices Heterogeneous algorithms 


  1. 1.
    Abu-Amsha, O., Vincent, J.-M.: An algorithm to bound functionals on Markov chains with large state space. In: 4th INFORMS Conference on Telecommunications, Boca Raton, Floride, E.U. INFORMS (1998)Google Scholar
  2. 2.
    Fourneau, J.-M., Pekergin, N.: An algorithmic approach to stochastic bounds. In: Calzarossa, M.C., Tucci, S. (eds.) Performance 2002. LNCS, vol. 2459, pp. 64–88. Springer, Heidelberg (2002) CrossRefGoogle Scholar
  3. 3.
    Stewart, W.J.: Introduction to the numerical Solution of Markov Chains. Princeton University Press, New Jersey (1995)Google Scholar
  4. 4.
    Bylina, J., Bylina, B., Karwacki, M.: A Markovian model of a network of two wireless devices. In: Kwiecień, A., Gaj, P., Stera, P. (eds.) CN 2012. CCIS, vol. 291, pp. 411–420. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  5. 5.
    Mamoun, M.B., Busic, A., Pekergin, N.: Generalized class C Markov chains and computation of closed-form bounding distributions. Probab. Eng. Inf. Sci. 21, 235–260 (2007)zbMATHGoogle Scholar
  6. 6.
    Busic, A., Fourneau, J.-M.: A matrix pattern compliant strong stochastic bound. In: 2005 IEEE/IPSJ International Symposium on Applications and the Internet Workshops (SAINT 2005 Workshops), Trento, Italy, pp. 260–263. IEEE Computer Society (2005)Google Scholar
  7. 7.
    Dayar, T., Pekergin, N., Younès, S.: Conditional steady-state bounds for a subset of states in Markov chains. In: Structured Markov Chain (SMCTools) Workshop in the 1st International Conference on Performance Evaluation Methodolgies and Tools, VALUETOOLS 2006, Pisa, Italy. ACM (2006)Google Scholar
  8. 8.
    Fourneau, J.-M., Le Coz, M., Pekergin, N., Quessette, F.: An open tool to compute stochastic bounds on steady-state distributions and rewards. In: 11th International Workshop on Modeling, Analysis, and Simulation of Computer and Telecommunication Systems (MASCOTS 2003), Orlando, FL, IEEE Computer Society (2003)Google Scholar
  9. 9.
    Fourneau, J.-M., Le Coz, M., Quessette, F.: Algorithms for an irreducible and lumpable strong stochastic bound. Linear Algebra Appl. 386, 167–185 (2004)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Thompson, C.J., Hahn, S., Oskin, M.: Using modern graphics architectures for general-purpose computing: a framework and analysis. In: Proceedings of the 35th Annual ACM/IEEE International Symposium on Microarchitecture, pp. 306–317. IEEE Computer Society Press, Los Alamitos (2002)Google Scholar
  11. 11.
    Kijima, M.: Markov Processes for Stochastic Modeling. Chapman & Hall, London (1997) CrossRefzbMATHGoogle Scholar
  12. 12.
    Muller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, New York (2002) Google Scholar
  13. 13.
    Shaked, M., Shantikumar, J.G.: Stochastic Orders and Their Applications. Academic Press, San Diego (1994) zbMATHGoogle Scholar
  14. 14.
    Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. Wiley, Berlin (1983) zbMATHGoogle Scholar
  15. 15.
    Bylina, B., Bylina, J., Karwacki, M.: Computational aspects of GPU-accelerated sparse matrix-vector multiplication for solving Markov models. Theor. Appl. Inform. 23(2), 127–145 (2011)Google Scholar
  16. 16.
    Bylina, J., Bylina, B., Karwacki, M.: An efficient representation on GPU for transition rate matrices for Markov chains. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Waśniewski, J. (eds.) PPAM 2013, Part I. LNCS, vol. 8384, pp. 663–672. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  17. 17.
    Bylina, B., Karwacki, M., Bylina, J.: A CPU-GPU hybrid approach to the uniformization method for solving Markovian models – a case study of a wireless network. In: Kwiecień, A., Gaj, P., Stera, P. (eds.) CN 2012. CCIS, vol. 291, pp. 401–410. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  18. 18.
    Lee, J., Samadi, M., Park, Y., Mahlke, S.: Transparent CPU-GPU collaboration for data-parallel kernels on heterogeneous systems. In: Proceedings of the 22nd International Conference on Parallel Architectures and Compilation Techniques (PACT), September 2013Google Scholar
  19. 19.
    Ohshima, S., Kise, K., Katagiri, T., Yuba, T.: Parallel processing of matrix multiplication in a CPU and GPU heterogeneous environment. In: Daydé, M., Palma, J.M.L.M., Coutinho, A.L.G.A., Pacitti, E., Lopes, J.C. (eds.) VECPAR 2006. LNCS, vol. 4395, pp. 305–318. Springer, Heidelberg (2007) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jarosław Bylina
    • 1
  • Jean-Michel Fourneau
    • 2
  • Marek Karwacki
    • 1
  • Nihal Pekergin
    • 3
  • Franck Quessette
    • 2
  1. 1.Institute of MathematicsMarie Curie-Skłodowska UniversityLublinPoland
  2. 2.PRiSMCNRS and Univ. Versailles St QuentinVersaillesFrance
  3. 3.LACLUPECCréteilFrance

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