A Matrix-Analytic Solution for Randomized Load Balancing Models with PH Service Times

  • Quan-Lin Li
  • John C. S. Lui
  • Yang Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6821)


In this paper, we provide a matrix-analytic solution for randomized load balancing models (also known as supermarket models) with phase-type (PH) service times. Generalizing the service times to the phase-type distribution makes analysis of the supermarket models more difficult and challenging than that of the exponential service time case which has been extensively discussed in the literature. We describe the supermarket model as a system of differential vector equations, provide a doubly exponential solution to the fixed point of the system of differential vector equations, and analyze the exponential convergence of the current location of the supermarket model to its fixed point.


  1. 1.
    Bramson, M., Lu, Y., Prabhakar, B.: Randomized load balancing with general service time distributions. In: Proceedings of the ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, pp. 275–286 (2010)Google Scholar
  2. 2.
    Dahlin, M.: Interpreting stale load information. IEEE Transactions on Parallel and Distributed Systems 11, 1033–1047 (1999)CrossRefGoogle Scholar
  3. 3.
    Harchol-Balter, M., Downey, A.B.: Exploiting process lifetime distributions for dynamic load balancing. ACM Transactions on Computer Systems 15, 253–285 (1997)CrossRefGoogle Scholar
  4. 4.
    Luczak, M., McDiarmid, C.: On the maximum queue length in the supermarket model. The Annals of Probability 34, 493–527 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Martin, J.B.: Point processes in fast Jackson networks. The Annals of Applied Probability 11, 650–663 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Martin, J.B., Suhov, Y.M.: Fast Jackson networks. The Annals of Applied Probability 9, 854–870 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Mitzenmacher, M.D.: The power of two choices in randomized load balancing. PhD thesis, University of California at Berkeley, Department of Computer Science, Berkeley, CA (1996)Google Scholar
  8. 8.
    Mitzenmacher, M.D.: Analyses of load stealing models using differential equations. In: Proceedings of the Tenth ACM Symposium on Parallel Algorithms and Architectures, pp. 212–221 (1998)Google Scholar
  9. 9.
    Mitzenmacher, M.D.: On the analysis of randomized load balancing schemes. Theory of Computing Systems 32, 361–386 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Mitzenmacher, M.D.: How useful is old information? IEEE Transactions on Parallel and Distributed Systems 11, 6–20 (2000)CrossRefGoogle Scholar
  11. 11.
    Mitzenmacher, M.D., Richa, A., Sitaraman, R.: The power of two random choices: a survey of techniques and results. In: Pardalos, P., Rajasekaran, S., Rolim, J. (eds.) Handbook of Randomized Computing, vol. 1, pp. 255–312 (2001)Google Scholar
  12. 12.
    Suhov, Y.M., Vvedenskaya, N.D.: Fast Jackson Networks with Dynamic Routing. Problems of Information Transmission 38, 136–153 (2002)CrossRefzbMATHGoogle Scholar
  13. 13.
    Telek, M., Heindl, A.: Matching moments for acyclic discrete and continuous phase-type distributions of second order. International Journal of Simulation: Systems, Science & Technology 3, 47–57 (2002)Google Scholar
  14. 14.
    Vvedenskaya, N.D., Dobrushin, R.L., Karpelevich, F.I.: Queueing system with selection of the shortest of two queues: An asymptotic approach. Problems of Information Transmissions 32, 20–34 (1996)MathSciNetGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2011

Authors and Affiliations

  • Quan-Lin Li
    • 1
  • John C. S. Lui
    • 2
  • Yang Wang
    • 3
  1. 1.School of Economics and Management SciencesYanshan UniversityChina
  2. 2.Department of Computer Science & EngineeringThe Chinese University of Hong KongChina
  3. 3.Department of Computer Science and TechnologyPeking UniversityChina

Personalised recommendations