Journal of Statistical Theory and Practice

, Volume 2, Issue 2, pp 279–292 | Cite as

Parallel Computing, Failure Recovery, and Extreme Values

  • Lars Nørvang AndersenEmail author
  • Søren Asmussen


A task of random size T is split into M subtasks of lengths T1, …, TM, each of which is sent to one out of M parallel processors. Each processor may fail at a random time before completing its allocated task, and then has to restart it from the beginning. If X1, …,XM are the total task times at the M processors, the overall total task time is then ZM = max1,…,MXi. Limit theorems as M → ∞ are given for ZM, allowing the distribution of T to depend on M. In some cases the limits are classical extreme value distributions, in others they are of a different type.


Cramér-Lundberg approximation failure recovery Fréchet distribution geometric sums Gumbel distribution heavy tails logarithmic asymptotics mixture distribution power tail RESTART triangular array 

AMS Subject Classification

Primary: 60G70 Secondary: 60F05, 68M20 


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  1. Asmussen, S., Fiorini, P., Lipsky, L., Rolski, T., Sheahan, R., 2007/8. On the distribution of total task times for tasks that must restart from the beginning if failure occurs. Mathematics of Operations Research (to appear).zbMATHGoogle Scholar
  2. Asmussen, S., Glynn, P.W., 2007. Stochastic Simulation: Algorithms and Analysis. Springer-VerlagCrossRefGoogle Scholar
  3. Bingham, N.H., Goldie, C.M., Teugels, J.L., 1987. Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
  4. Bobbio, A., Trivedi, K., 1990. Computation of the distribution of the completion time when the work requirement is a PH random variable. Stochastic Models 6, 133–150.MathSciNetCrossRefGoogle Scholar
  5. Castillo, X., Siewiorek, D.P., 1980. A performance-reliability model for computing systems. Proc FTCS-10, Silver Spring, MD, IEEE Computer Soc., 187–192.Google Scholar
  6. Chimento, Jr., P.F., Trivedi, K.S., 1993. The completion time of programs on processors subject to failure and repair. IEEE Trans. on Computers 42(1).Google Scholar
  7. Chlebus, B.S., De Prisco, R., Shvartsman, A.A., 2001. Performing tasks on synchronous restartable message-passing processors. Distributed Computing 14, 49–64.CrossRefGoogle Scholar
  8. De Prisco, R., Mayer, A., Yung, M., 1994. Time-optimal message-efficient work performance in the presence of faults. Proc. 13th ACM PODC, 161–172.Google Scholar
  9. Freitas, A.V., Hüsler, J., 2003. Conditions for the convergence of maxima of random triangular arrays. Extremes 6, 381–394.MathSciNetCrossRefGoogle Scholar
  10. Hoffmann-Jøgensen, J., 1994. Probability With a View Yoward Statistics, Vol. I. Chapman & Hall.Google Scholar
  11. Jelenkovic, P., Tan, V., 2007. Can retransmission of superexponential documents cause subexponential delays? Proc. IEEE Infocom2007, pp. 892–900, Anchorage, 6–12 May 2007.Google Scholar
  12. Kulkarni, V., Nicola, V., Trivedi, K., 1986. On modeling the performance and reliability of multimode systems. The Journal of Systems and Software 6, 175–183.CrossRefGoogle Scholar
  13. Kulkarni, V., Nicola, V., Trivedi, K., 1987. The completion time of a job on a multimode system. Advances in Applied Probability 19, 932–954.MathSciNetCrossRefGoogle Scholar
  14. Leadbetter, M.R., Lindgren, G., Rootzén, H., 1983. Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag.CrossRefGoogle Scholar
  15. Pickands, III, J., 1968. Moment convergence of sample extremes. Annals of Mathematical Statistics 39, 881–889.MathSciNetCrossRefGoogle Scholar
  16. Sheahan, R., Lipsky, L., Fiorini, P., Asmussen, S., 2006. On the distribution of task completion times for tasks that must restart from the beginning if failure occurs. ACM SIGMETRICS Performance Evaluation Review 34, 24–26.CrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2008

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAarhus University Ny MunkegadeAarhus CDenmark

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