Annals of Operations Research

, Volume 160, Issue 1, pp 227–255 | Cite as

Generalized parallel-server fork-join queues with dynamic task scheduling

  • Mark S. Squillante
  • Yanyong Zhang
  • Anand Sivasubramaniam
  • Natarajan Gautam
Article
First Online:
Received:
Accepted:

Abstract

This paper introduces a generalization of the classical parallel-server fork-join queueing system in which arriving customers fork into multiple tasks, every task is uniquely assigned to one of the set of single-server queues, and each task consists of multiple iterations of different stages of execution, including task vacations and communication among sibling tasks. Several classes of dynamic polices are considered for scheduling multiple tasks at each of the single-server queues to maintain effective server utilization. The paper presents an exact matrix-analytic analysis of generalized parallel-server fork-join queueing systems, for small instances of the stochastic model, and presents an approximate matrix-analytic analysis and fixed-point solution, for larger instances of the model.

Keywords

Quasi-birth-death processes Matrix-analytic methods Stochastic decomposition Parallel-server fork-join queues Stochastic networks 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Asmussen, S., Nerman, O., & Olsson, M. (1996). Fitting phase-type distributions via the EM algorithm. Scandinavian Journal of Statistics, 23, 419–441. Google Scholar
  2. Baccelli, F., Makowski, A. M., & Shwartz, A. (1989). The fork-join queue and related systems with synchronization constraints: Stochastic ordering and computable bounds. Advances in Applied Probability, 21, 629–660. CrossRefGoogle Scholar
  3. Bright, L., & Taylor, P. (1995). Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes. Stochastic Models, 11, 497–526. CrossRefGoogle Scholar
  4. Bright, L., & Taylor, P. (1997). Equilibrium distributions for level-dependent quasi-birth-and-death processes. In S. R. Chakravarthy & A. S. Alfa (Eds.), Lecture notes in pure and applied mathematics : Vol. 183. Matrix-analytic methods in stochastic models (pp. 359–375). New York: Dekker. Google Scholar
  5. Chapin, S. J., Cirne, W., Feitelson, D. G., Jones, J. P., Leutenegger, S. T., Schwiegelshohn, U., Smith, W., & Talby, D. (1999). Benchmarks and standards for the evaluation of parallel job schedulers. In D. G. Feitelson & L. Rudolph (Eds.), Lecture notes in computer science : Vol. 1659. Job scheduling strategies for parallel processing (pp. 67–90). Berlin: Springer. CrossRefGoogle Scholar
  6. Chen, R. J. (2001). A hybrid solution of fork/join synchronization in parallel queues. IEEE Transactions on Parallel and Distributed Systems, 12, 829–845. CrossRefGoogle Scholar
  7. Dusseau, A. C., Arpaci, R. H., & Culler, D. E. (1996). Effective distributed scheduling of parallel workloads. In Proceedings of the ACM SIGMETRICS 1996 conference on measurement and modeling of computer systems (pp. 25–36). Google Scholar
  8. Feldmann, A., & Whitt, W. (1998). Fitting mixtures of exponentials to long-tail distributions to analyze network performance models. Performance Evaluation, 31, 245–279. CrossRefGoogle Scholar
  9. Gamarnik, D., Jengte, N., Lu, Y., Ramachandran, B., Squillante, M. S., Radovanovic, A., Benayon, J., & Szaloky, V. (2006). Analysis of business processes using queueing analytics. Preprint. Google Scholar
  10. Islam, N., Prodromidis, A., & Squillante, M. S. (1996). Dynamic partitioning in different distributed-memory environments. In D. G. Feitelson & L. Rudolph (Eds.), Lecture notes in computer science : Vol. 1162. Job scheduling strategies for parallel processing (pp. 244–270). Berlin: Springer. CrossRefGoogle Scholar
  11. Kelly, F. P. (1991). Loss networks. Annals of Applied Probability, 1(3), 319–378. CrossRefGoogle Scholar
  12. Ko, S.-S., & Serfozo, R. F. (2004). Response times in M/M/s fork-join networks. Advances in Applied Probability, 36(3), 854–871. CrossRefGoogle Scholar
  13. Latouche, G., & Ramaswami, V. (1993). A logarithmic reduction algorithm for quasi-birth-and-death processes. Journal of Applied Probability, 30, 650–674. CrossRefGoogle Scholar
  14. Latouche, G., & Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modeling. Philadelphia: ASA-SIAM. Google Scholar
  15. Latouche, G., & Taylor, P. (2000). Advances in algorithmic methods for stochastic models. Neshanic Station: Notable. Google Scholar
  16. Little, J. D. C. (1961). A proof of the queuing formula L=λ W. Operations Research, 9, 383–387. CrossRefGoogle Scholar
  17. Nagar, S., Banerjee, A., Sivasubramaniam, A., & Das, C. R. (1999a). A closer look at coscheduling approaches for a network of workstations. In Proceedings of the eleventh annual ACM symposium on parallel algorithms and architectures (pp. 96–105). Google Scholar
  18. Nagar, S., Banerjee, A., Sivasubramaniam, A., & Das, C. R. (1999b). Alternatives to coscheduling a network of workstations. Journal of Parallel and Distributed Computing, 59(2), 302–327. CrossRefGoogle Scholar
  19. Naik, V. K., Setia, S. K., & Squillante, M. S. (1997). Processor allocation in multiprogrammed, distributed-memory parallel computer systems. Journal of Parallel and Distributed Computing, 46(1), 28–47. CrossRefGoogle Scholar
  20. Nelson, R. D., & Squillante, M. S. (2006). Parallel-server stochastic systems with dynamic affinity scheduling and load balancing. Preprint. Google Scholar
  21. Nelson, R. D., & Tantawi, A. N. (1988). Approximate analysis of fork/join synchronization in parallel queues. IEEE Transactions on Computers, 37(6), 739–743. CrossRefGoogle Scholar
  22. Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models: an algorithmic approach. Baltimore: Johns Hopkins Press. Google Scholar
  23. Neuts, M. F. (1989). Structured stochastic matrices of M/G/1 type and their applications. New York: Dekker. Google Scholar
  24. Riska, A., Squillante, M. S., Yu, S.-Z., Liu, Z., & Zhang, L. (2002). Matrix-analytic analysis of a MAP/PH/1 queue fitted to Web server data. In G. Latouche & P. Taylor (Eds.), Matrix-analytic methods: theory and applications (pp. 333–356). Singapore: World Scientific. Google Scholar
  25. Riska, A., Diev, V., & Smirni, E. (2004). An EM-based technique for approximating long-tailed data sets with PH distributions. Performance Evaluation, 55(1–2), 147–164. CrossRefGoogle Scholar
  26. Sobalvarro, P. G. (1997). Demand-based coscheduling of parallel jobs on multiprogrammed multiprocessors. Ph.D. thesis, Dept. of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, January 1997. Google Scholar
  27. Squillante, M. S. (2007). Stochastic analysis of multiserver systems. Performance Evaluation Review, 34(4), 44–51. CrossRefGoogle Scholar
  28. Squillante, M. S., & Nelson, R. D. (1991). Analysis of task migration in shared-memory multiprocessors. In Proceedings of ACM SIGMETRICS conference on measurement and modeling of computer systems (pp. 143–155). New York: ACM. Google Scholar
  29. Tan, X., & Knessl, C. (1996). A fork-join queueing model: Diffusion approximation, integral representations and asymptotics. Queueing Systems Theory and Applications, 22, 287–322. CrossRefGoogle Scholar
  30. Varma, S., & Makowski, A. M. (1994). Interpolation approximations for symmetric fork-join queues. Performance Evaluation, 20, 245–265. CrossRefGoogle Scholar
  31. Zhang, Y., Sivasubramaniam, A., Moreira, J., & Franke, H. (2000). A simulation-based study of scheduling mechanisms for a dynamic cluster environment. In Proceedings of the ACM 2000 international conference on supercomputing (pp. 100–109). Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Mark S. Squillante
    • 1
  • Yanyong Zhang
    • 2
  • Anand Sivasubramaniam
    • 3
  • Natarajan Gautam
    • 4
  1. 1.Mathematical Sciences DepartmentIBM Thomas J. Watson Research CenterYorktown HeightsUSA
  2. 2.Department of Electrical and Computer EngineeringRutgers UniversityPiscatawayUSA
  3. 3.Department of Computer Science and EngineeringPennsylvania State UniversityUniversity ParkUSA
  4. 4.Department of Industrial and Systems EngineeringTexas A&M UniversityCollege StationUSA

Personalised recommendations