Skip to main content

Stochastic Analysis and Optimization of Multiserver Systems

  • Chapter

Part of the book series: Autonomic Systems ((ASYS))

Abstract

Motivated by emerging trends and applications such as autonomic computing, this chapter presents an overview of some research in the stochastic analysis and optimization of multiserver systems. Our primary focus is on multiserver systems in general, since this research provides the mathematical methods and results that have been and will continue to be used for the stochastic analysis and/or optimization of existing and future multiserver systems arising in a wide variety of application domains including autonomic computing.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adan, I.J., Boxma, O.J., Resing, J.: Queueing models with multiple waiting lines. Queueing Syst. Theory Appl. 37, 65–98 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adan, I.J., Wessels, J., Zijm, W.: Analysis of the asymmetric shortest queue problem. Queueing Syst. Theory Appl. 8, 1–58 (1989)

    Article  MathSciNet  Google Scholar 

  3. Anselmi, J., Lu, Y., Sharma, M., Squillante, M.S.: Improved approximations for the Erlang loss model. Queueing Syst. Theory Appl. 63, 217–239 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Aven, O.I., Coffman, E.G. Jr., Kogan, Y.A.: Stochastic Analysis of Computer Storage. Dordrecht, Reidel (1987)

    MATH  Google Scholar 

  5. Avram, F., Bertsimas, D., Ricard, M.: Fluid models of sequencing problems in open queueing networks: An optimal control approach. In: F. Kelly, R. Williams (eds.) Stochastic Networks. IMA, vol. 71, pp. 199–234 (1995)

    Google Scholar 

  6. Baccelli, F., Makowski, A.M., Shwartz, A.: The fork-join queue and related systems with synchronization constraints: Stochastic ordering and computable bounds. Adv. Appl. Probab. 21, 629–660 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bell, S.L., Williams, R.J.: Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: Asymptotic optimality of a threshold policy. Ann. Appl. Probab. 11, 608–649 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bell, S.L., Williams, R.J.: Dynamic scheduling of a parallel server system in heavy traffic with complete resource pooling: Asymptotic optimality of a threshold policy. Electron. J. Probab. 10, 1044–1115 (2005)

    MathSciNet  Google Scholar 

  9. Bertsimas, D., Gamarnik, D., Tsitsiklis, J.: Performance of multiclass Markovian queueing networks via piecewise linear Lyapunov functions. Ann. Appl. Probab. 11(4), 1384–1428 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bhadra, S., Lu, Y., Squillante, M.S.: Optimal capacity planning in stochastic loss networks with time-varying workloads. In: Proceedings of ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems, pp. 227–238. ACM, New York (2007)

    Google Scholar 

  11. Bonald, T.: The Erlang model with non-Poisson call arrivals. In: Proceedings of Joint SIGMETRICS/Performance Conference on Measurement and Modeling of Computer Systems, pp. 276–286. ACM, New York (2006)

    Google Scholar 

  12. Boxma, O.J., Koole, G.M., Liu, Z.: Queueing-theoretic solution methods for models of parallel and distributed systems. In: O.J. Boxma, G.M. Koole (eds.) Performance Evaluation of Parallel and Distributed Systems, pp. 1–24. CWI Tract 105, Amsterdam (1994)

    Google Scholar 

  13. Bramson, M.: Instability of FIFO queueing networks. Ann. Appl. Probab. 4, 414–431 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chen, H., Yao, D.D.: Dynamic scheduling of a multiclass fluid network. Oper. Res. 41(6), 1104–1115 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chen, H., Yao, D.D.: Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization. Springer, Berlin (2001)

    MATH  Google Scholar 

  16. Cohen, J.W.: The Single Server Queue, 1st edn., North-Holland, Amsterdam (1969). Second edition, 1982

    MATH  Google Scholar 

  17. Cohen, J.W.: A two-queue, one-server model with priority for the longer queue. Queueing Syst. Theory Appl. 2(3), 261–283 (1987)

    Article  MATH  Google Scholar 

  18. Cohen, J.W.: Boundary value problems in queueing theory. Queueing Syst. Theory Appl. 3, 97–128 (1988)

    Article  MATH  Google Scholar 

  19. Cohen, J.W., Boxma, O.J.: Boundary Value Problems in Queueing System Analysis. North-Holland, Amsterdam (1983)

    MATH  Google Scholar 

  20. Courtois, P.J.: Decomposability. Academic Press, San Diego (1977)

    MATH  Google Scholar 

  21. Courtois, P.J., Semal, P.: Error bounds for the analysis by decomposition of non-negative matrices. In: Proceedings of International Workshop on Applied Mathematics and Performance/Reliability Models of Computer/Communication Systems, pp. 253–268 (1983)

    Google Scholar 

  22. Dai, J.G.: On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models. Ann. Appl. Probab. 5, 49–77 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  23. Dallery, Y., Frein, Y.: On decomposition methods for tandem queueing networks with blocking. Oper. Res. 41, 386–399 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  24. El Gamal, A., Mammen, J., Prabhakar, B., Shah, D.: Throughput-delay trade-off in wireless networks. In: Proc. IEEE Infocom, March 2004

    Google Scholar 

  25. El Gamal, A., Mammen, J., Prabhakar, B., Shah, D.: Optimal throughput-delay scaling in wireless networks—Part I: The fluid model. IEEE Trans. Inf. Theory 52(6), 2568–2592 (2006)

    Article  Google Scholar 

  26. El Gamal, A., Mammen, J., Prabhakar, B., Shah, D.: Optimal throughput-delay scaling in wireless networks—Part II: Constant-size packets. IEEE Trans. Inf. Theory 52(11), 5111–5116 (2006)

    Article  MathSciNet  Google Scholar 

  27. Fayolle, G., Iasnogorodski, R.: Two coupled processors: The reduction to a Reimann-Hilbert problem. Z. Wahrscheinlichkeitstheor. Verw. Geb. 47, 325–351 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  28. Fayolle, G., Iasnogorodski, R., Malyshev, V.: Random Walks in the Quarter-Plane: Algebraic Methods, Boundary Value Problems and Applications. Springer, Berlin (1999)

    MATH  Google Scholar 

  29. Fayolle, G., Malyshev, V.A., Menshikov, M.V.: Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  30. Flatto, L.: The longer queue model. Probab. Eng. Inf. Sci. 3, 537–559 (1989)

    Article  MATH  Google Scholar 

  31. Flatto, L., Hahn, S.: Two parallel queues created by arrivals with two demands. SIAM J. Appl. Math. 44, 1041–1053 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  32. Fleischer, L.K., Sethuraman, J.: Efficient algorithms for separated continuous linear programs: The multicommodity flow problem with holding costs and extensions. Math. Oper. Res. 30(4), 916–938 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  33. Foschini, G.J., Salz, J.: A basic dynamic routing problem and diffusion. IEEE Trans. Commun. 26(3), 320–327 (1978)

    Article  MATH  Google Scholar 

  34. Gamarnik, D., Squillante, M.S.: Analysis of stochastic online bin packing processes. Stoch. Models 21, 401–425 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  35. Gans, N., Koole, G., Mandelbaum, A.: Telephone call centers: Tutorial, review, and research prospects. Manuf. Serv. Oper. Manag. 5, 79–141 (2003)

    Article  Google Scholar 

  36. Gaver, D.P., Jr.: A waiting line with interrupted service, including priorities. J. R. Stat. Soc., Ser. B 24, 73–90 (1962)

    MATH  MathSciNet  Google Scholar 

  37. Gupta, P., Kumar, P.R.: The capacity of wireless networks. IEEE Trans. Inf. Theory 46(2), 388–404 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  38. Halfin, S., Whitt, W.: Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29, 567–588 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  39. Harchol-Balter, M., Li, C., Osogami, T., Scheller-Wolf, A., Squillante, M.S.: Cycle stealing under immediate dispatch task assignment. In: Proceedings of Annual ACM Symposium on Parallel Algorithms and Architectures, pp. 274–285, June 2003

    Google Scholar 

  40. Harchol-Balter, M., Li, C., Osogami, T., Scheller-Wolf, A., Squillante, M.S.: Task assignment with cycle stealing under central queue. In: Proceedings of International Conference on Distributed Computing Systems, pp. 628–637, May 2003

    Google Scholar 

  41. Harchol-Balter, M., Osogami, T., Scheller-Wolf, A., Wierman, A.: Multi-server queueing systems with multiple priority classes. Queueing Syst. Theory Appl. 51, 331–360 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  42. Harrison, J.M.: Brownian Motion and Stochastic Flow Systems. Wiley, New York (1985)

    MATH  Google Scholar 

  43. Harrison, J.M., Lopez, M.J.: Heavy traffic resource pooling in parallel-server systems. Queueing Syst. Theory Appl. 33, 339–368 (1989)

    Article  MathSciNet  Google Scholar 

  44. Harrison, J.M., Zeevi, A.: Dynamic scheduling of a multiclass queue in the Halfin and Whitt heavy traffic regime. Oper. Res. 52, 243–257 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  45. Jaiswal, N.K.: Priority Queues. Academic Press, San Diego (1968)

    MATH  Google Scholar 

  46. Jelenković, P., Momčilović, P., Squillante, M.S.: Buffer scalability of wireless networks. In: Proc. IEEE Infocom, April 2006

    Google Scholar 

  47. Jelenković, P., Momčilović, P., Squillante, M.S.: Scalability of wireless networks. IEEE/ACM Trans. Netw. 15(2) (2007)

    Google Scholar 

  48. Jovičić, A., Viswanath, P., Kulkarni, S.: Upper bounds to transport capacity of wireless networks. IEEE Trans. Inf. Theory 50(11), 2555–2565 (2004)

    Article  Google Scholar 

  49. Jung, K., Lu, Y., Shah, D., Sharma, M., Squillante, M.S.: Revisiting stochastic loss networks: Structures and algorithms. In: Proceedings of ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems, pp. 407–418. ACM, New York (2008)

    Google Scholar 

  50. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  51. Kelly, F.P.: Blocking probabilities in large circuit-switched networks. Adv. Appl. Probab. 18(2), 473–505 (1986)

    Article  MATH  Google Scholar 

  52. Kelly, F.P.: Loss networks. Ann. Appl. Probab. 1(3), 319–378 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  53. Kumar, P.R.: Re-entrant lines. Queueing Syst. Theory Appl. 13, 87–110 (1993)

    Article  MATH  Google Scholar 

  54. Kumar, P.R., Xie, L.-L.: A network information theory for wireless communications: Scaling laws and optimal operation. IEEE Trans. Inf. Theory 50(5), 748–767 (2004)

    Article  MathSciNet  Google Scholar 

  55. Kushner, H.J., Dupuis, P.: Numerical Methods for Stochastic Control Problems in Continuous Time. Springer, Berlin (1992)

    MATH  Google Scholar 

  56. Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM, Philadelphia (1999)

    MATH  Google Scholar 

  57. van Leeuwaarden, J.S., Squillante, M.S., Winands, E.M.: Quasi-birth-and-death processes, lattice path counting, and hypergeometric functions. J. Appl. Probab. 46(2), 507–520 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  58. Leveque, O., Telatar, E.: Information theoretic upper bounds on the capacity of ad hoc networks. IEEE Trans. Inf. Theory 51(3), 858–865 (2005)

    Article  MathSciNet  Google Scholar 

  59. Louth, G., Mitzenmacher, M., Kelly, F.: Computational complexity of loss networks. Theor. Comput. Sci. 125(1), 45–59 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  60. Meyn, S.P.: Sequencing and routing in multiclass queueing networks. Part I: Feedback regulation. SIAM J. Control Optim. 40, 741–776 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  61. Meyn, S.P.: Sequencing and routing in multiclass queueing networks. Part II: Workload relaxations. SIAM J. Control Optim. 42, 178–217 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  62. Meyn, S.P.: Dynamic safety-stocks for asymptotic optimality in stochastic networks. Queueing Syst. Theory Appl. 50, 255–297 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  63. Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer, Berlin (1993). Available at probability.ca/MT

    MATH  Google Scholar 

  64. Nelson, R.D., Philips, T.K.: An approximation for the mean response time for shortest queue routing with general interarrival and service times. Perform. Eval. 17, 123–139 (1993)

    Article  MATH  Google Scholar 

  65. Nelson, R.D., Squillante, M.S.: Parallel-server stochastic systems with dynamic affinity scheduling and load balancing. Preprint (2006)

    Google Scholar 

  66. Nelson, R.D., Tantawi, A.N.: Approximate analysis of fork/join synchronization in parallel queues. IEEE Trans. Comput. 37(6), 739–743 (1988)

    Article  Google Scholar 

  67. Neuts, M.F.: Moment formulas for the Markov renewal branching process. Adv. Appl. Probab. 8, 690–711 (1978)

    Article  MathSciNet  Google Scholar 

  68. Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore (1981)

    MATH  Google Scholar 

  69. Neuts, M.F.: Structured Stochastic Matrices of M/G/1 Type and Their Applications. Dekker, New York (1989)

    MATH  Google Scholar 

  70. Nowicki, T., Squillante, M.S., Wu, C.W.: Fundamentals of dynamic decentralized optimization in autonomic computing systems. In: Babaoglu, O., Jelasity, M., Montresor, A., Fetzer, C., Leonardi, S., van Moorsel, A., van Steen, M. (eds.) Self-star Properties in Complex Information Systems: Conceptual and Practical Foundations. Lecture Notes in Computer Science, vol. 3460, pp. 204–218. Springer, Berlin (2005)

    Chapter  Google Scholar 

  71. Osogami, T.: Analysis of Multi-server Systems via Dimensionality Reduction of Markov Chains. PhD thesis, Carnegie Mellon University (2005)

    Google Scholar 

  72. Osogami, T., Harchol-Balter, M., Scheller-Wolf, A.: Analysis of cycle stealing with switching times and thresholds. In: Proceedings of ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems, pp. 184–195. ACM, New York (2003)

    Google Scholar 

  73. Osogami, T., Harchol-Balter, M., Scheller-Wolf, A.: Analysis of cycle stealing with switching times and thresholds. Perform. Eval. 61, 347–369 (2005)

    Article  Google Scholar 

  74. Osogami, T., Harchol-Balter, M., Scheller-Wolf, A., Zhang, L.: Exploring threshold-based policies for load sharing. In: Proceedings of the 42nd Annual Allerton Conference on Communication, Control, and Computing (2004)

    Google Scholar 

  75. Peris, V.G., Squillante, M.S., Naik, V.K.: Analysis of the impact of memory in distributed parallel processing systems. In: Proceedings of ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems, pp. 5–18. ACM, New York (1994)

    Chapter  Google Scholar 

  76. Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., Mishchenko, E.: The Mathematical Theory of Optimal Processes. Interscience, New York (1962)

    MATH  Google Scholar 

  77. Prener, D.: Personal communication (2007)

    Google Scholar 

  78. Reiman, M.I., Simon, B.: An interpolation approximation for queueing systems with Poisson input. Oper. Res. 36(3), 454–469 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  79. Rybko, A.N., Stolyar, A.L.: Ergodicity of stochastic processes describing the operations of open queueing networks. Probl. Inf. Transm. 28, 199–220 (1992)

    MATH  MathSciNet  Google Scholar 

  80. Seierstad, A., Sydsieter, K.: Sufficient conditions in optimal control theory. Int. Econ. Rev. 18(2), 367–391 (1977)

    Article  MATH  Google Scholar 

  81. Sethuraman, J., Squillante, M.S.: Optimal stochastic scheduling in multiclass parallel queues. In: Proceedings of ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems, pp. 93–102. ACM, New York (1999)

    Google Scholar 

  82. Sethuraman, J., Squillante, M.S.: Analysis of parallel-server queues under spacesharing and timesharing disciplines. In: Latouche, G., Taylor, P. (eds.) Matrix-Analytic Methods: Theory and Applications, pp. 357–380. World Scientific, Singapore (2002)

    Google Scholar 

  83. Squillante, M.S.: Issues in Shared-Memory Multiprocessor Scheduling: A Performance Analysis. PhD thesis, Department of Computer Science, University of Washington, September 1990

    Google Scholar 

  84. Squillante, M.S.: Stochastic analysis of resource allocation in parallel processing systems. In: Gelenbe, E. (ed.) Computer System Performance Modeling in Perspective: A Tribute to the Work of Prof. K.C. Sevcik, pp. 227–256. Imperial College Press, London (2005)

    Google Scholar 

  85. Squillante, M.S., Tsoukatos, K.P.: Analysis of optimal scheduling in distributed parallel queueing systems. In: Proceedings of International Conference on Computer Communication, August 1995

    Google Scholar 

  86. Squillante, M.S., Wang, F., Papaefthymiou, M.: Stochastic analysis of gang scheduling in parallel and distributed systems. Perform. Eval. 27&28, 273–296 (1996)

    Google Scholar 

  87. Squillante, M.S., Zhang, Y., Sivasubramaniam, A., Gautam, N.: Generalized parallel-server fork-join queues with dynamic task scheduling. Ann. Oper. Res. 160, 227–255 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  88. Squillante, M.S., Zhang, Y., Sivasubramaniam, A., Gautam, N., Franke, H., Moreira, J.: Modeling and analysis of dynamic coscheduling in parallel and distributed environments. In: Proceedings of ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems, pp. 43–54. ACM, New York (2002)

    Google Scholar 

  89. Stewart, G.W.: Computable error bounds for aggregated Markov chains. J. ACM 30, 271–285 (1983)

    MATH  Google Scholar 

  90. Takagi, H.: Queueing Analysis—A Foundation of Performance Evaluation, vol. 1. North-Holland, New York (1991)

    MATH  Google Scholar 

  91. Varma, S., Makowski, A.M.: Interpolation approximations for symmetric fork-join queues. Perform. Eval. 20, 245–265 (1994)

    Article  Google Scholar 

  92. Whitt, W.: Blocking when service is required from several facilities simultaneously. AT&T Bell Lab. Techn. J. 64(8), 1807–1856 (1985)

    MATH  MathSciNet  Google Scholar 

  93. Whitt, W.: Stochastic-Process Limits. Springer, New York (2002)

    MATH  Google Scholar 

  94. Williams, R.J.: On dynamic scheduling of a parallel server system with complete resource pooling. Fields Inst. Commun. 28, 49–71 (2000)

    Google Scholar 

  95. Zheng, Y., Zipkin, P.H.: A queueing model to analyze the value of centralized inventory information. Oper. Res. 38, 296–307 (1990)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mark S. Squillante .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Basel AG

About this chapter

Cite this chapter

Squillante, M.S. (2010). Stochastic Analysis and Optimization of Multiserver Systems. In: Ardagna, D., Zhang, L. (eds) Run-time Models for Self-managing Systems and Applications. Autonomic Systems. Springer, Basel. https://doi.org/10.1007/978-3-0346-0433-8_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-0346-0433-8_1

  • Publisher Name: Springer, Basel

  • Print ISBN: 978-3-0346-0432-1

  • Online ISBN: 978-3-0346-0433-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics