Queueing Systems

, 68:385 | Cite as

The variance of departure processes: puzzling behavior and open problems

Article

Abstract

We consider the variability of queueing departure processes. Previous results have shown the so-called BRAVO effect occurring in M/M/1/K and GI/G/1 queues: Balancing Reduces Asymptotic Variance of Outputs. A factor of (1−2/π) appears in GI/G/1 and a factor of 1/3 appears in M/M/1/K, for large K. A missing piece in the puzzle is the GI/G/1/K queue: Is there a BRAVO effect? If so, what is the variability? Does 1/3 play a role?

This open problem paper addresses these questions by means of numeric and simulation results. We conjecture that at least for the case of light tailed distributions, the variability parameter is 1/3 multiplied by the sum of the squared coefficients of variations of the inter-arrival and service times.

Keywords

Queueing theory Loss systems Asymptotic variance rate BRAVO 

Mathematics Subject Classification (2000)

60J27 60K25 

References

  1. 1.
    Al Hanbali, A.: Busy period analysis of the level dependent PH/PH/1/K queue. EURANDOM Res. Rep., 2009-36 (2009) Google Scholar
  2. 2.
    Al Hanbali, A., Mandjes, M., Nazarathy, Y., Whitt, W.: The asymptotic variance of departures in critically loaded queue. Adv. Appl. Probab. 43(1), 243–263 (2011) CrossRefGoogle Scholar
  3. 3.
    Asmussen, S.: Applied Probability and Queues. Springer, Berlin (2003) Google Scholar
  4. 4.
    Berger, A.W., Whitt, W.: The Brownian approximation for rate-control throttles and the G/G/1/C queue. Discrete Event Dyn. Syst. Theory Appl. 2, 7–60 (1992) CrossRefGoogle Scholar
  5. 5.
    Breuer, L., Baum, D.: An Introduction to Queueing Theory and Matrix-Analytic Methods. Springer, Berlin (2005) Google Scholar
  6. 6.
    Burke, P.J.: The output of a queuing system. Oper. Res. 4(6), 699–704 (1956) CrossRefGoogle Scholar
  7. 7.
    Cox, D.R., Isham, V.: Point Processes. Chapman & Hall, London (1980) Google Scholar
  8. 8.
    Daley, D.J.: Queueing output processes. Adv. Appl. Probab. 8, 395–415 (1976) CrossRefGoogle Scholar
  9. 9.
    Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer, Berlin (2003) Google Scholar
  10. 10.
    Daley, J.D., Vesilo, R.: Long range dependence of point processes, with queueing examples. Stoch. Process. Appl. 70(2), 265–282 (1997) CrossRefGoogle Scholar
  11. 11.
    Disney, R.L., König, D.: Queueing networks: A survey of their random processes. SIAM Rev. 27(3), 335–403 (1985) CrossRefGoogle Scholar
  12. 12.
    Kerner, Y., Nazarathy, Y.: On the linear asymptote of the M/G/1 output variance curve. EURANDOM Res. Rep. (2011, in preparation) Google Scholar
  13. 13.
    Nazarathy, Y., Weiss, G.: The asymptotic variance rate of finite capacity birth–death queues. Queueing Syst. 59(2), 135–156 (2008) CrossRefGoogle Scholar
  14. 14.
    Nazarathy, Y., Weiss, G.: Positive Harris recurrence and diffusion scale analysis of a push pull queueing network. Perform. Eval. 67(4), 201–217 (2010) CrossRefGoogle Scholar
  15. 15.
    Tan, B.: Asymptotic variance rate of the output in production lines with finite buffers. Ann. Oper. Res. 93, 385–403 (2000) CrossRefGoogle Scholar
  16. 16.
    Whitt, W.: The queueing network analyzer. Bell Syst. Tech. J. 62(9), 2779–2815 (1983) Google Scholar
  17. 17.
    Whitt, W.: Approximations for departure processes and queues in series. Nav. Res. Logist. Q. 31(4), 499–521 (1984) CrossRefGoogle Scholar
  18. 18.
    Whitt, W.: Stochastic Process Limits. Springer, New York (2002) Google Scholar
  19. 19.
    Williams, R.J.: Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval. J. Appl. Probab. 29(4), 996–1002 (1992) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Mathematics, FEISSwinburne University of TechnologyHawthornAustralia

Personalised recommendations