Queueing Systems

, 59:135 | Cite as

The asymptotic variance rate of the output process of finite capacity birth-death queues



We analyze the output process of finite capacity birth-death Markovian queues. We develop a formula for the asymptotic variance rate of the form λ *+∑v i where λ * is the rate of outputs and v i are functions of the birth and death rates. We show that if the birth rates are non-increasing and the death rates are non-decreasing (as is common in many queueing systems) then the values of v i are strictly negative and thus the limiting index of dispersion of counts of the output process is less than unity.

In the M/M/1/K case, our formula evaluates to a closed form expression that shows the following phenomenon: When the system is balanced, i.e. the arrival and service rates are equal, \(\frac{\sum v_{i}}{\lambda^{*}}\) is minimal. The situation is similar for the M/M/c/K queue, the Erlang loss system and some PH/PH/1/K queues: In all these systems there is a pronounced decrease in the asymptotic variance rate when the system parameters are balanced.


Queueing theory Loss systems M/M/1/K MAP Asymptotic variance rate BRAVO 

Mathematics Subject Classification (2000)

60J27 60K25 


  1. 1.
    Asmussen, S.: Applied Probability and Queues. Springer, Berlin (2003) Google Scholar
  2. 2.
    Barnes, J.A., Disney, R.L.: Traffic processes in a class of finite Markovian queues. Queueing Syst. 6, 311–326 (1990) CrossRefGoogle Scholar
  3. 3.
    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
  4. 4.
    Branford, A.J.: On a property of finite-state birth and death processes. J. Appl. Probab. 23, 859–866 (1986) CrossRefGoogle Scholar
  5. 5.
    Breuer, L., Baum, D.: An Introduction to Queueing Theory and Matrix-Analytic Methods. Springer, Berlin (2005) Google Scholar
  6. 6.
    Chandramohan, J., Foley, R.D., Disney, R.L.: Thinning of point processes—covariance analysis. Adv. Appl. Probab. 17, 127–146 (1985) CrossRefGoogle Scholar
  7. 7.
    Cinlar, E., Disney, R.L.: Stream of overflows from a finite queue. Oper. Res. 15(1), 131–134 (1967) Google Scholar
  8. 8.
    Cox, D.R., Isham, V.: Point Processes. Chapman and Hall, London (1980) Google Scholar
  9. 9.
    Daley, D.J.: Queueing output processes. Adv. Appl. Probab. 8, 395–415 (1976) CrossRefGoogle Scholar
  10. 10.
    Disney, R.L., de Morais, P.R.: Covariance properties for the departure process of M/Ek/1/N queues. AIIE Trans. 8(2), 169–175 (1976) Google Scholar
  11. 11.
    Disney, R.L., Kiessler, P.C.: Traffic Processes in Queueing Networks—A Markov Renewal Approach. Johns Hopkins University Press, Baltimore (1987) Google Scholar
  12. 12.
    Disney, R.L., Konig, D.: Queueing networks: A survey of their random processes. SIAM Rev. 27(3), 335–403 (1985) CrossRefGoogle Scholar
  13. 13.
    Fischer, W., Meier-Hellstern, K.: The Markov-modulated Poisson process (MMPP) cookbook. Perform. Eval. 18, 149–171 (1992) CrossRefGoogle Scholar
  14. 14.
    Gershwin, S.B.: Variance of output of a tandem production system. In: Onvural, R., Akyildiz, I. (Eds.) Queueing Networks with Finite Capacity. Proceedings of the Second International Conference on Queueing Networks with Finite Capacity. Elsevier, Amsterdam (1993) Google Scholar
  15. 15.
    He, Q., Neuts, M.F.: Markov chains with marked transitions. Stoch. Process. Appl. 74, 37–52 (1998) CrossRefGoogle Scholar
  16. 16.
    Hendricks, K.B.: The output processes of serial production lines of exponential machines with finite buffers. Oper. Res. 40(6), 1139–1147 (1992) Google Scholar
  17. 17.
    Hendricks, K.B., McClain, J.O.: The output processes of serial production lines of general machines with finite buffers. Manag. Sci. 39(10), 1194–1201 (1993) Google Scholar
  18. 18.
    Keilson, J.: Markov Chain Models—Rarity and Exponentiality. Springer, Berlin (1979) Google Scholar
  19. 19.
    Kelly, F.: Reversibility and Stochastic Networks. Wiley, New York (1979) Google Scholar
  20. 20.
    Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia (1999) Google Scholar
  21. 21.
    Miltenburg, G.J.: Variance of the number of units produced on a transfer line with buffer inventories during a period of length T. Nav. Res. Logist. 34, 811–822 (1987) CrossRefGoogle Scholar
  22. 22.
    Naryana, S., Neuts, M.F.: The first two moment matrices of the counts for the Markovian arrival process. Stoch. Models 8(3), 459–477 (1992) CrossRefGoogle Scholar
  23. 23.
    Neuts, M.F., Li, J.: The input/output process of a queue. Appl. Stoch. Models Bus. Ind. 16, 11–21 (2000) CrossRefGoogle Scholar
  24. 24.
    Parthasarathy, P.R., Sudhesh, R.: The overflow process from a state-dependent queue. Int. J. Comput. Math. 82(9), 1073–1093 (2005) CrossRefGoogle Scholar
  25. 25.
    Parzen, E.: Stochastic Processes. Holden–Day, Oakland (1962) Google Scholar
  26. 26.
    Pourbabai, B.: Approximation of the overflow process from a G/M/N/K queueing system. Manag. Sci. 33(7), 931–938 (1987) CrossRefGoogle Scholar
  27. 27.
    Reynolds, J.F.: The covariance structure of queues and related processes—a survey of recent work. Adv. Appl. Probab. 7, 383–415 (1975) CrossRefGoogle Scholar
  28. 28.
    Rudemo, M.: Point processes generated by transitions of Markov chains. Adv. Appl. Probab. 5, 262–286 (1973) CrossRefGoogle Scholar
  29. 29.
    Tan, B.: Variance of the output as a function of time: Production line dynamics. Eur. J. Oper. Res. 177(3), 470–484 (1999) CrossRefGoogle Scholar
  30. 30.
    Tan, B.: Asymptotic variance rate of the output in production lines with finite buffers. Ann. Oper. Res. 93, 385–403 (2000) CrossRefGoogle Scholar
  31. 31.
    van Doorn, E.A.: On the overflow process from a finite Markovian queue. Perform. Eval. 4, 233–240 (1984) CrossRefGoogle Scholar
  32. 32.
    Whitt, W.: Approximating a point process by a renewal process, I: Two basic methods. Oper. Res. 30(1), 125–147 (1982) Google Scholar
  33. 33.
    Whitt, W.: The queueing network analyzer. Bell Syst. Tech. J. 62(9), 2779–2815 (1983) Google Scholar
  34. 34.
    Whitt, W.: Asymptotic formulas for Markov processes with applications to simulation. Oper. Res. 40(2), 279–291 (1992) Google Scholar
  35. 35.
    Whitt, W.: Stochastic-Process Limits, an Introduction to Stochastic-Process Limits and their Application to Queues. Springer, Berlin (2001) Google Scholar
  36. 36.
    Whitt, W.: Heavy traffic limits for loss proportions in single-server queues. Queueing Syst. 46, 507–536 (2004) CrossRefGoogle Scholar
  37. 37.
    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 2008

Authors and Affiliations

  1. 1.Department of StatisticsThe University of HaifaMount CarmelIsrael

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