Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Asmussen, S.: Applied Probability and Queues. Springer, Berlin (2003)
Barnes, J.A., Disney, R.L.: Traffic processes in a class of finite Markovian queues. Queueing Syst. 6, 311–326 (1990)
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)
Branford, A.J.: On a property of finite-state birth and death processes. J. Appl. Probab. 23, 859–866 (1986)
Breuer, L., Baum, D.: An Introduction to Queueing Theory and Matrix-Analytic Methods. Springer, Berlin (2005)
Chandramohan, J., Foley, R.D., Disney, R.L.: Thinning of point processes—covariance analysis. Adv. Appl. Probab. 17, 127–146 (1985)
Cinlar, E., Disney, R.L.: Stream of overflows from a finite queue. Oper. Res. 15(1), 131–134 (1967)
Cox, D.R., Isham, V.: Point Processes. Chapman and Hall, London (1980)
Daley, D.J.: Queueing output processes. Adv. Appl. Probab. 8, 395–415 (1976)
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)
Disney, R.L., Kiessler, P.C.: Traffic Processes in Queueing Networks—A Markov Renewal Approach. Johns Hopkins University Press, Baltimore (1987)
Disney, R.L., Konig, D.: Queueing networks: A survey of their random processes. SIAM Rev. 27(3), 335–403 (1985)
Fischer, W., Meier-Hellstern, K.: The Markov-modulated Poisson process (MMPP) cookbook. Perform. Eval. 18, 149–171 (1992)
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)
He, Q., Neuts, M.F.: Markov chains with marked transitions. Stoch. Process. Appl. 74, 37–52 (1998)
Hendricks, K.B.: The output processes of serial production lines of exponential machines with finite buffers. Oper. Res. 40(6), 1139–1147 (1992)
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)
Keilson, J.: Markov Chain Models—Rarity and Exponentiality. Springer, Berlin (1979)
Kelly, F.: Reversibility and Stochastic Networks. Wiley, New York (1979)
Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia (1999)
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)
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)
Neuts, M.F., Li, J.: The input/output process of a queue. Appl. Stoch. Models Bus. Ind. 16, 11–21 (2000)
Parthasarathy, P.R., Sudhesh, R.: The overflow process from a state-dependent queue. Int. J. Comput. Math. 82(9), 1073–1093 (2005)
Parzen, E.: Stochastic Processes. Holden–Day, Oakland (1962)
Pourbabai, B.: Approximation of the overflow process from a G/M/N/K queueing system. Manag. Sci. 33(7), 931–938 (1987)
Reynolds, J.F.: The covariance structure of queues and related processes—a survey of recent work. Adv. Appl. Probab. 7, 383–415 (1975)
Rudemo, M.: Point processes generated by transitions of Markov chains. Adv. Appl. Probab. 5, 262–286 (1973)
Tan, B.: Variance of the output as a function of time: Production line dynamics. Eur. J. Oper. Res. 177(3), 470–484 (1999)
Tan, B.: Asymptotic variance rate of the output in production lines with finite buffers. Ann. Oper. Res. 93, 385–403 (2000)
van Doorn, E.A.: On the overflow process from a finite Markovian queue. Perform. Eval. 4, 233–240 (1984)
Whitt, W.: Approximating a point process by a renewal process, I: Two basic methods. Oper. Res. 30(1), 125–147 (1982)
Whitt, W.: The queueing network analyzer. Bell Syst. Tech. J. 62(9), 2779–2815 (1983)
Whitt, W.: Asymptotic formulas for Markov processes with applications to simulation. Oper. Res. 40(2), 279–291 (1992)
Whitt, W.: Stochastic-Process Limits, an Introduction to Stochastic-Process Limits and their Application to Queues. Springer, Berlin (2001)
Whitt, W.: Heavy traffic limits for loss proportions in single-server queues. Queueing Syst. 46, 507–536 (2004)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by Israel Science Foundation Grant 249/02 and 454/05 and by European Network of Excellence Euro-NGI.
Rights and permissions
About this article
Cite this article
Nazarathy, Y., Weiss, G. The asymptotic variance rate of the output process of finite capacity birth-death queues. Queueing Syst 59, 135–156 (2008). https://doi.org/10.1007/s11134-008-9079-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-008-9079-4