Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T05:18:50.923Z Has data issue: false hasContentIssue false
  • English
  • Français

Strong Approximations of Irreducible Closed Queueing Networks

Part of: Operations research and management science Stochastic processes Special processes Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

Hanqin Zhang
Hanqin Zhang*
Affiliation:
Academia Sinica
Get access Rights & Permissions [Opens in a new window]

Abstract

A sequence of irreducible closed queueing networks is considered in this paper. We obtain that the queue length processes can be approximated by reflected Brownian motions. Using these approximations, we get rates of convergence of the distributions of queue lengths.

Keywords

CLOSED QUEUEING NETWORKREFLECTED BROWNIAN MOTIONSTRONG APPROXIMATIONRATE OF CONVERGENCELEVY-PROHOROV DISTANCE

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 60K25: Queueing theory 60G17: Sample path properties
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 90B10: Network models, deterministic 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 2 , June 1997 , pp. 498 - 522
Copyright
Copyright © Applied Probability Trust 1997 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abate, J. and Whitt, W. (1987a) Transient behavior of regulated Brownian motion I: starting at the origin. Adv. Appl. Prob. 19, 560–598.CrossRefGoogle Scholar
Abate, J. and Whitt, W. (1987b) Transient behavior of regulated Brownian motion II: non-zero initial conditions. Adv. Appl. Prob. 19, 599–631.Google Scholar
Berman, A. and Plemmons, R. J. (1979) Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York.Google Scholar
Chen, H. and Mandelbaum, A. (1991a) Leontief systems, RBV's and RBM's. In Proc. Imperial College Workshop on Applied Stochastic Processes. ed. Davis, M. H. A. and Elliott, R. J. Gordon and Breach, New York.Google Scholar
Chen, H. and Mandelbaum, A. (1991b) Discrete flow networks: bottleneck analysis and fluid approximations. Math. Operat. Res. 16, 408–446.Google Scholar
Chen, H. and Mandelbaum, A. (1991c) Stochastic discrete flow networks: diffusion approximations and bottlenecks. Ann. Prob. 19, 1463–1519.Google Scholar
Csörgö, M., Deheuvels, P. and Horvath, L. (1987) An approximation of stopped sums with applications in queueing theory. Adv. Appl. Prob. 19, 674–690.Google Scholar
Csörgo, M., HorváTh, L. and Steinebach, J. (1987) Invariance principles for renewal processes. Ann. Prob. 15, 1441–1460.Google Scholar
Csörgo, M. and Revesz, P. (1981) Strong Approximations in Probability and Statistics. Academic Press, New York.Google Scholar
Einmahl, U. (1989) Extensions of results of Komlós, Major and Tusnády to the multivariate case. J. Multivar. Anal. 28, 20–68.CrossRefGoogle Scholar
Feller, W. (1968) An Introduction to Probability Theory and Its Applications. Vol. I. 3rd edn. Wiley, New York.Google Scholar
Glynn, P. W. and Whitt, W. (1991a) A new view of the heavy traffic limit theorems for infinite-server queues. Adv. Appl. Prob. 23, 188–209.Google Scholar
Glynn, P. W. and Whitt, W. (1999b) Departures from many queues. Ann. Appl. Prob. 1, 546–572.Google Scholar
Harrison, J. M. (1985) Brownian Motion and Stochastic Systems. Wiley, New York.Google Scholar
Harrison, J. M. and Reiman, M. L. (1981) Reflected Brownian motion on an orthant. Ann. Prob. 9, 302–308.Google Scholar
HorváTh, L. (1992) Strong approximations of open queueing networks. Math. Operat. Res. 17, 487–508.Google Scholar
Komlós, J., Major, P. and Tusnády, G. (1975) An approximation of partial sums of independent R.V.'s and the sample DF. I. Z. Wahrscheinlichkeitsth. 32, 111–131.Google Scholar
Komlós, J., Major, P. and Tusnády, G. (1976) An approximation of partial sums of independent R.V.'s and the sample DF. II. Z. Wahrscheinlichkeitsth. 34, 33–58.CrossRefGoogle Scholar
Prohorov, Yu. V. (1956) Convergence of random processes and limit theorems in probability theory. Theory Prob. Appl. 1, 157–214.CrossRefGoogle Scholar
Reiman, M. L. (1984) Open queueing networks in heavy traffic. Math. Operat. Res. 9, 441–458.Google Scholar
Strassen, V. (1965) The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439.Google Scholar