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An asymptotically exact decomposition of coupled Brownian systems

Part of: Special processes Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Kerry W. Fendick
Kerry W. Fendick*
Affiliation:
AT&T Bell Laboratories
*
∗ Postal address: AT&T Bell Laboratories, Room IF-401, 101 Crawfords Corner Road, Holmdel, NJ 07733–3030, USA.
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Abstract

Brownian flow systems, i.e. multidimensional Brownian motion with regulating barriers, can model queueing and inventory systems in which the behavior of different queues is correlated because of shared input processes. The behavior of such systems is typically difficult to describe exactly. We show how Brownian models of such systems, conditioned on one queue length exceeding a large value, decompose asymptotically into smaller subsystems. This conditioning induces a change in drift of the system's net input process and its components. The results here are analogous to results for jump-Markov queues recently obtained by Shwartz and Weiss. The Brownian setting leads to a simple description of the component processes' asymptotic behaviour, as well as to explicit distributional results.

Keywords

STOCHASTIC PROCESSESCONDITIONAL LIMIT THEOREMSREGULATED BROWNIAN MOTION

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 4 , December 1993 , pp. 819 - 834
Copyright
Copyright © Applied Probability Trust 1993 

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References

Abate, J, and Whitt, W. (1987) Transient behavior of regulated Brownian motion, I: starting at the origin. Adv. Appl. Prob. 19, 560–598.Google Scholar
Anantharam, V. (1989) How large delays build up in a GI/G/1 queue. QUESTA 5, 345–368.Google Scholar
Anick, D., Mitra, D. and Sondhi, M. M. (1982) Stochastic theory of a data-handling system with multiple sources. Bell System Tech. J. 61, 1871–1894.Google Scholar
Asmussen, S. (1982) Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/G/1 queue. Adv. Appl. Prob. 14, 143–170.Google Scholar
Berger, A. W. and Whitt, W. (1991) The Brownian approximation for rate-control throttles and the G/G/C queue. Discrete Event Dynamic Systems.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Billingsley, P. (1971) Weak Convergence of Measures: Applications in Probability. Society for Industrial and Applied Mathematics, Philadelphia.Google Scholar
Dai, J. G. and Harrison, J. M. (1991) Steady-state analysis of RBM in a rectangle: numerical methods and queueing applications. Ann. Appl. Prob. 1, 16–35.CrossRefGoogle Scholar
Dai, J. G. and Harrison, J. M. (1992) Reflected Brownian motion in an orthant: numerical methods for steady-state analysis. Ann. Appl. Prob. 2, 65–86.Google Scholar
Durrett, R. (1980) Conditioned limit theorems for random walks with negative drift. Z. Wahrscheinlichkeitsth. 52, 277–287.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York.Google Scholar
Flatto, L. (1985) Two parallel queues created by arrivals with two demands, II. SIAM J. Appl. Math. 45, 861–878.Google Scholar
Flatto, L. and Hahn, S. (1984) Two parallel queues created by arrivals with two demands, I. SIAM J. Appl. Math 44, 1041–1053.Google Scholar
Foschini, G. J. (1982) Equilibria for diffusion models for pairs of communicating computers - symmetric case. IEEE Trans. Inf. Theory 28, 273–284.Google Scholar
Glynn, P. W. (1990) Diffusion approximations. In Handbooks in Operations Research and Management Science 2. Elsevier, North-Holland, Amsterdam.Google Scholar
Harrison, J. M. (1985) Brownian Motion and Stochastic Flow Systems. Wiley, New York.Google Scholar
Harrison, J. M. and Nguyen, V. (1990) The QNET method for two-moment analysis of open queueing networks. QUESTA 6, 1–32.Google Scholar
Harrison, J. M. and Reiman, M. I. (1981a) On the distribution of multidimensional reflected Brownian motion. SIAM J. Appl. Math. 41, 345–361.CrossRefGoogle Scholar
Harrison, J. M. and Reimann, M. I. (1981b) Reflected Brownian motion on an orthant. Ann. Prob. 9, 302–308.Google Scholar
Iglehart, D. L. and Whitt, W. (1970) Multichannel queues in heavy traffic, I and II. Adv. Appl. Prob. 2, 150–177, 355-369.Google Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Karlin, S. and Taylor, H. M. (1981) A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kent, J. (1978) Time-reversible diffusions. Adv. Appl. Prob. 10, 819–835. Addendum 11 (1979), 888.Google Scholar
Knessl, C. (1991) On the diffusion approximation to a fork and join queueing model. SIAM J. Appl. Math. 51, 187–213.Google Scholar
Knessl, C. (1992) On the diffusion approximation to two parallel queues with processor sharing. IEEE Trans. Autom. Control. To appear.Google Scholar
Liptser, R. S. and Shiryayev, A. N. (1974) Statistics of Random Processes, I, translated by Aries, A. B. Springer-Verlag, Berlin.Google Scholar
Newell, G. F. (1982) Applications of Queueing Theory, 2nd edn. Chapman and Hall, London.Google Scholar
Shwartz, A. and Weiss, A. (1993) Induced rare events: analysis via large deviations and time reversal. Adv. Appl. Prob. 25 (3).CrossRefGoogle Scholar
Whitt, W. (1982) Approximating a point process by a renewal process, I: two basic methods. Operat. Res. 30, 125–147.CrossRefGoogle Scholar
Wright, P. E. (1992) Two parallel processors with coupled inputs. Adv. Appl. Prob. 24, 986–1007.Google Scholar