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Stability and structural properties of stochastic storage networks

Part of: Special processes Operations research and management science Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Offer Kella  and
Ward Whitt
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
Ward Whitt*
Affiliation:
AT&T Laboratories
*
∗Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. email:[email protected]
∗∗Postal address: AT&T Laboratories, Room 2C-178, 600 Mountain Avenue, Murray Hill, NJ 07974–0636, USA. email:[email protected]
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Abstract

We establish stability, monotonicity, concavity and subadditivity properties for open stochastic storage networks in which the driving process has stationary increments. A principal example is a stochastic fluid network in which the external inputs are random but all internal flows are deterministic. For the general model, the multi-dimensional content process is tight under the natural stability condition. The multi-dimensional content process is also stochastically increasing when the process starts at the origin, implying convergence to a proper limit under the natural stability condition. In addition, the content process is monotone in its initial conditions. Hence, when any content process with non-zero initial conditions hits the origin, it couples with the content process starting at the origin. However, in general, a tight content process need not hit the origin.

Keywords

STABILITYFLUID NETWORKSSTATIONARY INCREMENTSLEVY PROCESSREFLECTED PROCESSSTOCHASTICALLY INCREASINGTIGHTNESSSTOCHASTIC ORDERBOUNDS

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G10: Stationary processes 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 4 , December 1996 , pp. 1169 - 1180
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

This work was partially supported by Grant No. 92–00035 from the United States Israel Binational Science Foundation.

References

Baccelli, F. and Foss, S. (1994) Ergodicity of Jackson-type queueing networks. Queueing Systems 17, 5–72.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Borovkov, A. A. (1976) Stochastic Processes in Queueing Theory. Springer, New York.CrossRefGoogle Scholar
Chen, H. and Mandelbaum, A. (1989) 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 Whitt, W. (1993) Diffusion approximations for open queueing networks with service interruptions. Queueing Systems 13, 335–359.CrossRefGoogle Scholar
Dai, J. (1995) On the positive Harris recurrence for multiclass queueing networks: A unified approach via fluid limits. Ann. Appl. Prob. 5, 49–77.CrossRefGoogle Scholar
Dai, J. and Weiss, G. (1996) Stability and instability of fluid models for certain re-entrant lines. Math. Operat. Res. 21, 115–134.CrossRefGoogle Scholar
Glynn, P. W. and Whitt, W. (1988) Ordinary CLT and WLLN versions of L = ? W. Math. Operat. Res. 13, 674–692.CrossRefGoogle Scholar
Harrison, J. M. and Reiman, M. I. (1981) Reflected Brownian motion on an orthant. Ann. Prob. 9, 302–308.CrossRefGoogle Scholar
Kaspi, H. and Kella, O. (1996) Stability of feed-forward fluid networks with Lévy input. J. Appl. Prob. 33, 513–523.CrossRefGoogle Scholar
Kella, O. (1992) Concavity and reflected Lévy processes. J. Appl. Prob. 29, 209–215.CrossRefGoogle Scholar
Kella, O. (1993) Parallel and tandem fluid networks with dependent Lévy inputs. Ann. Appl. Prob. 3, 682–695.CrossRefGoogle Scholar
Kella, O. (1996) Stability and non-product form of stochastic fluid networks with Lévy inputs. Ann. Appl. Prob. 6, 186–199.CrossRefGoogle Scholar
Kella, O. and Sverchkov, ?. (1994) On concavity of the mean function and stochastic ordering for reflected processes with stationary increments. J. Appl. Prob. 31, 1140–1142.CrossRefGoogle Scholar
Kella, O. and Whitt, W. (1992a) A tandem fluid network with Lévy input. In Queues and Related Models. ed. Basawa, I. and Bhat, U. Oxford University Press, Oxford. pp. 112–128.Google Scholar
Kella, O. and Whitt, W. (1992b) Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29, 296–403.CrossRefGoogle Scholar
Kumar, P. R. (1993) Re-entrant lines. Queueing Systems 13, 87–110.CrossRefGoogle Scholar
Kumar, P. R. and Meyn, S. P. (1995) Stability of queueing networks and scheduling policies. IEEE Trans. Automat. Control 40, 251–260.CrossRefGoogle Scholar
Loynes, R. M. (1962) The stability of a queue with non-independent interarrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.CrossRefGoogle Scholar