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Stochastic storage networks: stationarity and the feedforward case

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Offer Kella
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
*
∗Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email: [email protected]
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Abstract

We show that for a certain storage network the backward content process is increasing, and when the net input process has stationary increments then, under natural stability conditions, the content process has a stationary version under which the cumulative lost capacities have stationary increments. Moreover, for the feedforward case, we show that under some minimal conditions, two content processes with net input processes which differ only by initial conditions can be coupled in finite time and that the difference of two content processes vanishes in the limit if the difference of the net input processes monotonically approaches a constant. As a consequence, it is shown that for the natural stability conditions, when the net input process has stationary increments, the distribution of the content process converges in total variation to a proper limit, independent of initial conditions.

Keywords

STABILITYFLUID NETWORKSSTORAGE NETWORKSREFLECTED PROCESSES

MSC classification

Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 90B05: Inventory, storage, reservoirs 90B15: Network models, stochastic
Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997 , pp. 498 - 507
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

Supported in part by grant 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
Baccelli, F. and Foss, S. (1995) On the saturation rule for the stability of queues. J. Appl. Prob. 32, 494–507.CrossRefGoogle 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 Mandelbaum, A. (1991) Discrete flow networks: bottleneck analysis and fluid approximations. Math. Operat. Res. 16, 408–146.Google Scholar
Chen, H. and Whitt, W. (1993) Diffusion approximations for open queueing networks with service interruptions. Queueing Systems 13, 335–350.CrossRefGoogle Scholar
Dai, J. G. (1995) On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann. Appl. Prob. 5, 49–77.CrossRefGoogle Scholar
Dupuis, P. and Williams, R. J. (1994) Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Prob. 22, 680–702.Google Scholar
Kaspi, H. and Kella, O. (1996) Stability of feed-forward fluid networks with Lévy input. J. Appl. Prob. 33, 513–522.Google Scholar
Kella, O. (1996) Stability and non-product form of stochastic fluid networks with Lévy inputs. Ann. Appl. Prob. 6, 186–199.Google Scholar
Kella, O. and Sverchkov, M. (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. (1996) Stability and structural properties of stochastic storage networks. J. Appl. Prob. 33, 1169–1180.Google Scholar
Reiman, M. I. (1984) Open queueing networks in heavy traffic. Math. Operat. Res. 9, 441–458.Google Scholar