Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T20:27:37.126Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of feed-forward fluid networks with Lévy input

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  14 July 2016

Haya Kaspi  and
Offer Kella
Haya Kaspi*
Affiliation:
Technion — Israel Institute of Technology
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
*
∗Postal address: Faculty of Industrial Engineering and Management, Technion — Israel Institute of Technology, Haifa 32000, Israel.
∗∗Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a stochastic fluid network with independent subordinator inputs to the various stations and deterministic internal flow which is of feed-forward type. We show that under suitable conditions the process of fluid contents in the station has a limiting distribution, where the limit holds in total variation and is independent of the initial condition. We also show that this limiting distribution is of product form only for trivial setups.

Keywords

FLUID NETWORKSSTORAGE MODELSSUBORDINATORLEVY PROCESS

MSC classification

Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 90B05: Inventory, storage, reservoirs 90B15: Network models, stochastic
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 2 , September 1996 , pp. 513 - 522
Copyright
Copyright © Applied Probability Trust 1996 

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.)

Footnotes

Supported in part by grant 92–00035 from the United States - Israel Binational Science Foundation.

References

Anick, D., Mitra, D. and Sondhi, M. M. (1982) Stochastic theory of a data handling system. Bell Sys. Tech. J 61, 1871–1894.CrossRefGoogle Scholar
Chen, H. and Mandelbaum, A. (1991) Discrete flow networks: bottleneck analysis and fluid approximations. Math. Operat. Res. 16, 408–446.CrossRefGoogle Scholar
Chen, H. and Yao, D. D. (1992) A fluid model for systems with random disruptions. Operat. Res. 40 (Supplement 2), 239–247.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
Dai, J. G. and Meyn, S. (1994) Stability and convergence of moments for multiclass queueing networks via fluid limits models. Preprint. CrossRefGoogle Scholar
Down, D. and Meyn, S. (1992) Stability of multiclass queueing networks. Proc. 26th Annual Conf. on Information Sciences and Systems. Princeton.Google Scholar
Down, D. and Meyn, S. (1993) Stability of acyclic multiclass queueing networks. Preprint. Google Scholar
Gaver, D. P. Jr. and Miller, R. G. Jr. (1962) Limiting distributions for some storage problems. In Studies in Applied Probability and Management Sciences , pp. 110–126. ed. Arrow, K. J., Karlin, S. and Scarf, H. Stanford University Press, Stanford, CA.Google Scholar
Glickmann, H. (1993) Stochastic Fluid Networks. M.Sc. thesis, Hebrew University.Google Scholar
Kaspi, H. (1984) Storage processes with Markov additive input and output. Math. Operat. Res. 9, 424–440.CrossRefGoogle Scholar
Kella, O. (1993) Parallel and tandem fluid networks with dependent Lévy inputs. Ann. Appl. Prob. 3, 682–695.CrossRefGoogle Scholar
Kella, O. and Whitt, W. (1992a) A storage model with a two-stage random environment. Operat. Res. 40 (supplement 2), 257–262.CrossRefGoogle Scholar
Kella, O, and Whitt, W. (1992b) A tandem fluid network with Lévy input. In Queues and Related Models , pp. 112–128. ed. Basawa, I. and Bhat, U. Oxford University Press, Oxford.Google Scholar
Kella, O. and Whitt, W. (1992C) Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29, 296–403.CrossRefGoogle Scholar
Meyer, R. R., Rothkopf, M. H. and Smith, S. A. (1979) Reliability and inventory in a production-storage system. Management Sci. 25, 799–807.CrossRefGoogle Scholar
Meyer, R. R., Rothkopf, M. H. and Smith, S. A. (1983) Erratum to 'Reliability and inventory in a production-storage system. Management Sci. 29, 1346.CrossRefGoogle Scholar
Miller, R. G. Jr. (1963) Continuous time stochastic storage processes with random linear inputs and outputs. J. Math. Mech. 12, 275–291.Google Scholar
Mitra, D. (1988) Stochastic theory of a fluid model of multiple failure-susceptible producers and consumers coupled by a buffer. Adv. Appl. Prob. 20, 646–676.CrossRefGoogle Scholar
Newell, G. F. (1982) Applications of Queueing Theory. Chapman and Hall, London.CrossRefGoogle Scholar