Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T16:32:21.572Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-product form of two-dimensional fluid networks with dependent Lévy inputs

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

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 address: mskella@mscc.huji.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the stationary distribution of a two-dimensional stochastic fluid network with (possibly dependent) Lévy inputs does not have product form other than in truly obvious cases. This is in contrast to queueing networks, where product form exists for non-obvious situations in which the inputs are independent, and for Brownian networks, where it typically exists for cases where the driving processes are actually dependent.

Keywords

Stochastic fluid networksLévy processreflected processnon-product form

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 90B05: Inventory, storage, reservoirs 90B15: Network models, stochastic
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 1117 - 1122
Copyright
Copyright © by the Applied Probability Trust 2000 

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 794/97 from the Israel Science Foundation.

References

[1] 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.Google Scholar
[2] Gnedenko, B. W., and Kolmogorov, A. N. (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, New York.Google Scholar
[3] Harrison, J. M., and Reiman, M. (1981). On the distribution of multidimensional reflected Brownian motion. SIAM J. Appl. Math. 41, 345–361.Google Scholar
[4] Harrison, J. M., and Williams, R. J. (1987). Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Prob. 15, 115–137.Google Scholar
[5] Kaspi, H., and Kella, O. (1996). Stability of feedforward fluid networks with Lévy inputs. J. Appl. Prob. 33, 513–522.Google Scholar
[6] Kella, O. (1993). Parallel and tandem fluid networks with dependent Lévy inputs. Ann. Appl. Prob. 3, 682–695.Google Scholar
[7] Kella, O. (1996). Stability and nonproduct form of stochastic fluid networks with Lévy inputs. Ann. Appl. Prob. 6, 186–199.Google Scholar
[8] Kella, O. (1997). Stochastic storage networks: stationarity and the feedforward case. J. Appl. Prob. 34, 498–507.Google Scholar
[9] Kella, O., and Whitt, W. (1992). A tandem fluid network with Lévy input. In Queueing and Related Models, eds Bhat, U. N. and Basawa, I. V. Oxford University Press, pp. 112–128.Google Scholar
[10] Kella, O., and Whitt, W. (1992). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29, 396–403.Google Scholar
[11] Kella, O., and Whitt, W. (1996). Stability and structural properties of stochastic fluid networks. J. Appl. Prob. 33, 1169–1180.Google Scholar
[12] Kella, O., and Whitt, W. (1999). Linear stochastic fluid networks. J. Appl. Prob. 36, 244–260.Google Scholar