Quasi-Product Forms for Levy-Driven Fluid Networks

We study stochastic tree fluid networks driven by a multidimensional Lévy process. We are interested in (the joint distribution of) the steady-state content in each of the buffers, the busy periods, and the idle periods. To investigate these fluid networks, we relate the above three quantities to fl...

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Bibliographic Details
Published in:Mathematics of operations research 2007-08, Vol.32 (3), p.629-647
Main Authors: Debicki, K, Dieker, A. B, Rolski, T
Format: Article
Language:English
Subjects:
Analysis
buffer content
busy period
Componentwise operations
dimensional Lévy process
Equilibrium flow
Fluctuation theorem
idle period
Laplace transformation
Laplace transforms
Lévy-driven fluid network
Markov processes
Mathematical theorems
Mathematical vectors
multidimensional Skorokhod problem
Poisson process
quasi-product form
Queuing theory
Random variables
splitting time
Stochastic models
Studies
tree fluid network
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Summary:We study stochastic tree fluid networks driven by a multidimensional Lévy process. We are interested in (the joint distribution of) the steady-state content in each of the buffers, the busy periods, and the idle periods. To investigate these fluid networks, we relate the above three quantities to fluctuations of the input Lévy process by solving a multidimensional Skorokhod reflection problem. This leads to the analysis of the distribution of the componentwise maximums, the corresponding epochs at which they are attained, and the beginning of the first last-passage excursion. Using the notion of splitting times, we are able to find their Laplace transforms. It turns out that, if the components of the Lévy process are "ordered," the Laplace transform has a so-called quasi-product form. The theory is illustrated by working out special cases, such as tandem networks and priority queues.
ISSN:0364-765X
1526-5471
DOI:10.1287/moor.1070.0259