Abstract
We consider a Lévy stochastic network as a regulated multidimensional Lévy process. The reflection direction is constant on each boundary of the positive orthant and the corresponding reflection matrix corresponds to a single-class network. We use the representation of the Lévy process and Itô's formula to arrive at some equations for the steady-state process; the latter is shown to exist, under natural stability conditions. We specialize first to the class of Lévy processes with non-negative jumps and then add the assumption of self-similarity. We show that the stationary distribution of the network corresponding the the latter process does not has product form (except in trivial cases). Finally, we derive asymptotic bounds for two-dimensional Lévy stochastic network.
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Konstantopoulos, T., Last, G. & Lin, SJ. On a Class of Lévy Stochastic Networks. Queueing Systems 46, 409–437 (2004). https://doi.org/10.1023/B:QUES.0000027993.51077.f2
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DOI: https://doi.org/10.1023/B:QUES.0000027993.51077.f2