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Recurrence and transience of open Jackson networks with balanced nodes
Abstract
Consider an open Jackson network with Poisson arrivals and exponential services, in which no node is overloaded and exactly~k0 nodes have load equal to~$1$. We prove that the queue-length process is recurrent if and only if k0≤ 2. For that, we show that the network satisfies a sector condition, so its capacities are comparable to those of its symmetrisation; the latter is a reversible network whose recurrence/transience can be dealt with by standard means. We also discuss what the Lyapunov-function method gives in this setting; in particular, we show that every balanced queue is empty at arbitrarily large times, also in the transient case.
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BibTeX
@misc{popov2026recurrence,
title = {{Recurrence and transience of open Jackson networks with balanced nodes}},
author = {Serguei Popov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31183},
eprint = {2609.31183},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31183},
}
RIS
TY - UNPB
TI - Recurrence and transience of open Jackson networks with balanced nodes
AU - Serguei Popov
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31183
UR - https://arxiv.org/abs/2609.31183
AB - Consider an open Jackson network with Poisson arrivals and exponential services, in which no node is overloaded and exactly~k_0 nodes have load equal to~$1$. We prove that the queue-length process is recurrent if and only if k_0≤ 2. For that, we show that the network satisfies a sector condition, so its capacities are comparable to those of its symmetrisation; the latter is a reversible network whose recurrence/transience can be dealt with by standard means. We also discuss what the Lyapunov-function method gives in this setting; in particular, we show that every balanced queue is empty at arbitrarily large times, also in the transient case.
ER -