On the Lack of Periodicity of Walker Satellite Constellation Routing Tables
Abstract
In a referential that rotates with Earth, the dynamics of the configurations of satellites in a Delta Walker constellation can be analyzed as a dynamical system as a function of a translation on the torus. This paper extends this type of analysis to routing over a Walker constellation, leveraging fixed ground relays. It considers any deterministic, time-invariant routing rule on such a collection of satellites and relays. The main object of interest is the routing table associated with this rule, which specifies, for all source-destination pairs, a route made of satellites and relays between them, for instance the shortest, together with angular information on next hop at each step of the route, which is essential for beamforming in this context. The main result is that, for each such routing rule, the routing table inherits the dichotomy of the torus flow: when the ratio of the Earth-spin and satellite angular speeds is rational, the routing table process is periodic with the same period as the constellation. When it is irrational, the table is non periodic but admits ergodic long-run averages which can be computed as ensemble averages w.r.t. an explicit invariant measure. Simulations with mixed satellite--gateway greedy routing illustrate these findings in terms of both temporal and spectral properties of the Walker routing table time series. The paper also discusses how to define metrics that cope with such non periodic fluctuations when present. This is illustrated by a comparison of the round-trip times between two far away cities as offered by a Walker constellation using greedy routing on one side, and by the currently available terrestrial fiber on the other side. It is shown how ergodicity can be used to make this comparison between the time varying instantaneous round trip times of the constellation and the constant round trip time of the fiber network meaningful.
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BibTeX
@misc{choi2026lack,
title = {{On the Lack of Periodicity of Walker Satellite Constellation Routing Tables}},
author = {Chang-Sik Choi and François Baccelli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36566},
eprint = {2609.36566},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36566},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On the Lack of Periodicity of Walker Satellite Constellation Routing Tables
AU - Chang-Sik Choi
AU - François Baccelli
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36566
UR - https://arxiv.org/abs/2609.36566
AB - In a referential that rotates with Earth, the dynamics of the configurations of satellites in a Delta Walker constellation can be analyzed as a dynamical system as a function of a translation on the torus. This paper extends this type of analysis to routing over a Walker constellation, leveraging fixed ground relays. It considers any deterministic, time-invariant routing rule on such a collection of satellites and relays. The main object of interest is the routing table associated with this rule, which specifies, for all source-destination pairs, a route made of satellites and relays between them, for instance the shortest, together with angular information on next hop at each step of the route, which is essential for beamforming in this context. The main result is that, for each such routing rule, the routing table inherits the dichotomy of the torus flow: when the ratio of the Earth-spin and satellite angular speeds is rational, the routing table process is periodic with the same period as the constellation. When it is irrational, the table is non periodic but admits ergodic long-run averages which can be computed as ensemble averages w.r.t. an explicit invariant measure. Simulations with mixed satellite--gateway greedy routing illustrate these findings in terms of both temporal and spectral properties of the Walker routing table time series. The paper also discusses how to define metrics that cope with such non periodic fluctuations when present. This is illustrated by a comparison of the round-trip times between two far away cities as offered by a Walker constellation using greedy routing on one side, and by the currently available terrestrial fiber on the other side. It is shown how ergodicity can be used to make this comparison between the time varying instantaneous round trip times of the constellation and the constant round trip time of the fiber network meaningful.
ER -