December 2022 Crossover times in bipartite networks with activity constraints and time-varying switching rates
Sem Borst, Frank den Hollander, Francesca Romana Nardi, Siamak Taati
Author Affiliations +
Ann. Appl. Probab. 32(6): 4279-4314 (December 2022). DOI: 10.1214/22-AAP1787


In this paper, we study the performance of a bipartite network in which customers arrive at the nodes of the network, but not all nodes are able to serve their customers at all times. Each node can be either active or inactive, and two nodes connected by a bond cannot be active simultaneously. This situation arises in wireless random-access networks where, due to destructive interference, stations that are close to each other cannot use the same frequency band.

We consider a model where the network is bipartite, the active nodes switch themselves off at rate 1 and the inactive nodes switch themselves on at a rate that depends on time and on which half of the bipartite network they are in. An inactive node cannot become active when one of the nodes it is connected to by a bond is active. The switching protocol allows the nodes to share activity among each other. In the limit as the activation rate becomes large, we compute the crossover time between the two states where one-half of the network is active and the other half is inactive. This allows us to assess the overall activity of the network depending on the switching protocol. Our results make use of the metastability analysis for hard-core interacting particle models on finite bipartite graphs derived in an earlier paper. They are valid for a large class of bipartite networks, subject to certain assumptions. Proofs rely on a comparison with switching protocols that are not time varying, through coupling techniques.

Funding Statement

The research in this paper was supported through NWO Gravitation Grant 024.002.003–NETWORKS.
ST was also supported through NWO Grant 612.001.409.


Francesca Nardi sadly passed away before the publication of this article.


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Sem Borst. Frank den Hollander. Francesca Romana Nardi. Siamak Taati. "Crossover times in bipartite networks with activity constraints and time-varying switching rates." Ann. Appl. Probab. 32 (6) 4279 - 4314, December 2022.


Received: 1 December 2020; Revised: 1 December 2021; Published: December 2022
First available in Project Euclid: 6 December 2022

MathSciNet: MR4522352
zbMATH: 1514.60100
Digital Object Identifier: 10.1214/22-AAP1787

Primary: 60K25 , 60K30 , 60K35 , 90B15 , 90B18

Keywords: metastability , switching protocols , Wireless random-access networks

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 6 • December 2022
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