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April 2022 Metastability in loss networks with dynamic alternative routing
Sam Olesker-Taylor
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Ann. Appl. Probab. 32(2): 1362-1399 (April 2022). DOI: 10.1214/21-AAP1711

Abstract

Consider N stations interconnected with links, each of capacity K, forming a complete graph. Calls arrive to each link at rate λ and depart at rate 1. If a call arrives to a link xy, connecting stations x and y, which is at capacity, then a third station z is chosen uniformly at random and the call is attempted to be routed via z: if both links xz and zy have spare capacity, then the call is held simultaneously on these two; otherwise the call is lost.

We analyse an approximation of this model. We show rigorously that there are three phases according to the traffic intensity α:=λ/K: for α(0,αc)(1,), the system has mixing time logarithmic in the number of links n:=N2; for α(αc,1) the system has mixing time exponential in n, the number of links. Here αc:=13(51013)0.937 is an explicit critical threshold with a simple interpretation. We also consider allowing multiple rerouting attempts. This has little effect on the overall behaviour; it does not remove the metastability phase.

Finally, we add trunk reservation: in this, some number σ of circuits are reserved; a rerouting attempt is only accepted if at least σ+1 circuits are available. We show that if σ is chosen sufficiently large, depending only on α, not K or n, then the metastability phase is removed.

Funding Statement

The author was supported by EPSRC Doctoral Training Grant #1885554.

Acknowledgements

The question of studying mixing times for this model was originally raised by Nathanaël Berestycki. I would like to thank Perla Sousi, my Ph.D. supervisor, for reading this paper and giving lots of constructive feedback. I would also like to thank Frank Kelly, for numerous helpful discussions on this work and related stochastic networks discussions. He introduced me to the topic through his Cambridge Part III lecture course and his book [12] with Elena Yudovina; I have become thoroughly interested in the topic as a result.

I also thank the anonymous referee for helpful comments which improved the clarity and presentation of the paper. They also alerted me to the analogous metastable behaviour exhibited by the Chayes–Machta dynamics in the random cluster model and to the references [1, 3].

Citation

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Sam Olesker-Taylor. "Metastability in loss networks with dynamic alternative routing." Ann. Appl. Probab. 32 (2) 1362 - 1399, April 2022. https://doi.org/10.1214/21-AAP1711

Information

Received: 1 August 2020; Revised: 1 April 2021; Published: April 2022
First available in Project Euclid: 28 April 2022

Digital Object Identifier: 10.1214/21-AAP1711

Subjects:
Primary: 60K20 , 60K25 , 60K30 , 90B15 , 90B18 , 90B22

Keywords: dynamic alternative routing , Loss network , metastability , Mixing times

Rights: Copyright © 2022 Institute of Mathematical Statistics

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