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 , 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 and 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 : for , the system has mixing time logarithmic in the number of links ; for the system has mixing time exponential in n, the number of links. Here 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 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.
The author was supported by EPSRC Doctoral Training Grant #1885554.
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  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].
"Metastability in loss networks with dynamic alternative routing." Ann. Appl. Probab. 32 (2) 1362 - 1399, April 2022. https://doi.org/10.1214/21-AAP1711