Metastability in Loss Networks with Dynamic Alternative Routing

TL;DR

This paper analyzes a dynamic routing model in loss networks, revealing three phases of system behavior based on traffic intensity, and shows how modifications like trunk reservation can eliminate metastability.

Contribution

It provides a rigorous analysis of phase transitions and metastability in a complex loss network model with dynamic rerouting and reservation strategies.

Findings

Three distinct phases identified based on traffic intensity.

Metastability occurs in an intermediate traffic range.

Trunk reservation can eliminate metastability.

Abstract

Consider $N$ stations interconnected with links, each of capacity $K$, forming a complete graph. Calls arrive to each link at rate $\lambda$ and depart at rate $1$. If a call arrives to a link $x y$, 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 $x z$ and $z y$ 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 $\alpha := \lambda/K$: for $\alpha \in (0,\alpha_c) \cup (1,\infty)$, the system has mixing time logarithmic in the number of links $n := \binom N2$; for $\alpha \in (\alpha_c,1)$ the system has mixing time exponential in $n$, the number of links. Here $\alpha_c := \tfrac13 (5 \sqrt{10} - 13) \approx 0.937$ 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 $\sigma$ of circuits are reserved; a rerouting attempt is only accepted if at least $\sigma+1$ circuits are available. We show that if $\sigma$ is chosen sufficiently large, depending only on $\alpha$, not $K$ or $n$, then the metastability phase is removed.