  We consider the model of a token-based joint auto-scaling and load balancing
strategy, proposed in a recent paper by Mukherjee, Dhara, Borst, and van
Leeuwaarden (SIGMETRICS '17, arXiv:1703.08373), which offers an efficient
scalable implementation and yet achieves asymptotically optimal steady-state
delay performance and energy consumption as the number of servers $N\to\infty$.
In the above work, the asymptotic results are obtained under the assumption
that the queues have fixed-size finite buffers, and therefore the fundamental
question of stability of the proposed scheme with infinite buffers was left
open. In this paper, we address this fundamental stability question. The system
stability under the usual subcritical load assumption is not automatic.
Moreover, the stability may not even hold for all $N$. The key challenge stems
from the fact that the process lacks monotonicity, which has been the powerful
primary tool for establishing stability in load balancing models. We develop a
novel method to prove that the subcritically loaded system is stable for large
enough $N$, and establish convergence of steady-state distributions to the
optimal one, as $N \to \infty$. The method goes beyond the state of the art
techniques -- it uses an induction-based idea and a "weak monotonicity"
property of the model; this technique is of independent interest and may have
broader applicability.
