Stability of storage processes with general release rates
Miha Brešar, Aleksandar Mijatović, Nikola Sandrić
TL;DR
The paper analyzes stability and tail behavior for a broad class of storage processes driven by subordinator inputs and governed by a general release rate $r$. Using a combination of Foster–Lyapunov drift conditions and coupling arguments, the authors derive subgeometric and geometric ergodicity results in total variation, as well as convergence in Wasserstein distances, with rates encoded by functions of the release-rate and the Lévy measure tail $\bar{\nu}$. They connect the tail of the stationary distribution to $\bar{\nu}$ and $r$, and provide upper and lower bounds that capture polynomial to uniform ergodicity across various regimes, including finite and infinite activity inputs. These results extend classical storage-model theory (Brockwell, Meyn–Tweedie, Yamazato) and offer a unified framework for analyzing stability and tail behavior in both TV and Wasserstein metrics, with broad implications for queueing and storage systems. The work also clarifies when Wasserstein distances are preferable due to irregularities in total-variation convergence and furnishes a general methodological toolkit (Lyapunov methods and synchronous coupling) for similar jump-into-continuous-state-space models.
Abstract
This paper quantifies the ergodicity and the rate of decay of the tail of the stationary distribution for a broad class of storage models, encompassing constant, linear, and power-type release rates with both finite and infinite activity input process. Our results are expressed in terms of the asymptotics of the release rate, the tail-decay rate of the Lévy measure of the input process and its (possibly infinite) first moment. Our framework unifies and significantly extends classical results on the stability of storage models. Under certain regularity assumptions, we also provide upper bounds on the stability in the Wasserstein distance.