Stein's method for models with general clocks: A tutorial
Anton Braverman, Ziv Scully
TL;DR
This paper provides a tutorial on using the generator-comparison branch of Stein’s method to obtain non-asymptotic diffusion-approximation error bounds for Markovian service-system models driven by general clocks. It develops a systematic approach based on the BAR and Palm inversion to extract diffusion generators from compensated processes, yielding a one-dimensional reflected Brownian motion (RBM) diffusion for G/G/1 and JSQ, with parameters $\theta = \mu\delta^2$ and $\sigma^2 = \delta^2(\lambda c_U^2 + \mu c_S^2)$ (and analogous expressions for multi-server cases). The tandem-queue example extends the framework to a two-dimensional RBM with oblique reflection, highlighting the need for new Stein-factor bounds for higher dimensions and oblique-derivative Poisson problems. The main contributions are a clear, multi-clock extension of the generator approach, explicit diffusion-operator extractions, and structured error-term bounds that depend on the first three moments of clock distributions, along with a candid discussion of open challenges and future directions. This work provides a practical foundation for non-asymptotic diffusion approximations in complex queueing networks with general clocks, enabling more precise performance guarantees in realistic settings.
Abstract
Diffusion approximations are widely used in the analysis of service systems, providing tractable insights into complex models. While heavy-traffic limit theorems justify these approximations asymptotically, they do not quantify the error when the system is not in the limit regime. This paper presents a tutorial on the generator comparison approach of Stein's method for analyzing diffusion approximations in Markovian models where state transitions are governed by general clocks, which extends the well-established theory for continuous-time Markov chains and enables non-asymptotic error bounds for these approximations. Building on recent work that applies this method to single-clock systems, we develop a framework for handling models with multiple general clocks. Our approach is illustrated through canonical queueing systems, including the G/G/1 queue, the join-the-shortest-queue system, and the tandem queue. We highlight the role of the Palm inversion formula and the compensated queue-length process in extracting the diffusion generator. Most of our error terms depend only on the first three moments of the general clock distribution. The rest require deeper, model-specific, insight to bound, but could in theory also depend on only the first three moments.