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Abstract:We investigate the steady-state diffusion-approximation error for continuous-time queueing systems with generally distributed primitives. A common picture emerges after analyzing a number of canonical systems: the error decomposes into interior and boundary terms. The former are simpler to handle and can be bounded using only low-order moments of the system's primitives -- when the approximation error is measured using the Wasserstein distance, three moments suffice. The boundary terms are inherently more delicate: while crude bounds are easy to obtain, sharper (e.g., order optimal) bounds require deeper, model specific, insights.
Methodologically, we extend the generator comparison approach of Stein's method to piecewise-deterministic Markov processes (PDMPs). The discontinuous nature of the PDMP at jump times necessitates using the basic adjoint relationship (BAR), instead of the infinitesimal generator, to characterize the stationary distribution. A second-order Taylor expansion of the BAR jump terms, coupled with a Palm-inversion step that converts event-averaged quantities into time averages, yields the candidate diffusion generator and a transparent interior/boundary error decomposition. In parallel, we show how the prelimit generator approach -- working with the Poisson equation of the queueing system instead of the diffusion process -- offers a promising avenue for bounding the challenging boundary terms.

Submission history

From: Anton Braverman [view email]
[v1] Wed, 17 Jul 2024 16:37:48 UTC (50 KB)
[v2] Tue, 30 Jul 2024 18:23:10 UTC (54 KB)
[v3] Tue, 2 Sep 2025 16:32:48 UTC (80 KB)
[v4] Tue, 30 Jun 2026 17:06:53 UTC (95 KB)