Stein's method for models with general clocks: A tutorial
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910877262807040 |
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| author | Braverman, Anton Scully, Ziv |
| author_facet | Braverman, Anton Scully, Ziv |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12189 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stein's method for models with general clocks: A tutorial Braverman, Anton Scully, Ziv Probability Primary 60J25, 60F99, secondary 60K25, 60J60 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. |
| title | Stein's method for models with general clocks: A tutorial |
| topic | Probability Primary 60J25, 60F99, secondary 60K25, 60J60 |
| url | https://arxiv.org/abs/2503.12189 |