Heavy traffic limit with discontinuous coefficients via a non-standard semimartingale decomposition
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914204833808384 |
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| author | Atar, Rami Miyazawa, Masakiyo |
| author_facet | Atar, Rami Miyazawa, Masakiyo |
| contents | This paper studies a single server queue in heavy traffic, with general inter-arrival and service time distributions, where arrival and service rates vary discontinuously as a function of the (diffusively scaled) queue length. It is proved that the weak limit is given by the unique-in-law solution to a stochastic differential equation in $[0,\infty)$ with discontinuous drift and diffusion coefficients. The main tool is a semimartingale decomposition for point processes introduced in \cite{dal-miy}, which is distinct from the Doob-Meyer decomposition of a counting process. Whereas the use of this tool is demonstrated here for a particular model, we believe it may be useful for investigating the scaling limits of queueing models very broadly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16467 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Heavy traffic limit with discontinuous coefficients via a non-standard semimartingale decomposition Atar, Rami Miyazawa, Masakiyo Probability 60K25, 60H10, 60H30 This paper studies a single server queue in heavy traffic, with general inter-arrival and service time distributions, where arrival and service rates vary discontinuously as a function of the (diffusively scaled) queue length. It is proved that the weak limit is given by the unique-in-law solution to a stochastic differential equation in $[0,\infty)$ with discontinuous drift and diffusion coefficients. The main tool is a semimartingale decomposition for point processes introduced in \cite{dal-miy}, which is distinct from the Doob-Meyer decomposition of a counting process. Whereas the use of this tool is demonstrated here for a particular model, we believe it may be useful for investigating the scaling limits of queueing models very broadly. |
| title | Heavy traffic limit with discontinuous coefficients via a non-standard semimartingale decomposition |
| topic | Probability 60K25, 60H10, 60H30 |
| url | https://arxiv.org/abs/2502.16467 |