Diffusion Limits for Measure-Valued Queueing Models, With Application to General Random Order of Service Queues
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917553830363136 |
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| author | Loeser, Eva H |
| author_facet | Loeser, Eva H |
| contents | Currently, there is no general theory for deriving diffusion approximations of queueing systems with high- or infinite-dimensional state descriptors. In this paper, we explore one path for deriving diffusion limit equations of queueing models. The method hinges on a martingale decomposition of dynamics driven by time-changed renewal processes, which are a common feature of many queueing models. We then prove a central limit theorem for models decomposed in this way, which gives the form of stochastic differential equations (SDEs) that will be satisfied in the diffusion limit of such a system. We demonstrate the approach on a multiclass, multi-server random order of service queue with reneging and generally distributed interarrival, service, and patience times. In this setting, the state descriptor is measure-valued and the workload is nonlinear in the fluid limit. We prove tightness and uniqueness of limit points, and derive SDEs satisfied by the diffusion limit. Our results offer a broadly applicable method for diffusion approximations in complex queueing systems without relying on dimension reductions or model-specific structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10633 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diffusion Limits for Measure-Valued Queueing Models, With Application to General Random Order of Service Queues Loeser, Eva H Probability 60K25, 60F05, 60B10 Currently, there is no general theory for deriving diffusion approximations of queueing systems with high- or infinite-dimensional state descriptors. In this paper, we explore one path for deriving diffusion limit equations of queueing models. The method hinges on a martingale decomposition of dynamics driven by time-changed renewal processes, which are a common feature of many queueing models. We then prove a central limit theorem for models decomposed in this way, which gives the form of stochastic differential equations (SDEs) that will be satisfied in the diffusion limit of such a system. We demonstrate the approach on a multiclass, multi-server random order of service queue with reneging and generally distributed interarrival, service, and patience times. In this setting, the state descriptor is measure-valued and the workload is nonlinear in the fluid limit. We prove tightness and uniqueness of limit points, and derive SDEs satisfied by the diffusion limit. Our results offer a broadly applicable method for diffusion approximations in complex queueing systems without relying on dimension reductions or model-specific structure. |
| title | Diffusion Limits for Measure-Valued Queueing Models, With Application to General Random Order of Service Queues |
| topic | Probability 60K25, 60F05, 60B10 |
| url | https://arxiv.org/abs/2504.10633 |