A restless time-fractional multiclass queue
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908901119623168 |
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| author | Georgiou, Nicos Scalas, Enrico Vysotsky, Vladislav |
| author_facet | Georgiou, Nicos Scalas, Enrico Vysotsky, Vladislav |
| contents | We study a single-server priority queue with a finite number of classes, in which the arrivals follow a fractional Poisson process of index $α\in (0,1]$ and the service completions are triggered by an independent fractional Poisson process of index $β\in (0,1]$. Each of the customers arriving is assigned at random to one of the priority classes. This assignment is independent of the rest of the system and follows a fixed probability distribution.
Using a time-change representation of a fractional Poisson process, we first give a multinomial thinning decomposition: the total number of arrivals in each class are independent standard Poisson processes of appropriate intensities, time-changed by a common independent random clock that is the inverse of an $α$-stable subordinator. This yields a process-level law of large numbers and a functional central limit theorem for the process of arrivals.
For the queueing system itself, we identify process-level scaling limits for the cumulative and individual queue lengths of the classes. We also prove that the queue gets empty infinitely often when $α\le β$, which does include the critical case $α= β$. A final example shows how the model can be extended to a continuum of classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18461 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A restless time-fractional multiclass queue Georgiou, Nicos Scalas, Enrico Vysotsky, Vladislav Probability 60K25, 60F17, 60G55, 60B51 We study a single-server priority queue with a finite number of classes, in which the arrivals follow a fractional Poisson process of index $α\in (0,1]$ and the service completions are triggered by an independent fractional Poisson process of index $β\in (0,1]$. Each of the customers arriving is assigned at random to one of the priority classes. This assignment is independent of the rest of the system and follows a fixed probability distribution. Using a time-change representation of a fractional Poisson process, we first give a multinomial thinning decomposition: the total number of arrivals in each class are independent standard Poisson processes of appropriate intensities, time-changed by a common independent random clock that is the inverse of an $α$-stable subordinator. This yields a process-level law of large numbers and a functional central limit theorem for the process of arrivals. For the queueing system itself, we identify process-level scaling limits for the cumulative and individual queue lengths of the classes. We also prove that the queue gets empty infinitely often when $α\le β$, which does include the critical case $α= β$. A final example shows how the model can be extended to a continuum of classes. |
| title | A restless time-fractional multiclass queue |
| topic | Probability 60K25, 60F17, 60G55, 60B51 |
| url | https://arxiv.org/abs/2510.18461 |