Heavy-tail asymptotics for the length of a busy period in a Generalised Jackson Network
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913917594238976 |
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| author | Foss, Sergey Miyazawa, Masakiyo Yuan, Linglong |
| author_facet | Foss, Sergey Miyazawa, Masakiyo Yuan, Linglong |
| contents | We consider a Generalised Jackson Network with finitely many servers, a renewal input and $i.i.d.$ service times at each queue. We assume the network to be stable and, in addition, the distribution of the inter-arrival times to have unbounded support. This implies that the length of a typical busy period $B$, which is the time between two successive idle periods, is finite a.s. and has a finite mean.
We assume that the distributions of the service times with the heaviest tails belong to the class of so-called intermediate regularly varying distributions. We obtain the exact asymptotics for the probability ${\mathbb P} (B>x)$, as $x\to\infty$. For that, we show that the Principle of a Single Big Jump holds: $B$ takes a large value mainly due to a single unusually large service time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23310 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Heavy-tail asymptotics for the length of a busy period in a Generalised Jackson Network Foss, Sergey Miyazawa, Masakiyo Yuan, Linglong Probability 60K25, 60F10, 60K20 We consider a Generalised Jackson Network with finitely many servers, a renewal input and $i.i.d.$ service times at each queue. We assume the network to be stable and, in addition, the distribution of the inter-arrival times to have unbounded support. This implies that the length of a typical busy period $B$, which is the time between two successive idle periods, is finite a.s. and has a finite mean. We assume that the distributions of the service times with the heaviest tails belong to the class of so-called intermediate regularly varying distributions. We obtain the exact asymptotics for the probability ${\mathbb P} (B>x)$, as $x\to\infty$. For that, we show that the Principle of a Single Big Jump holds: $B$ takes a large value mainly due to a single unusually large service time. |
| title | Heavy-tail asymptotics for the length of a busy period in a Generalised Jackson Network |
| topic | Probability 60K25, 60F10, 60K20 |
| url | https://arxiv.org/abs/2506.23310 |