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Samenvatting
In this paper, we study the maximum waiting time $\max_{i\leq N}W_i(\cdot)$ in an $N$-server fork-join queue with heavy-tailed services as $N\to\infty$. The service times are the product of two random variables. One random variable has a regularly varying tail probability and is the same among all $N$ servers, and one random variable is Weibull distributed and is independent and identically distributed among all servers. This setup has the physical interpretation that if a job has a large size, then all the subtasks have large sizes, with some variability described by the Weibull-distributed part. We prove that after a temporal and spatial scaling, the maximum waiting time process converges in $D[0,T]$ to the supremum of an extremal process with negative drift. The temporal and spatial scaling are of order $\tilde{L}(b_N)b_N^{\frac{\beta}{(\beta-1)}}$, where $\beta$ is the shape parameter in the regularly varying distribution, $\tilde{L}(x)$ is a slowly varying function, and $(b_N,N\geq 1)$ is a sequence for which holds that $\max_{i\leq N}A_i/b_N\overset{\mathbb{P}}{\longrightarrow}1$, as $N\to\infty$, where $A_i$ are i.i.d.\ Weibull-distributed random variables. Finally, we prove steady-state convergence.
Originele taal-2 | Engels |
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Artikelnummer | 2211.02313 |
Aantal pagina's | 24 |
Tijdschrift | arXiv |
Volume | 2022 |
DOI's | |
Status | Gepubliceerd - 4 nov. 2022 |
Vingerafdruk
Duik in de onderzoeksthema's van 'Maximum waiting time in heavy-tailed fork-join queues'. Samen vormen ze een unieke vingerafdruk.Projecten
- 1 Afgelopen
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Complexity in Transport and Logistics
Vlasiou, M. (Project Manager) & Schol, C. (Projectmedewerker)
28/08/17 → 30/09/22
Project: Onderzoek direct
Impact
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Stochactic processes on interacting networks
Vlasiou, M. (Content manager)
Impact: Research Topic/Theme (at group level)
Onderzoekersoutput
- 4 Tijdschriftartikel
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Large fork-join queues with nearly deterministic arrival and service times
Schol, D., Vlasiou, M. & Zwart, B., 1 mei 2022, In: Mathematics of Operations Research. 47, 2, blz. 1335-1364 30 blz.Onderzoeksoutput: Bijdrage aan tijdschrift › Tijdschriftartikel › Academic › peer review
Open AccessBestand5 Citaten (Scopus)72 Downloads (Pure) -
Tail Asymptotics for the Delay in a Brownian Fork-Join Queue
Schol, D., Vlasiou, M. & Zwart, B., 9 aug. 2022, In: arXiv. 2022, 24 blz., 2208.04796.Onderzoeksoutput: Bijdrage aan tijdschrift › Tijdschriftartikel › Academic
Open AccessBestand33 Downloads (Pure) -
Extreme-value theory for large fork-join queues, with an application to high-tech supply chains
Meijer, M., Schol, D., Jaarsveld, W. V., Vlasiou, M. & Zwart, B., 19 mei 2021, In: arXiv. 2021, 40 blz., 2105.09189.Onderzoeksoutput: Bijdrage aan tijdschrift › Tijdschriftartikel › Academic
Open AccessBestand57 Downloads (Pure)