2022/11/04 by Dennis Schol, Schol, Dennis, Maria Vlasiou +3
Business, Management and Accounting · Decision Sciences · #60G70 #60K25 #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.2211.02313
openalex publication_date 2022/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this paper, we study the maximum waiting time maxi≤ NWi(⋅) in an N-server fork-join queue with heavy-tailed services as N→∞. 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 L(bN)bN^\fracβ(β-1), where β is the shape parameter in the regularly varying distribution, L(x) is a slowly varying function, and (bN,N≥ 1) is a sequence for which holds that maxi≤ NAi/bN\oversetℙ\longrightarrow1, as N→∞, where Ai are i.i.d. Weibull-distributed random variables. Finally, we prove steady-state convergence.