Work

Extreme values for the waiting time in large fork-join queues

Year: 2025

Type: article

Abstract: We prove that the scaled maximum steady-state waiting time and the scaled maximum steady-state queue length among N GI/GI/1-queues in the N-server fork-join queue converge to a normally distributed random variable as $$N\rightarrow \infty $$ N → ∞ . The maximum steady-state waiting time in this queueing system scales around $$\frac{1}{\gamma }\log N$$ 1 γ log N , where $$\gamma $$ γ is determined by the cumulant generating function $$\Lambda $$ Λ of the service times distribution and solves the Cramér–Lundberg equation with stochastic service times and deterministic interarrival times. This value $$\frac{1}{\gamma }\log N$$ 1 γ log N is reached at a certain hitting time. The number of arrivals until that hitting time satisfies the central limit theorem, with standard deviation $$\frac{\sigma _A}{\sqrt{\Lambda '(\gamma )\gamma }}$$ σ A Λ ′ ( γ ) γ . By using the distributional form of Little’s law, we can extend this result to the maximum queue length. Finally, we extend these results to a fork-join queue with different classes of servers.

Source: Queueing Systems

DOI: 10.1007/s11134-025-09937-2

Authors: Dennis Schol, Maria Vlasiou, Bert Zwart

Institutions: Eindhoven University of Technology, University of Twente, Centrum Wiskunde & Informatica

Language: English

Topic: Advanced Queuing Theory Analysis

Subfield: Management Information Systems

Field: Business, Management and Accounting

Domain: Social Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funder: Stichting voor de Technische Wetenschappen