arXiv subjects

Dennis Schol

Publications and source records attributed to Dennis Schol.

5 recordsLinked to original sources

arXiv2024

Optimization of inventory and capacity in large-scale assembly systems using extreme-value theory

High-tech systems are typically produced in two stages: 1) Production of components using specialized equipment and staff; 2) System assembly/integration. Component production capacity is subject to fluctuations, causing a high risk of shortages of at least one component, which results in costly delays. Companies hedge this risk by strategic investments in excess production capacity and in buffer inventories of components. To optimize these, it is crucial to characterize the relation between component shortage risk and capacity and inventory investments. We suppose that component production capacity and produce demand are normally distributed over finite time intervals, and we accordingly model the production system as a symmetric fork-join queueing network with $N$ statistically identical queues with a common arrival process and independent service processes. Assuming a symmetric cost structure, we subsequently apply extreme value theory to gain analytic insights into this optimization problem. We derive several new results for this queueing network, notably that the scaled maximum of $N$ steady-state queue lengths converges in distribution to a Gaussian random variable. These results translate into asymptotically optimal methods to dimension the system. Tests on a range of problems reveal that these methods typically work well for systems of moderate size.

math.PR↗

arXiv2023

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

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 $N\to\infty$. The maximum steady-state waiting time in this queueing system scales around $\frac{1}γ\log N$, where $γ$ is determined by the cumulant generating function $Λ$ of the service distribution and solves the Cramér-Lundberg equation with stochastic service times and deterministic inter-arrival times. This value $\frac{1}γ\log N$ is reached at a certain hitting time. The number of arrivals until that hitting time satisfies the central limit theorem, with standard deviation $\frac{σ_A}{\sqrt{Λ'(γ)γ}}$. By using distributional 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.

math.PR↗

arXiv2022

Maximum waiting time in heavy-tailed fork-join queues

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 $\tilde{L}(b_N)b_N^{\fracβ{(β-1)}}$, where $β$ 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.

math.PR↗

arXiv2022

Tail Asymptotics for the Delay in a Brownian Fork-Join Queue

In this paper, we study the tail behavior of $\max_{i\leq N}\sup_{s>0}\left(W_i(s)+W_A(s)-βs\right)$ as $N\to\infty$, with $(W_i,i\leq N)$ i.i.d. Brownian motions and $W_A$ an independent Brownian motion. This random variable can be seen as the maximum of $N$ mutually dependent Brownian queues, which in turn can be interpreted as the backlog in a Brownian fork-join queue. In previous work, we have shown that this random variable centers around $\frac{σ^2}{2β}\log N$. Here, we analyze the rare-event that this random variable reaches the value $(\frac{σ^2}{2β}+a)\log N$, with $a>0$. It turns out that its probability behaves roughly as a power law with $N$, where the exponent depends on $a$. However, there are three regimes, around a critical point $a^{\star}$; namely, $0 a^{\star}$. The latter regime exhibits a form of asymptotic independence, while the first regime reveals highly irregular behavior with a clear dependence structure among the $N$ suprema, with a nontrivial transition at $a=a^{\star}$.

math.PR↗

arXiv2021

Large fork-join queues with nearly deterministic arrival and service times

In this paper, we study an $N$ server fork-join queue with nearly deterministic arrival and service times. Specifically, we present a fluid limit for the maximum queue length as $N\to\infty$. This fluid limit depends on the initial number of tasks. In order to prove these results, we develop extreme value theory and diffusion approximations for the queue lengths.

math.PR↗