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 [Formula: see text]. 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.

ON THE APPLICATION OF THE EXTREME VALUE THEORY IN SHIP’S STRENGTH CALCULATIONS

The problem undertaken by the Author is fundamen- tal for ship safety and was discussed many times in the past (e.g. Boistov, 2000). However, the discus- sions did not dispel the Author’s or my doubts for the same reason: “If the calculations with the Ex- treme Value Theory of the probability of exceeding of the design hull girder bending moment are cor- rect, ships would suffer more severe casualties than previously observed.”

ON THE APPLICATION OF THE EXTREME VALUE THEORY IN SHIP’S STRENGTH CALCULATIONS

A procedure is proposed for application of the extreme value theory (EVT) approach considering not only the maximal value of the corresponding random variable but also its probability of exceedance. It substantially reduces the probability of exceedance of any given limit value used in the case when traditional EVT is applied. Examples are provided to illustrate its application when records of the random process are available.

Numerical solution of a Gamma - integral equation using a higher order composite Newton-Cotes formulas

The paper aims at solving a complex equation with Gamma - integral. The solution is the infected size (p) at equilibrium. The approaches are both numerical and analytical methods. As a numerical method, the higher-order composite Newton-Cotes formula is developed and implemented. The results show that the infected size (p) increases along with the shape parameter (k). But the increase has two phases: an increasing rate phase and a decreasing rate phase; both phases can be explained by the instantaneous death rate characteristics of the Gamma distribution hazard function. As an analytical method, the Extreme Value Theory consolidates the numerical solutions of the infected size (p) when k ≥ 1 and provides a solution limit ( p = 1 − 1 2 R ) as k goes to +∞.