Joseph G. Voelkel | Searcharxiv

arXiv2018

Replace-after-Fixed-or-Random-Time Extensions of the Poisson Process

We analyze extensions of the Poisson process in which any interarrival time that exceeds a fixed value $r$ is counted as an interarrival of duration $r$. In the engineering application that initiated this work, one part is tested at a time, and $N(t)$ is the number of parts that, by time $t$, have either failed, or if they have reached age $r$ while still functioning, have been replaced. We refer to $\{N(t), t \geq 0\}$ as a replace-after-fixed-time process. We extend this idea to the case where the replacement time for the process is itself random, and refer to the resulting doubly stochastic process as a replace-after-random-time process.

math.PR↗

arXiv2017

The Design of Order-of-Addition Experiments

We introduce systematic methods to create optimal designs for order-of-addition (OofA) experiments, those that study the order in which $m$ components are applied---for example, the order in which chemicals are added to a reaction or layers are added to a film. Full designs require $m!$ runs, so we investigate design fractions. Balance criteria for creating such designs employ an extension of orthogonal arrays (OA's) to OofA-OA's. A connection is made between $D$-efficient and OofA-OA designs. Necessary conditions are found for the number of runs needed to create OofA-OA's of strengths 2 and 3. We create a number of new, optimal, designs: 12-run OofA-OA's in 4 and 5 components, 24-run OofA-OA's in 5 and 6 components, and near OofA-OA's in 7 components. We extend these designs to include (a) process factors, and (b) the common case in which component orderings are restricted. We also suggest how such designs may be analyzed.

stat.ME↗