Multilayer parking with screening on a random tree

In this paper we present a multilayer particle deposition model on a random tree. We derive the time dependent densities of the first and second layer analytically and show that in all trees the limiting density of the first layer exceeds the density in the second layer. We also provide a procedure to calculate higher layer densities and prove that random trees have a higher limiting density in the first layer than regular trees. Finally, we compare densities between the first and second layer and between regular and random trees.Comment: 15 pages, 2 figure

Parking on a Random Tree

Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Renyi's parking problem, alternatively called blocking RSA: at every vertex of the tree a particle (or car) arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet. We provide an explicit expression for the so-called parking constant in terms of the generating function.Comment: 7 page

Parking On A Random Rooted Plane Tree

In this paper, we investigate a parking process on a uniform random rooted plane tree with $n$ vertices. Every vertex of the tree has a parking space for a single car. Cars arrive at independent uniformly random vertices of the tree. If the parking space at a vertex is unoccupied when a car arrives there, it parks. If not, the car drives towards the root and parks in the first empty space it encounters (if there is one). We are interested in asymptotics of the probability of the event that all cars can park when $\lfloor \alpha n \rfloor$ cars arrive, for $\alpha > 0$. We observe that there is a phase transition at $\alpha_c := \sqrt{2} -1$: if $\alpha < \alpha_c$ then the event has positive probability, whereas for $\alpha > \alpha_c$ it has probability 0. Analogous results have been proved by Lackner and Panholzer, Goldschmidt and Przykucki, and Jones for different underlying random tree models.Comment: 12 page

Parking on a Random Tree

Abstract Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Rényi&apos;s parking problem, alternatively called blocking RSA (random sequential adsorption): at every vertex of the tree a particle (or &quot;car&quot;) arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet. We provide an explicit expression for the so-called parking constant in terms of the generating function. That is, the occupation probability, averaged over dynamics and the probability distribution of the random trees converges in the large-time limit to (1 − α 2 )/2 with 1 α xdx G(x) = 1

Evaluating the Efficacy of Magnetometer-Based Vehicle Sensors

The issue of parking is more at the forefront of urban development than one might believe. In fact, academic studies have shown that roughly 30% of city traffic is due to drivers circling city blocks attempting to find an open spot. Due to such congestion people often avoid urban centers and downtown areas for shopping or dining because parking is such a hassle and assumed to be unavailable. If drivers knew where parking was available in real time, they could proceed directly to open spaces as opposed to their congestion-inducing attempts to park. A better solution would guide drivers to available parking and may help re-vitalize downtown areas. The problem of knowing whether an available spot exists, however, is complex. This thesis is an investigation and analysis of the efficacy of magnetometers as vehicle sensors for on-street (non-garage) parking. While many solutions to detecting available parking have been tried, we focused on magnetometer-based vehicle sensors placed in each parking spot. We built a sensor comprised of a low-cost magnetometer, a radio, a micro-controller, and a battery on a custom printed circuit board. Our idea is that such a sensor could be placed in each parking space and monitor for vehicles. When a vehicle arrives, the magnetometer detects a change in the magnetic environment, then radios the presence of the vehicle in a space to a central server that aggregates and disseminates parking data to drivers and city officials. City officials could use this data to craft better parking policies and prices. Drivers could then use GPS coordinates and the aggregated space availability in navigation apps to proceed directly to open spaces. Our hope is that this work will provide a foundation for others to learn from our insights into the reliability, stability, and accuracy of such parking sensors. After obtaining permission from Dartmouth Parking and Transportation Services, we conducted experiments in a surface lot on Dartmouth College\u27s campus, and as such, we limited our data collection to a single grid-like parking arrangement to gain deeper insight to one common mode of parking. The analysis of the collected data leverages machine learning via sci-kit learn to form a robust detection algorithm for whether a vehicle is in a space. We utilized four detection algorithms in total, one via a simple magnitude threshold, another using Gaussian Bayes classification, a decision tree classification model, and finally a random forest model. All of these methods succeeded in correctly detecting the status of a parking spot with accuracy well above 90%. Our best classification model, which uses a decision tree, correctly predicted parking space occupancy with 99% accuracy. In our experiments we show that these sensors are stable and do not drift from their initial reading. Our detection algorithms show that they are an accurate option for vehicle detection. Finally, we show that the placement of a sensor is not crucial, so long as the sensor is centrally placed in a parking spot

Merging costs for the additive Marcus-Lushnikov process, and Union-Find algorithms

Starting with a monodisperse configuration with $n$ size-1 particles, an additive Marcus-Lushnikov process evolves until it reaches its final state (a unique particle with mass $n$). At each of the $n-1$ steps of its evolution, a merging cost is incurred, that depends on the sizes of the two particles involved, and on an independent random factor. This paper deals with the asymptotic behaviour of the cumulated costs up to the $k$th clustering, under various regimes for $(n,k)$, with applications to the study of Union--Find algorithms.Comment: 28 pages, 1 figur