Analysis of Push-type Epidemic Data Dissemination in Fully Connected Networks

Consider a fully connected network of nodes, some of which have a piece of data to be disseminated to the whole network. We analyze the following push-type epidemic algorithm: in each push round, every node that has the data, i.e., every infected node, randomly chooses $c \in {\mathbb Z}_+$ other nodes in the network and transmits, i.e., pushes, the data to them. We write this round as a random walk whose each step corresponds to a random selection of one of the infected nodes; this gives recursive formulas for the distribution and the moments of the number of newly infected nodes in a push round. We use the formula for the distribution to compute the expected number of rounds so that a given percentage of the network is infected and continue a numerical comparison of the push algorithm and the pull algorithm (where the susceptible nodes randomly choose peers) initiated in an earlier work. We then derive the fluid and diffusion limits of the random walk as the network size goes to $\infty$ and deduce a number of properties of the push algorithm: 1) the number of newly infected nodes in a push round, and the number of random selections needed so that a given percent of the network is infected, are both asymptotically normal 2) for large networks, starting with a nonzero proportion of infected nodes, a pull round infects slightly more nodes on average 3) the number of rounds until a given proportion $\lambda$ of the network is infected converges to a constant for almost all $\lambda \in (0,1)$. Numerical examples for theoretical results are provided.Comment: 28 pages, 5 figure

Application of Stochastic Flows to the Sticky Brownian Motion Equation

We show how the theory of stochastic flows allows to recover in an elementary way a well known result of Warren on the sticky Brownian motion equation

An optimal stopping problem for spectrally negative Markov additive processes

Previous authors have considered optimal stopping problems driven by the running maximum of a spectrally negative L\'evy process $X$, as well as of a one-dimensional diffusion. Many of the aforementioned results are either implicitly or explicitly dependent on Peskir's maximality principle. In this article, we are interested in understanding how some of the main ideas from these previous works can be brought into the setting of problems driven by the maximum of a class of Markov additive processes (more precisely Markov modulated L\'evy processes). Similarly to previous works in the L\'evy setting, the optimal stopping boundary is characterised by a system of ordinary first-order differential equations, one for each state of the modulating component of the Markov additive process. Moreover, whereas scale functions played an important role in the previously mentioned work, we work instead with scale matrices for Markov additive processes here. We exemplify our calculations in the setting of the Shepp-Shiryaev optimal stopping problem, as well as a family of capped maximum optimal stopping problems.Comment: 31 page

MAPPING OF VEHICLE EMISSIONS IN ZONGULDAK PROVINCE

Increase in the rate of individual vehicle use and decrease in the usage habits of public transportation results in significant increase in traffic-related emissions. One of the most important factor among these is the number of vehicles that increases day by day especially in developing countries. In Zonguldak, the number of vehicles have also increased by 21% in the last five years. For that reason we focused on to determine the effects of urban transportation on air quality in the city center of Zonguldak/Turkey. The main objective of the study was to determine pollutant emissions in different parts of a highway and present it on emission maps. In this context, hourly vehicle counts were conducted at Zonguldak D010 highway in four zones in the coastal area. In addition, speed counts were carried out in the same zones. Emissions were calculated by the obtained data and IPCC guidance was used for these calculations. The IPCC guidelines include main headings such as energy, industrial processes, agriculture and waste. In this study, emission related data under the head of “energy” were used. Emission intensity maps for Zonguldak province were established by using the obtained values. The results of the study show that fuel consumption was the highest between 08:00 and 09:00 a.m. It decreases between 12:00 and 13:00 at noon and then tends to increase again between 18:00 and 19:00 p.m. Pollutant emissions were also higher in the morning and evening hours, depending on fuel consumption. In this study only the main arterial road was selected as study area. In future studies, by choosing whole road segments, the study area can be expanded and more accurate and reliable results can be obtained