Coalescence times for critical Galton-Watson processes with immigration

Let $X^I_n$ be the coalescence time of two particles picked at random from the $n$th generation of a critical Galton-Watson process with immigration, and let $A^I_n$ be the coalescence time of the whole population in the $n$th generation. In this paper, we study the limiting behaviors of $X^I_n$ and $A^I_n$ as $n\to\infty$

An efficient MapReduce-based parallel clustering algorithm for distributed traffic subarea division

Traffic subarea division is vital for traffic system management and traffic network analysis in intelligent transportation systems (ITSs). Since existing methods may not be suitable for big traffic data processing, this paper presents a MapReduce-based Parallel Three-Phase K -Means (Par3PKM) algorithm for solving traffic subarea division problem on a widely adopted Hadoop distributed computing platform. Specifically, we first modify the distance metric and initialization strategy of K -Means and then employ a MapReduce paradigm to redesign the optimized K -Means algorithm for parallel clustering of large-scale taxi trajectories. Moreover, we propose a boundary identifying method to connect the borders of clustering results for each cluster. Finally, we divide traffic subarea of Beijing based on real-world trajectory data sets generated by 12,000 taxis in a period of one month using the proposed approach. Experimental evaluation results indicate that when compared with K -Means, Par2PK-Means, and ParCLARA, Par3PKM achieves higher efficiency, more accuracy, and better scalability and can effectively divide traffic subarea with big taxi trajectory data