Relaxation Times of Information Gathering in an Epidemically Information Sharing Process with Randomly Moving Sensors in a Bounded Space

In the context of sensor and ad hoc networks, wireless networking systems with mobile sensors have been studied enthusiastically for the applications to habitat and environment monitoring, and information communications between vehicles and portable devices. Since mobile objects move randomly and communicate each other wirelessly only on close ranges, it is possible to happen frequent disconnection all over the place on the network. The basic properties of information communications in these poor networking systems are not well understood. For example, ZebraNet is known as a habitat monitoring system that collects biological information of zebras wandering on a broad savanna with wireless sensors. As time passes, sensors collect the information,but here we have a simple question "How much time should we wait until the information is gathered enough ?" To answer the question theoretically, we introduced a mathematical model that N sensors randomly move in a d-dimensional square lattice bounded by system size L. The sensors share information epidemically when they are at the same position, which expresses sensors\´ irregular motions and close-range communications. Using this model, we numerically investigated the properties of probability distribution functions of information gathering times on the basis of some known theoretical results from the first passage process. We found that the distribution function of information gathering times generally obeys a exponential distribution at the tail with the relaxation time ¿ depending on d, L, and N. Furthermore, we also found the non-trivial scaling relations of tau with L and N, which finally gives the answer to the question about the latency of information gatherings with using mobile sensors.