Expected Number of Area Exits, Entrances and Crossovers for the Investigation of Area Based Push Service Utilization

Area based services enable the provisioning of various useful services for users of smart and transportable wireless mobile clients in defined geographical areas. By periodically tracking the clients' positions, special events can be generated if they change their spatial relation to an area. The events can be utilized for the triggering of certain push services, sending e.g. a warning if a client approaches a hazardous area, or an advertisement if he enters the area of a shop. This paper presents an analytical investigation of the expected utilization of the service (i.e. the number of events), the type of information being capital for a proper dimensioning and billing of the services. By means of geometrical derivations, we determine the probabilities for mobile clients to enter or leave an area, investigating different area sizes and shapes (circular, rectangular and hexagonal). Additionally, we evaluate the cases of failed detections and provide some numerical results for different client velocity distributions.