My model is like the arrival model, and the arrived-rate is related with the time. In most situation, we can choose the Poisson distribution for describing arrival-model time with specific arrived-rate (It can calculate the possibility of the event occurred specific times until specific moment). But in my model, the arrival rate increasing (may be exponential growth) depends on time flying until the specific time. I want to know, under this condition, which model/function is suitable for the arrived-rate?

The Model Example:

There are 50 students in a class. Every morning, the Mathematics class will be started at 9:00. So everyone should arrive classroom before 9:00. For every student, he/she do not have to go to the classroom too early, so he/she could choose the suitable time arrived classroom. For most of students, the time they arrived is always closely to 9:00 (we assume that every student does not like to go to school. they want to delay the arrived-classroom time, but not be late). Under this condition, the arrived-rate of student arriving classroom is increasing with the time flying until 9:00 , and it quickly fall to 0 at 9:00. So, how to choose the model/function for this arrival-rate?

Thanks.