I encountered a practical problem and want a help (my Google skill fail T^T). Basically, this is a birthday problem in continuous space, is it hard to solve?
Problem Statement
A computer has N cores and there are M tasks within a day. Each core can only process one job at a time and each take T seconds to complete. And tasks can arrive at anytime (continuous time) with uniform probability.
Q1. Given N,M,T. What is the probability that the event of all cores being occupied (processing a task) happen during a day?
Q2. Given M,T. Find N such that P(not all cores being occupied) > 0.9, 0.99, 0.999, 0.9999, 0.99999?
If there are approximation methods, the background information is that T is small e.g. 0.1 second, and M is much larger 10^7.