Why does he times tau by N when calculating the bus arrival times in the beginning?
replies(1):
One way to simulate "random" arrival time is to draw uniform points in the interval [0, N * tau].
It turns out the inter-arrival time generated this way is approximately exponential:
1. the difference of consecutive ordered uniformly distribution random variables follows a Beta(1, N) distribution [1].
2. As N goes to infinity, N * Beta(1, N) converges to Exponential(1) [2].
3. Since we scale the rand() by N * tau, the inter arrival time will follow an Exponential(1 / tau) distribution (as N goes to infinity), which has an expected value of tau [3].
Edit: I just realized that the author did mention this simulation is only an approximation in the side note.
[1] https://en.wikipedia.org/wiki/Order_statistic#The_joint_dist...
[2] https://en.wikipedia.org/wiki/Beta_distribution#Special_and_...
[3] https://en.wikipedia.org/wiki/Exponential_distribution#Relat...