If the bus arrives on time, the arrival time would be [tau, 2 * tau, ..., N * tau]. One way to simulate "random" arrival time is to draw uniform points in the

If the bus arrives on time, the arrival time would be [tau, 2 * tau, ..., N * tau]. 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_distribution_of_the_order_statistics_of_the_uniform_distribution https://en.wikipedia.org/wiki/Order_statistic#The_joint_dist... [2] https://en.wikipedia.org/wiki/Beta_distribution#Special_and_limiting_cases https://en.wikipedia.org/wiki/Beta_distribution#Special_and_... [3] https://en.wikipedia.org/wiki/Exponential_distribution#Related_distributions https://en.wikipedia.org/wiki/Exponential_distribution#Relat...