I am curious how to simulate time-dependent probability models?
To clarify my question, imagine that some cafe has N customers. Each customer could order coffee(if he is not drinking it right now). If he had coffee at the moment $t_0$, than at time $t$ he is going to order coffee with the probability $p(t) = \exp(-\alpha_i (t - t_0))$. It takes $t_{make} \sim |N(m_0, \sigma_0)|$ to make coffee. Customer is drinking coffee for $t_{drink} \sim |N(m_1, \sigma_1)|$.
So, I am curious in a probability distribution for a waiting time at some moment $T$ for a customer $i$.
Of course, I can do a Monte Carlo and approximate it with an empirical distribution, but I am not interested in solving this exact problem. I would like to learn some general approaches to solve similar problems.
I would be grateful if you share your thoughts, links to books and articles.