Imagine you run a small call center. Customers arrive, wait on hold, get served, and hang up. At any moment the system is in some state—maybe three people waiting, two being served. Over time the state jumps around randomly. If you want to predict the chance of finding five waiting at noon, you need a way to combine differential equations and probability. That is exactly what this chapter gives you: we build continuous‑time Markov chains and solve them with matrix exponentials.