Overview

This topic closes the unit by lifting every major piece of discrete-time Markov chain theory to continuous time, using exactly the exponential-distribution and Poisson-process tools built in the previous topic. The generator matrix $Q$ replaces the transition matrix $p$; the embedded jump chain shows every CTMC is built from a discrete chain plus exponential holding times; and classification, stationary distributions, detailed balance, and convergence all carry over — continuous time even simplifies the convergence theory, since exponential holding times automatically rule out periodicity. The topic's second half specialises this machinery to queueing theory, applied probability's most celebrated application, closing with Burke's theorem (a stable queue's departures are themselves Poisson) and Jackson networks (an entire network of interacting queues has an independent, product-form stationary distribution).