Applied Probability (Lent 2021)

Time and Location: Online

Applied Probability is a course in Part II of the Cambridge Tripos.

Schedule

  • Finite-state continuous-time Markov chains: basic properties. Q-matrix (or generator), backward and forward equations. The homogeneous Poisson process and its properties (thinning, superposition). Birth and death processes.
  • General continuous-time Markov chains. Jump chains. Explosion. Minimal Chains. Communicating classes. Hitting times and probabilities. Recurrence and transience. Positive and null recurrence. Convergence to equilibrium. Reversibility.
  • Applications: the M/M/1 and M/M/∞ queues. Burke’s theorem. Jackson’s theorem for queueing networks. The M/G/1 queue.
  • Renewal theory: renewal theorems, equilibrium theory (proof of convergence only in discrete time). Renewal-reward processes. Little’s formula.
  • Spatial Poisson processes in d dimensions. The superposition, mapping, and colouring theorems. Renyi’s theorem. Applications including Olbers’ paradox.

Lecture notes (updated throughout the term)

Videos available on Moodle

Example Sheets

References