Week 1: | Mon, Jan 7 Introduction to the course, discussion of policy, dates of exams, other administrative issues. Introduction to Markov chains. Problem Set 1 assigned, due on Wednesday, Jan 23, at 8:30 AM |
Wed, Jan 9 Introduction to stationary distributions, Gambler's ruin, hitting times, recurrent and transient states |
Fri, Jan 11 discussion of Theorems in Ch 1 of Hoel, Port, Stone |
Week 2: | Mon, Jan 14 proof of Theorems in Ch 1 of Hoel, Port, Stone |
Wed, Jan 16 Example 1.14 from Durrett extension of ideas: recurrent vs transient expected values, long-run averages; start of discussion of birth-death chains |
Fri, Jan 18 more discussion of birth-death chains |
Week 3: | Mon, Jan 21 (no lecture) Martin Luther King Jr. Day |
Wed, Jan 23 Problem Set 1 due today at 8:30 AM Problem Set 2 assigned, due on Wednesday, Feb 6, at 8:30 AM |
Fri, Jan 25 branching processes: introduction and examples |
Week 4: | Mon, Jan 28 when are branching processes transient or recurrent? (with proofs) |
Wed, Jan 30 queuing processes: introduction and examples |
Fri, Feb 1 when are queuing processes transient or recurrent? (with proofs) |
Week 5: | Mon, Feb 4 overview of steady states and connections to renewal theory |
Wed, Feb 6 discussion about expected proportion of the time a chain is in a state Problem Set 2 due today at 8:30 AM |
Fri, Feb 8 Midterm Exam 1 |
Week 6: | Mon, Feb 11 proofs about expected proportion of the time a chain is in a state Problem Set 3 assigned, due on Monday, Feb 25, at 8:30 AM |
Wed, Feb 13 stationary distributions |
Fri, Feb 15 stationary distributions |
Week 7: | Mon, Feb 18 examples about expected return time in a Markov chain (examples of positive vs null recurrence) |
Wed, Feb 20 in-depth discussion of expected return time in queuing processes |
Fri, Feb 22 introduction to Poisson Processes |
Week 8: | Mon, Feb 25 more about equivalent definitions of Poisson processes; comparison to order statistics of uniformly distributed random variables Problem Set 3 due today at 8:30 AM Problem Set 4 assigned, due on Wed, Mar 6, at 8:30 AM |
Wed, Feb 27 interarrival times in a Poisson process |
Fri, Mar 1 conditional distribution of arrival times in a Poisson process |
Week 9: | Mon, Mar 4 discussion of M/G/1 busy period including 3 lemmas needed for the proof |
Wed, Mar 6 Problem Set 4 due today at 8:30 AM end of the proof for M/G/1 busy period Introduction to nonhomogeneous Poisson processes |
Fri, Mar 8 Midterm Exam 2 |
Spring Break: | Mon, Mar 11 (no lecture) Spring Break |
Mon, Mar 13 (no lecture) Spring Break |
Mon, Mar 15 (no lecture) Spring Break |
Week 10: | Mon, Mar 18 Problem Set 5 assigned, due on Mon, Apr 1, at 8:30 AM Introduction to renewal processes |
Wed, Mar 20 Elementary Renewal Theorem |
Fri, Mar 22 Blackwell's Theorem and the Key Renewal Theorem |
Week 11: | Mon, Mar 25 Examples of the Key Renewal Theorem |
Wed, Mar 27 alternating renewal processes; limiting mean excess |
Fri, Mar 29 age-dependent branching processes |
Week 12: | Mon, Apr 1 Problem Set 5 due today at 8:30 AM Problem Set 6 assigned, due on Wednesday, Apr 10, at 8:30 AM continuous time birth and death processes |
Wed, Apr 3 Kolmogorov's forward and backward equations for continuous time Markov chains |
Fri, Apr 5 Q-matrices for continuous time Markov chains |
Week 13: | Mon, Apr 8 introduction to martingales |
Wed, Apr 10 Problem Set 6 due today at 8:30 AM stopping times |
Fri, Apr 12 Midterm Exam 3 |
Week 14: | Mon, Apr 15 Problem Set 6 due today at 8:30 AM Problem Set 7 assigned, due on Friday, Apr 15, at 8:30 AM examples of martingales |
Wed, Apr 17 introduction to Brownian Motion and Gaussian processes |
Fri, Apr 19 more about Brownian Motion; expected values, covariances; Brownian bridges |
Week 15: | Mon, Apr 22 4 variations on Brownian motion |
Wed, Apr 24 Brownian motion with drift |
Fri, Apr 26 Problem Set 7 due today at 8:30 AM review for Final Exam |
Final exam date/time/location: Tuesday, April 30, 7 PM to 9 PM, ME 1009 |