Math 180C Course Topics

Below is a list of the topics that will be covered in Math 180C, together with references to the relevant sections in the textbooks by Durrett, Pinsky and Karlin, and Dobrow. We will spend approximately the first three weeks of the quarter on continuous-time Markov chains (topics 1-6), the next two weeks on renewal theory (topics 7-10), the following two weeks on queueing theory (topics 11-14), and the last three weeks on Brownian motion (topics 15-21).

Topic Durrett   Pinsky-Karlin   Dobrow
1 The exponential distribution   2.1 1.4.2, 1.5.2 N/A
2 Birth and death processes N/A 6.1-6.3 N/A
3 General continuous-time Markov chains 4.1 6.6 7.1-7.2
4 First-jump conditioning and hitting probabilities 4.4 6.5 N/A
5 Kolmogorov's forward and backward equations 4.2 6.3.3, 6.6 7.3
6 Stationary distributions and long-run behavior 4.3 6.4 7.4
7 Introduction to renewal processes 3.1 7.1-7.3 N/A
8 Asymptotic results for renewal processes 3.1 7.4 N/A
9 Renewal reward processes 3.1 7.5 N/A
10 Age and residual life 3.3 N/A N/A
11 The M/M/1 queue 4.5.1 9.1-9.2 7.6
12 Other queues with exponential service times 4.5.2 9.2 7.6
13 Queues with general service times 3.2 9.3 N/A
14 Queues in tandem 4.6 9.5 N/A
15 Definition and basic properties of Brownian motion   N/A 8.1 8.1-8.2
16 Brownian motion as a Gaussian process N/A 8.1 8.3
17 The reflection principle N/A 8.2 8.4
18 Zeros of Brownian motion N/A 8.2 8.4
19 Brownian motion with drift N/A 8.4 8.5
20 Geometric Brownian motion N/A 8.4 8.5
21 Other processes derived from Brownian motion N/A 8.3 8.5