What math do I need to understand Markov Process and queueing theory? The objective of the course is to learn how to model and compare elementary queues like the M/M/1 queue.  
I'd like to know what are the minimal math knowledge I need to catch up with to understand all this?
 A: You should take a look at the following book for basic probability theory, or a similar book,
Elementary Probability Theory, by Kai Lai Chung, Farid Aitsahlia 
This book begins from the basics, sets, probability, counting, random variables, conditioning, independence, pretty standard stuff in probability theory. 
If your calculus is rusty, you might want to take a look at something like 
The Calculus Lifesaver: All the Tools You Need to Excel at Calculus, by Adrian Banner
If you do not know any calculus, depending on the course, you might not get very far.
I think these are the absolute minimum requirements needed for a course like the one you are in.
A: Queueing theory is not often taught as a first year course, so you need linear algebra, calculus and probability theory at the level of a second year university student.
The book you reference states

The book assmes some background in elementary probability and some background in either electrical engineering or computer science, but aside from this, the material is self contained.

Here are some examples of prerequisites of courses that sound similar to yours at other institutions.
Stanford MS&E 121

Students should have a working knowledge of calculus at the level of Math 51, including differentiation and integration of functions of a single variable. Students are also expected to have had previous exposure to basic probability (at the level of either MS&E 120 or Stat 116). Exposure to matrix notation and basic linear algebra is also important. 

Pittsburgh IE3085

IE 2072: Probability
  This course develops important probability concepts for advanced undergraduate and first-year graduate students. Topics include measure-theoretic preliminaries, counting methods, conditional probability and expectation, Bayes' theorem, random variables and distributions, discrete and continuous transforms, functions of random variables, limit theorems, including strong and weak laws of large numbers, central limit theorem(s), and stochastic ordering concepts. Prerequisite(s): Multivariable calculus and one course in probability.
IE 2084: Stochastic Processes 
  The primary objective of this course is to provide graduate students with a strong foundation in the theory and applications of stochastic processes. The course emphasizes processes most relevant to industrial engineering and operations research. Specific topics include discrete- and continuous-time Markov chains, the Poisson process and its variants, renewal and Markov-renewal theory, regenerative and Markov-regenerative processes, Brownian motion and martingales. Applications in queueing, reliability, inventory and finance will be discussed. 

Colorado ECEN 5612

An undergraduate course in probability theory.

