Preparing for the Stochastic Modelling midterm at VU Amsterdam? It's no secret—it's tough! But, breathe easy. This summary has you covered for the first midterm. Don't drown in the sea of notes, streamline your revision with this guide.
, 1 Introduction to Markov Chains
1.1 Why Markov Chains?
• Markov Chains are tractable: analysis is feasible with many standard
procedures.
• They can predict time-average behavior, such as average costs per day or
the fraction of lost customers.
• Many real-world scenarios can be modeled using Markov Chains.
1.2 Characteristics of Discrete-time Markov Chains
1.2.1 Discrete-time
A DTMC is a sequence of random variables {X0 , X1 , X2 , . . .} where the index
n represents discrete time units.
1.2.2 Countable State Space
All possible values of Xn for n = 0, 1, 2, . . . are in a countable set S, termed the
state space.
1.2.3 Markov Chains
A sequence of random variables is a Markov Chain if it possesses the Markov
property: the next state depends only on the present state, not on any prior
history.
1.2.4 Time-homogeneity
A DTMC is time-homogeneous if its transition probabilities remain constant
over time.
1.3 Transition Diagram and Matrix
1.3.1 Transition Diagram
The transition diagram is a graphical representation of a DTMC. It provides a
visual overview of the system’s dynamics, making it easier to understand and
analyze.
• States: Each state in the state space S is represented as a node or circle
in the diagram.
• Transitions: Arrows between states represent possible transitions. The
direction of the arrow indicates the direction of the transition.
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