Question

A small computer lab has 2 terminals. The number of students working in this lab is recorded at the end of every hour. A computer assistant notices the following pattern: If there are 0 or 1 students in a lab, then the number of students in 1 hour has a 50-50% chance to increase by 1 or remain unchanged.If there are 2 students in a lab, then the number of students in 1 hour has a 50-50% chance to decrease by 1 or remain unchanged. (a) Write the transition probability matrix for this Markov chain.

Answer 1

The transition probability matrix for the Markov chain with three states in the computer lab setting, based on the provided information, is a 3x3 matrix with rows indicating the probabilities of staying in the current state or moving to a different state.

Step-by-step explanation:

To answer the student's question about creating a transition probability matrix for a Markov chain in a computer lab setting with 0, 1, or 2 users, we consider three states for this system: State 0 (no students), State 1 (one student), and State 2 (two students). The transition probabilities are based on the given scenario:

• From State 0, there is a 50% chance to go to State 1 and a 50% chance to stay at State 0.
• From State 1, there is also a 50% chance to go to State 2 and a 50% chance to remain in State 1.
• From State 2, there is a 50% chance to go to State 1 and a 50% chance to remain in State 2.

The transition probability matrix P is therefore:

[ P = begin{bmatrix} 0.5 & 0.5 & 0 \ 0 & 0.5 & 0.5 \ 0 & 0.5 & 0.5 \end{bmatrix} \]

Each row of the matrix corresponds to the probabilities of transitioning from a particular state to another state (including itself) after one hour.