Question

a student society at a large university campus decides to create a pool of bicycles that can be used by the members of the society. bicycles can be borrowed or returned at the residence, the library, or the athletic center. the first day, 200 marked bicycles are left at each location. at the end of the day, at the residence, there are 160 bicycles that started at the residence, 40 that started at the library, and 60 that started at the athletic center. at the library, there are 20 that started at the residence, 140 that started at the library, and 40 that started at the athletic center. at the athletic center, there are 20 that started at the residence, 20 that started at the library, and 100 that started at the athletic center. if this pattern is repeated every day, what is the steady-state distribution of bicycles?

Answer 1

To find the steady-state distribution of bicycles in the system created by the student society, we would typically calculate the eigenvector corresponding to the eigenvalue 1 of the transition matrix derived from the daily bicycle movements.

Step-by-step explanation:

The student society's pattern forms a square matrix that describes the behavior of the bicycle movements, which can be interpreted as a transition matrix in a Markov Chain. To find the steady-state distribution of bicycles, we need to solve for the eigenvector corresponding to the eigenvalue 1 of the transition matrix.

This involves setting up the equations based on the movements, converting these equations to matrix form, and applying linear algebra techniques to solve for the steady-state vector.

Initially, we have 200 bicycles at each location, and the transition probabilities can be determined from the movements recorded at the end of the day: 80% (0.8) of bicycles at the residence stay there, while 20% (0.1) from the library and 30% (0.3) from the athletic center move there, and so on for the other locations.