A bike-share company in Ithaca has three docking stations where customers can pick up or deposit a bike: station A is located downtown near the Commons, station B is in the heart of Collegetown, and station C is outside the Statler hotel. The daily rate of migration for bikes moving between the three stations is depicted in Figure 1 (e.g. the green arrow indicates migration from Station A to Station B). 

(a) Model the daily migration of bikes between the three stations as a Markov chain: write down the transition matrix P (which maps the initial distribution of bikes among the docking stations to the distribution of bikes among the stations one day later) and find the associated steady-state vector q. 

(b) Suppose that the distribution of bikes between stations A, B, and C is initially provided by the entries of the vector xo. After two days, the distribution of bikes is X2 = (0.6 0.2 0.27. What was the initial distribution xo of bikes among the three stations? stations? Compare the entries of the vectors X2 and xo with the entries of the steady-state vector q.

1/10 Statler Hotel The Commons 1/5 2/3 1/2 1/31 1/10 Collegetown