Evaluation of Markov chains to describe movements on tiling

This study investigate movements of molecule on the biological cell via the cell walls at any given time. Specifically, we examined the movement of a particle in tiling, i.e. in hexagonal and square tiling. The specific questions we posed includes (i) whether particles moves faster in hexagonal tiling or in square tiling (ii) whether the starting point of particles affect the movement toward attainment of stationary distribution. We employed the transitional probabilities and stationary distribution to derive expected passage time to state j from state i , and the expected recurrence time to state i in both hexagonal and square tiling. We also employed aggregation of state symmetries to reduce the number of state spaces to overcome the problems (i.e. the difficulty to perform algebraic computation) associated with large transition matrix. This approach leads to formation of a new Markov chain X t that retains the original Markov chains properties, i.e. by aggregation of states with the same stochastic behavior to the process. Graphical visualization for how fast the equilibrium is attained with different values of the probability parameter p in both tilings is also provided. Due to difficulties in obtaining some analytical results, numerical simulation were performed to obtains useful results like expected passage time and recurrence time.