Six positions, numbered 0 to 5, are arranged on a circle. A particle moves around the...

Transcribed Image Text:

Six positions, numbered 0 to 5, are arranged on a circle. A particle moves around the circle as follows. With probability 1/3 the particle moves counter-clockwise one position, with probability 1/3, the particle moves clockwise one position, and with probability 1/3 the particle remains at its current position. If the particle starts off at position 0, what is the distribution of the position after 5 transitions? What about after 50 transitions? If the particle starts off at a random position, uniformly distributed from 0 to 5, what is the distribution of the position after 1 transition? How do your answers change if the particle moves counter-clockwise two positions with probability 1/7, counter-clockwise one positions with proba- bility 2/7, and with probability 4/7 the particle moves 1 position clockwise? Identify the communications classes and state for each class whether it is transient or recurrent, and whether it is periodic (if so, give the period). Six positions, numbered 0 to 5, are arranged on a circle. A particle moves around the circle as follows. With probability 1/3 the particle moves counter-clockwise one position, with probability 1/3, the particle moves clockwise one position, and with probability 1/3 the particle remains at its current position. If the particle starts off at position 0, what is the distribution of the position after 5 transitions? What about after 50 transitions? If the particle starts off at a random position, uniformly distributed from 0 to 5, what is the distribution of the position after 1 transition? How do your answers change if the particle moves counter-clockwise two positions with probability 1/7, counter-clockwise one positions with proba- bility 2/7, and with probability 4/7 the particle moves 1 position clockwise? Identify the communications classes and state for each class whether it is transient or recurrent, and whether it is periodic (if so, give the period).