(a) The random walk and Markov Chain are commonly applied in building (20 marks) the simulation models. Evaluate these two techniques in business applications. (b) There are two modes, namely Mode I and Mode II of a machine. Every hour, it remains in the same mode or switches to a different mode according to the transition probability matrix M that is shown below: Both the columns and rows in the matrix are labelled in the same order of 'Mode I' and 'Mode II' respectively. (i) Convert M into its associated transition diagram and interpret the (5 marks) stochastic matrix M. (ii) If the system is in Mode I at 8:00am, what is the probability that it (2 marks) will be in Mode II at 10:00 am on the same day? (iii) After 100 hours (or in the long run), evaluate the probability that (3 marks) the machine remains in Mode I. (a) The random walk and Markov Chain are commonly applied in building (20 marks) the simulation models. Evaluate these two techniques in business applications. (b) There are two modes, namely Mode I and Mode II of a machine. Every hour, it remains in the same mode or switches to a different mode according to the transition probability matrix M that is shown below: Both the columns and rows in the matrix are labelled in the same order of 'Mode I' and 'Mode II' respectively. (i) Convert M into its associated transition diagram and interpret the (5 marks) stochastic matrix M. (ii) If the system is in Mode I at 8:00am, what is the probability that it (2 marks) will be in Mode II at 10:00 am on the same day? (iii) After 100 hours (or in the long run), evaluate the probability that (3 marks) the machine remains in Mode