1. (30 points) Markov chains: Consider the first three letters of your FIRST name. Each letter is given a numerical integer value starting with A = 1 to Z = 26. Write the numeric values for the first three letters of your first name: Consider the following homogeneous Markov chain in which state 1 corresponds to the first letter of your first name, state 2 corresponds to the second letter in your first name, and state 3 corresponds to the third letter of your first name. 1 - Poo 1 - PI Poo 2 2 1 1 - P11 1 - P22 2 1 - Poo P 22 2 Given the integer value k of a letter, an average visit time of T= 10k units of time is associated with each letter. The transition probability of remaining in the same state corresponding to a letter K is PKK = 1/(k+ 2), where k is the integer corresponding to letter K. For example, for letter A we have TA= 10 and PMA = 1/3. While the system is in a given state corresponding to letter K, the system is successful with probability Pk = 1/k, where k is the is the integer corresponding to letter K. The system throughput is defined to be the ratio of the average time the system spends being successful while in State 1 divided by the average cycle time needed to go from State 1 and back to that state. (a) Compute the state probabilities associated with the Markov chain above (b) Compute the throughput of the system