1. You are about to present a new proposal for funding to the company shareholders. Assume that whether the proposal is approved depends on whether they approved the previous three proposals they received from your team. We assume that proposals are either approved or not approved.

(a) Show how this system may be analyzed by using Markov chain. How many states are needed? List your states.

(b) Suppose that if all of the last three proposals were approved, then this new proposal will be approved with probability 0.85. If none of the last three proposals were approved, then this new proposal will be approved with probability 0.25. In any other case whether or not the proposal was approved will, with probability 0.7, be the same as whether or not the last proposal was approved. Determine Pfor this Markov chain.

2. Professor Lavieri's server may either be up or down. If the server is up and working at the beginning of a day, it continues to work until the next day with probability 0.9, and it fails with probability 0.1. The IOE technical support team is very efficient; if the server is down, repairs will begin the next morning. It takes exactly 3 days for the repairs. (i.e. if the server is down on Monday, it will be up by Friday morning)

(a) Model this as a Markov Chain. Make sure to clearly define your states and give the transition probability matrix.

(b) For the Markov Chain above: draw the state transition diagram, specify the classes, determine whether they are transient or recurrent, determine the period of each class and write whether or not the chain is irreducible.

(c) It is Friday morning and the server has been down since Thursday. Professor Lavieri's student must decide whether to run her code on Wednesday or Thursday next week. On which day would she have the highest probability of having a server that is up. Indicate the probability the server will be up on each of these two days.