A system is defined to have three states: (a) working; (b) under repair; (c) waiting for a new task. a) Suppose that if the system was working yesterday, today the probability to break is 0.1 and the probability to go to waiting is 0.2; if the system was under repair yesterday, then today the system will get repaired and become waiting state with the probability of 0.1. A broken system will never be brought directly to work in one step. If the system was waiting yesterday, there will be 0.9 probability to get into working. A waiting system will never break directly. Describe the system as a Markov process and find out the probability of the system at working status after a long time? b) Consider the same system in continuous time. Assume the state transition takes an exponential amount of time with constant rate. Suppose the failure rate (i.e. rate from working to repair) is 0.02 per day, the rate from waiting to operating is 1 per day, the rate from operating to waiting is 0.1 per day and the repair rate (i.e. rate from repair to waiting) is denoted as r. Describe the system as a Markov Chain and find out the repair rate r so that the system in steady state will be in the repair state less than 5% of the time. A system is defined to have three states: (a) working; (b) under repair; (c) waiting for a new task. a) Suppose that if the system was working yesterday, today the probability to break is 0.1 and the probability to go to waiting is 0.2; if the system was under repair yesterday, then today the system will get repaired and become waiting state with the probability of 0.1. A broken system will never be brought directly to work in one step. If the system was waiting yesterday, there will be 0.9 probability to get into working. A waiting system will never break directly. Describe the system as a Markov process and find out the probability of the system at working status after a long time? b) Consider the same system in continuous time. Assume the state transition takes an exponential amount of time with constant rate. Suppose the failure rate (i.e. rate from working to repair) is 0.02 per day, the rate from waiting to operating is 1 per day, the rate from operating to waiting is 0.1 per day and the repair rate (i.e. rate from repair to waiting) is denoted as r. Describe the system as a Markov Chain and find out the repair rate r so that the system in steady state will be in the repair state less than 5% of the time