Markov Processes;

A machine shop operates two identical machines, which are supervised by one oper- ator. Each machine requires the operator's attention at random points in time. The probability that the machine requires service in a period of 5 minutes is p = 0.4. The operator is able to service the machine in 5 minutes. Let us approximate the situation by assuming that a machine requires service always at the beginning of a 5 minute period. (5 ) a) Construct a transition diagram for this problem and find the transition matrix associated with it. (8 ) b) If both machines are operating at 8AM, find the marginal probabilities after 5 minutes and 10 minutes. (8) c) Find the steady state probabilities for this process.A video recorder manufacturer is so certain of its quality control, that it is offering a complete replacement warranty if the set fails within two years. Based upon complied data, the company has noted that only 1 percent of its recorders fail during the first year, whereas 5 percent of the recorders that survive the first year will fail during the second year. The warranty does not cover replaced recorders. (13) a) Formulate the the evolution of the status of a recorder as a DTMC whose states include two absorption states that involve needing to honor the warranty or having the recorder survive the warranty period. Then construct the transition matrix. (12 ) b) Find the probability that the manufacturer will have to honor the warranty.Assume that the probability of rain tomorrow is a if it is raining today, an that the probability of its being clear tomorrow is 3, if it is clear today. (a) Determine the one-step transition matrix of the DTMC. (b) Find the two-step transition matrix. (c) Find the steady-state probabilities