Question

Please show your work! Fully characterize all Markov chain models. Thanks!!

2. As an intern in a manufacturing firm, you are assigned to model the hourly operating status of a particular station on an assembly line as a Markov chain. You determine that the station's status can be described using the following state descriptions: busy, idle, malfunctioning, or undergoing maintenance. You talk to the operator who tells you several things. First, the station never malfunctions when it's idle, only when it's busy. When it's busy, it will malfunction 5% of the time, become idle 30% of the time, and continue to be busy otherwise. When the station is idle, it will become busy 75% of the time and remain idle otherwise. If the station has malfunctioned, the maintenance team will arrive within the hour 95% of the time. If the machine is undergoing maintenance, it will be repaired and become busy again therwise, it continues to undergo maintenance. Define the Markov chain model for the status of the manufacturing station. Include the transition matrix and transition diagram.

Answer 1

The states are: Busy, Idle, Malfunctioning, or Undergoing Maintenance. Consider the states as, Busy: 1 Idle: 2 Malfunctioning: 3 Undergoing Maintenance: 4 First, the station never malfunctions when it's idle, only when it's busy. P32 = P31 = 0. When its busy, it will malfunction 5% of the time, become idle 30% of the time, and continue to be busy otherwise. Pag = 0.05, P12 = 0.30. P = 0.65. When the station is idle, it will become busy 75% of the time and remain idle otherwise. P2 = 0.75, P22 = 0.25. If the station has malfunctioned, the maintenance team will arrive within the hour 95% of the time. P34 = 0.95. Since, Px2 = Px1 = 0 and P34 = 0.95 then pag = 0.05. If the machine is undergoing maintenance, it will be repaired and become busy again within the hour 80% of the time. Otherwise, it continuous to undergo maintenance. P41 = 0.80, P-4 = 0.20.

The transition matrix is given as follows: 1 2 3 4 1[0.65 0.30 0.05 01 2 0.75 0.250 0 3 0 0 0.05 0.95 4 0.80 0 0 0.20 The transition diagram is given as follows: 0.65 0.05 0.30 0.95 0.75 0.80