ISE-230 Fall 2015 - Prof. B. Defourny HW 7 - Continuous-Time Markov Chains Consider two identical 3D printers, labeled A and B. The printers are used 24/7. When they are working, each printer produces 100 objects per day. Due to the intense utilization, each printer has a constant rate of failure of 2 failures per week. Moreover, in 25% of the cases, the failure is due to a common cause - which also implies that the 2 printers fail at the same time. There is a worker on duty who repairs the machines when they fail. The worker can only repair the machines one at a time. The repair time for one printer follows an exponential distribution. In average the repair time for one printer is half a day. 1. Show that this problem can be represented as a continuous-time Markov Chain using 4 distinct states: a) Explain the meaning of each state. b) Compute the transition rates among the states. You need to be extra careful regarding the units, the impact of the common cause, and the deployment of the worker. For this reason, you are asked to justify each rate with a narrative to relate to the problem description. c) Draw the rate diagram for the continuous-time Markov Chain. 2. Compute the steady-state probabilities of each state: a) Write the balance equations and the normalization equation b) Select an independent system of 4 equations in matrix form, and then solve it to find the probabilities (Calculator or Matlab). 3. a) Determine the fraction of the time both machines are operating. b) Determine the fraction of the time only one machine is operating. c) Determine the total expected number of objects that are 3-D printed per week. ISE-230 Fall 2015 - Prof. B. Defourny HW 7 - Continuous-Time Markov Chains Consider two identical 3D printers, labeled A and B. The printers are used 24/7. When they are working, each printer produces 100 objects per day. Due to the intense utilization, each printer has a constant rate of failure of 2 failures per week. Moreover, in 25% of the cases, the failure is due to a common cause - which also implies that the 2 printers fail at the same time. There is a worker on duty who repairs the machines when they fail. The worker can only repair the machines one at a time. The repair time for one printer follows an exponential distribution. In average the repair time for one printer is half a day. 1. Show that this problem can be represented as a continuous-time Markov Chain using 4 distinct states: a) Explain the meaning of each state. b) Compute the transition rates among the states. You need to be extra careful regarding the units, the impact of the common cause, and the deployment of the worker. For this reason, you are asked to justify each rate with a narrative to relate to the problem description. c) Draw the rate diagram for the continuous-time Markov Chain. 2. Compute the steady-state probabilities of each state: a) Write the balance equations and the normalization equation b) Select an independent system of 4 equations in matrix form, and then solve it to find the probabilities (Calculator or Matlab). 3. a) Determine the fraction of the time both machines are operating. b) Determine the fraction of the time only one machine is operating. c) Determine the total expected number of objects that are 3-D printed per week. ISE-230 Fall 2015 - Prof. B. Defourny HW 7 - Continuous-Time Markov Chains Consider two identical 3D printers, labeled A and B. The printers are used 24/7. When they are working, each printer produces 100 objects per day. Due to the intense utilization, each printer has a constant rate of failure of 2 failures per week. Moreover, in 25% of the cases, the failure is due to a common cause - which also implies that the 2 printers fail at the same time. There is a worker on duty who repairs the machines when they fail. The worker can only repair the machines one at a time. The repair time for one printer follows an exponential distribution. In average the repair time for one printer is half a day. 1. Show that this problem can be represented as a continuous-time Markov Chain using 4 distinct states: a) Explain the meaning of each state. b) Compute the transition rates among the states. You need to be extra careful regarding the units, the impact of the common cause, and the deployment of the worker. For this reason, you are asked to justify each rate with a narrative to relate to the problem description. c) Draw the rate diagram for the continuous-time Markov Chain. 2. Compute the steady-state probabilities of each state: a) Write the balance equations and the normalization equation b) Select an independent system of 4 equations in matrix form, and then solve it to find the probabilities (Calculator or Matlab). 3. a) Determine the fraction of the time both machines are operating. b) Determine the fraction of the time only one machine is operating. c) Determine the total expected number of objects that are 3-D printed per week. ISE-230 Fall 2015 - Prof. B. Defourny HW 7 - Continuous-Time Markov Chains Consider two identical 3D printers, labeled A and B. The printers are used 24/7. When they are working, each printer produces 100 objects per day. Due to the intense utilization, each printer has a constant rate of failure of 2 failures per week. Moreover, in 25% of the cases, the failure is due to a common cause - which also implies that the 2 printers fail at the same time. There is a worker on duty who repairs the machines when they fail. The worker can only repair the machines one at a time. The repair time for one printer follows an exponential distribution. In average the repair time for one printer is half a day. 1. Show that this problem can be represented as a continuous-time Markov Chain using 4 distinct states: a) Explain the meaning of each state. b) Compute the transition rates among the states. You need to be extra careful regarding the units, the impact of the common cause, and the deployment of the worker. For this reason, you are asked to justify each rate with a narrative to relate to the problem description. c) Draw the rate diagram for the continuous-time Markov Chain. 2. Compute the steady-state probabilities of each state: a) Write the balance equations and the normalization equation b) Select an independent system of 4 equations in matrix form, and then solve it to find the probabilities (Calculator or Matlab). 3. a) Determine the fraction of the time both machines are operating. b) Determine the fraction of the time only one machine is operating. c) Determine the total expected number of objects that are 3-D printed per week