I am working on a problem that combines Negative Exponential distributions and their impact on the throughput of a system. The output I want to create is a range of Jobs Per Hour given as a probability distribution (and/or 95% confidence interval). The system to be modeled is simple, with two machines connected by a single buffer. Inputs (variables) include cycle time, the Mean Time Before Failure (MTBF) and Mean Time to Repair (MTTR) of each machine, as well as Buffer Size (can be zero) and Time Delay (per part, modeling an index time per position) in the Buffer between the machines. The Negative Exponential's Probability Distribution Function for the MTBF and MTTR values are F(x) = (1/MTBF) x e^((-1/MTBF) x Time). Time is given in minutes. Example, (40/60) minute cycle time (40 seconds), Machine 1 MTBF = 200 minutes / MTTR = 5 minutes, Buffer Size between 0 and 10 units (with a 10 sec delay per part), Machine 2 MTBF = 150 minutes / MTTR = 7 minutes. In this system with no downtime, the throughput is 3600 seconds / 40 second cycle time = 90 Jobs Per Hour Gross. Using the failure inputs for the two machines, the throughput is reduced. By adding a buffer, the throughput recovers. Any insight on a robust approach to model these variables and creating a probability distribution for a range of throughput values would be greatly appreciated.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an alternative browser.
You should upgrade or use an alternative browser.
Reliability in Production Systems
- Thread starter jmpehrson
- Start date
I am working on a problem that combines Negative Exponential distributions and their impact on the throughput of a system. The output I want to create is a range of Jobs Per Hour given as a probability distribution (and/or 95% confidence interval). The system to be modeled is simple, with two machines connected by a single buffer. Inputs (variables) include cycle time, the Mean Time Before Failure (MTBF) and Mean Time to Repair (MTTR) of each machine, as well as Buffer Size (can be zero) and Time Delay (per part, modeling an index time per position) in the Buffer between the machines. The Negative Exponential's Probability Distribution Function for the MTBF and MTTR values are F(x) = (1/MTBF) x e^((-1/MTBF) x Time). Time is given in minutes. Example, (40/60) minute cycle time (40 seconds), Machine 1 MTBF = 200 minutes / MTTR = 5 minutes, Buffer Size between 0 and 10 units (with a 10 sec delay per part), Machine 2 MTBF = 150 minutes / MTTR = 7 minutes. In this system with no downtime, the throughput is 3600 seconds / 40 second cycle time = 90 Jobs Per Hour Gross. Using the failure inputs for the two machines, the throughput is reduced. By adding a buffer, the throughput recovers. Any insight on a robust approach to model these variables and creating a probability distribution for a range of throughput values would be greatly appreciated.
I'm not quite sure about the process so let me talk my way through a cycle and you correct it (or say it is ok): 'Raw material' for a job enters machine 1 and is partially processed in 30 seconds and passed to machine 2 where the process is completed in an additional 10 seconds. Passage of a particular partially processed job from machine 1 to machine 2 is delayed by (10/60)n minutes where n is the buffer size, 0\(\displaystyle \le\)n\(\displaystyle \le\)10.
As a general comment, I would investigate the possibility of a 'repair/maintenance' on both machines when either fails so that you wouldn't have, for example, a failure at a hundred and fifty minutes on machine 1, a repair, and then a failure on machine 2 forty five minutes later. That would depend, in part, on the cost of the repairs/maintenance and the value of the processed jobs of course.
Last edited:
Clarification on Cycle Time
To clarify the cycle time of 40 seconds, every 40 seconds Machine 1 receives a part, does work, and releases the part to the buffer. It takes 10 seconds to move through the buffer. Simultaneously, in a 40 second cycle time, Machine 2 takes a part from the buffer, does work, and releases the part. The buffer serves to minimize the impact to throughput if either machine is down. If Machine 1 is down, Machine 2 can continue to work until the buffer is emptied or Machine 1 is repaired, and if Machine 2 is down, Machine 1 can continue to work until the buffer is filled or Machine 2 is repaired.
I'm not quite sure about the process so let me talk my way through a cycle and you correct it (or say it is ok): 'Raw material' for a job enters machine 1 and is partially processed in 30 seconds and passed to machine 2 where the process is completed in an additional 10 seconds. Passage of a particular partially processed job from machine 1 to machine 2 is delayed by (10/60)n minutes where n is the buffer size, 0\(\displaystyle \le\)n\(\displaystyle \le\)10.
As a general comment, I would investigate the possibility of a 'repair/maintenance' on both machines when either fails so that you wouldn't have, for example, a failure at a hundred and fifty minutes on machine 1, a repair, and then a failure on machine 2 forty five minutes later. That would depend, in part, on the cost of the repairs/maintenance and the value of the processed jobs of course.
To clarify the cycle time of 40 seconds, every 40 seconds Machine 1 receives a part, does work, and releases the part to the buffer. It takes 10 seconds to move through the buffer. Simultaneously, in a 40 second cycle time, Machine 2 takes a part from the buffer, does work, and releases the part. The buffer serves to minimize the impact to throughput if either machine is down. If Machine 1 is down, Machine 2 can continue to work until the buffer is emptied or Machine 1 is repaired, and if Machine 2 is down, Machine 1 can continue to work until the buffer is filled or Machine 2 is repaired.
Last edited by a moderator: