# Deterministic model of throughput in a (single tread) production line

I am looking for a simple and canonical model of throughput (number of batches per unit of time) of a sequential and deterministic production line. For this simple model, I am searching for theorems relating throughput to the throughput of the slowest step and lead time (time to transform inputs into the final product).

The production line (P) is sequential and single threaded, comprises S steps. Each step, s, comprises a throughput per worker and a number of workers. The throughput per worker in each step is deterministic.

Is there a canonical model for this?

I found references for Jackson networks in this Operations Research textbook, such as this, but these seem much more complex and flexible than I am looking for (every node can connect to any other node, the productivity of each step is stochastic). I suppose this is related to work on queues, but I was neve exposed to such theories.

This is for a research project. Any references will be appreciated.

• It seems rather simple, the throughput of the entire line will be the throughput of its slowest step. Every other step must slow production to match the slowest - those before to prevent the built-up of inventory, and those after because they don't receive input any faster. Mar 30, 2019 at 0:58
• @PaulSinclair, I agree. But is there a proof for this? Or how to go about proving this? Mar 31, 2019 at 7:43
• I gave the reasoning for the proof in the comment above. Anything more is just window dressing. If the steps before run at full capacity, their output will just pile up, as it cannot be cleared any faster by the slowest step, and the steps after simply experience downtime as they wait for the slowest step to produce another output for them to work on. Mar 31, 2019 at 17:02
• @PaulSinclair, would you like to write your comments above as an answer, so I can mark this question as resolved? Apr 8, 2019 at 2:21
• If you are satisfied I answered your question, I will. I was making comments because I wasn't sure if this was what you needed. Apr 8, 2019 at 23:52