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> You might think this system has a capacity of 1/t (100% utilization) but if you actually try that you'd find that the length of the queue becomes infinite I'

> You might think this system has a capacity of 1/t (100% utilization) but if you actually try that you'd find that the length of the queue becomes infinite I'm not following. 1/t = Facilities's rate of work t = Time to process a unit If you had an infinite queue of objects (with t time required) being processed at 1/t (rate), your workload/utilization is 100%, right? You seem to be saying otherwise. Also, what do you mean by "the length of the queue becomes infinite"? Is there some property of a random Poisson process that has a growth bias? Can I think of it like flipping heads/tails where you might go on a run, however, for a large number of outcomes, it should converge on a number of outcomes in the middle?

vaidhy 4y ago
What is being implied is that you start with a finite queue size and there is a random poisson process at the rate of t adding items to the queue. Intuitively, you would expect the queue size to be finite and stable. When you simulate this, more often than not, the queue tends to grow to infinity. In practice, it means that your team will burn out much faster.
- caminante 4y ago
  Thank you. I figured this is one of those things where a simulation really proves the intuition.
hansvm 4y ago
> length of the queue becomes infinite If you simulated the queue forever and took a running average (for any finite length of time that's just an integral, and you take the limit as t->infinity) of the queue length, the result would "be infinite" -- for ever possible real number the result of such a limit of it existed would be greater. > is poisson special It's easiest to work out with elementary mathematical tools with a poisson process, but otherwise no. There's a decent overview [0] that derives the expected length, and showing that it grows without bound up to 100% is suggestive of the result (combine that with the fact that the mean length is strictly monotonic increasing where it exists and is infinite at >100% utilization, and that shores up the mathematical loose ends). For the poisson process in particular, the "balance principle" in that link I think will serve to provide the right intuition about the problem. > should converge on the number of outcomes in the middle That's the problem. We have infinite head and tail flips, but when the length is 0 only half of those matter. Adding up those one-sided differences still wouldn't have to be infinite per se, but 100% utilization is the threshold at which it gets pushed over the edge (large differences are "rare" to get to in the first place but you'll spend time inverse to that probability in their vicinity). [0] https://www.win.tue.nl/~iadan/que/h4.pdf https://www.win.tue.nl/~iadan/que/h4.pdf
- caminante 4y ago
  > There's a decent overview [0] that derives the expected length, and showing that it grows without bound up to 100% is suggestive of the result (combine that with the fact that the mean length is strictly monotonic increasing where it exists and is infinite at >100% utilization, and that shores up the mathematical loose ends). Ah! I can process this and it makes crude sense to me. Thank you!

vaidhy 4y ago

What is being implied is that you start with a finite queue size and there is a random poisson process at the rate of t adding items to the queue. Intuitively, you would expect the queue size to be finite and stable. When you simulate this, more often than not, the queue tends to grow to infinity. In practice, it means that your team will burn out much faster.

caminante 4y ago
Thank you. I figured this is one of those things where a simulation really proves the intuition.

hansvm 4y ago

> length of the queue becomes infinite If you simulated the queue forever and took a running average (for any finite length of time that's just an integral, and you take the limit as t->infinity) of the queue length, the result would "be infinite" -- for ever possible real number the result of such a limit of it existed would be greater. > is poisson special It's easiest to work out with elementary mathematical tools with a poisson process, but otherwise no. There's a decent overview [0] that derives the expected length, and showing that it grows without bound up to 100% is suggestive of the result (combine that with the fact that the mean length is strictly monotonic increasing where it exists and is infinite at >100% utilization, and that shores up the mathematical loose ends). For the poisson process in particular, the "balance principle" in that link I think will serve to provide the right intuition about the problem. > should converge on the number of outcomes in the middle That's the problem. We have infinite head and tail flips, but when the length is 0 only half of those matter. Adding up those one-sided differences still wouldn't have to be infinite per se, but 100% utilization is the threshold at which it gets pushed over the edge (large differences are "rare" to get to in the first place but you'll spend time inverse to that probability in their vicinity). [0] https://www.win.tue.nl/~iadan/que/h4.pdf https://www.win.tue.nl/~iadan/que/h4.pdf

caminante 4y ago
> There's a decent overview [0] that derives the expected length, and showing that it grows without bound up to 100% is suggestive of the result (combine that with the fact that the mean length is strictly monotonic increasing where it exists and is infinite at >100% utilization, and that shores up the mathematical loose ends). Ah! I can process this and it makes crude sense to me. Thank you!