The following problem came up in my work.(adsbygoogle = window.adsbygoogle || []).push({});

You have a tube, open at the top. Raindrops fall into the mouth of the tube at a mean rate i per second, 0 <= i < 1, in a Poisson process. There's a hole in the bottom of the tube. When there's water in the tube, it flows out at a constant rate of 1 drop/sec. (Ignore that this is not physically plausible. You can imagine that there's a peristaltic pump hooked up if it makes you feel better.) What is the mean amount of water in the tube at steady-state?

I came up with a complicated solution to this (which I will describe, if anyone wishes) in the form of a sum. But when I evaluated my solution numerically, I found that, within round-off error, computed values equaled the simple result <v> = i/(2(1 - i)). This is also plausible and has correct limiting behavior at 0 and 1. Since then I've been trying hard without success to come up with a proof or even fairly good rationale for that formula.

Thanks in advance for any ideas.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Poisson inflow, constant outflow question

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**