REPLY Age of material in a stock (SD6408)

[SDMAIL Jack Homer]

Posted by  "Jack Homer" <jhomer at comcast.net>

Tom Fiddaman writes:
"If you play with this, you'll find the expected steady state response. Replacing
stuff_in with a pulse, a ramp, and exponential growth is interesting. It becomes
apparent that for particular inputs, it is possible to create a rule of thumb,
but I'll leave that for someone else to tackle."

I addressed this topic in an appendix on co-flows in my 1983 doctoral thesis, 
looking at step changes and exponential growth in particular.  If the fractional 
growth rate in the stock is 'G' and the average time-to-outflow is 'To', then 
with straightforward algebra one may show that the steady-state age in the stock 
'As' is expressed simply as:

As = 1 / (G + (1/To))

In the case of zero stock growth (G=0), this reduces, as one would expect, to 
As = To.  Positive growth in the stock leads to As<To, while negative growth in 
the stock leads to As>To.

This derivation assumes a first-order (single stock) process, with perfect mixing 
as Tom notes.

Jack Homer
Posted by  "Jack Homer" <jhomer at comcast.net>
posting date  Tue, 17 Apr 2007 09:02:42 -0400