REPLY Age of material in a stock (SD6411)

[SDMAIL Tom Fiddaman]

Posted by  Tom Fiddaman <tom at ventanasystems.com>

It's worth noting a few consequences of Jack's analysis:

- The fractional growth rate G is only constant for exponential growth 
or decay.  Growth in the stock leading to As<To  means the average age 
of material is younger than you'd expect from Little's Law because a 
disproportionate amount of material arrived recently. Little's law 
remains a good approximation as long as G is small compared to 1/To - 
that is, as long as growth is slow with respect to the lifetime of 
the stock.

- It would seem that As becomes negative when growth is negative, 
with -G > 1/To. Fortunately that can't happen; the inflow to the stock 
could quickly approach zero at some large negative growth rate, but the 
stock can decay at most as fast as the time constant of the outflow, To. 
In that case, As = 1/0, and if you run the model with zero input, you will 
find that the average residence time does grow without bound. This makes 
sense - if you look in a box that's had zero inflow for a long time, 
anything in there has clearly been there for a long time.

- If the inflow is an increasing ramp, the stock comes into a different 
steady state. In this case the rate of increase in the stock is constant 
at S*To (where S is the slope of the input ramp), while the stock grows 
without bound. That means G (net increase rate divided by the stock) falls 
to zero, so after an initial transient,  Little's Law holds as usual, 
As = To.

Tom
Posted by  Tom Fiddaman <tom at ventanasystems.com>
posting date  Wed, 18 Apr 2007 08:59:25 -0600