It has become a problem that nowadays control system is getting complex. To describe, calculate, and optimize production, 3 classical mathematical models are introduced.

1.Discrete event simulation: very accurate but hard to calculate complex models. Probability distribution calculation needed. Distribution functions calculability limited by sample accessibility.

2.Queueing networks: fairly accurate (using average data) but hard to calculate complex models. Only works with steady models (time infinite).

3.Fluid Models (ODE): [latex]\frac{dq(t)}{dt}=\lambda(t)-\sigma(t)[/latex]
It describes queue change rate is influx rate minus outflux rate.
It is easy to calculate. But it uses average data (no possibility included), which may lead to trivial results. And [latex]\lambda(t)[/latex] and [latex]\sigma(t)[/latex] are not easy to find.

So the author introduced a new model: Continuum Model
It uses a totally new view. The model doesn’t put too much attention on the physical position. A new attribute is introduced for each item, stage x, from 0% to 100%. Then the completion of a item can be involved into mathematical calculation.
The core equation is:
[latex]\frac{\partial \rho (x,t)}{\partial t}+\frac{\partial}{\partial t}(v(\rho)\rho (x,t))=0[/latex]
The function [latex]\rho (x,t)[/latex]is the density, when the stage is x and time is t.

It took me some time to understand the equation. I think it is just like the flood in Yangtze River now. The height of water is the function [latex]\rho (x,t)[/latex], over position x and time t.
The equation is setup up like this because the change of [latex]\rho (x,t)[/latex] over time(x fixed) is just like the change of the height of water. It is caused by the different velocity of water over position x. The different velocity is describe as the [latex]-\frac{\partial}{\partial t}(v(\rho)\rho (x,t))[/latex].
Here the author uses [latex]v(\rho)[/latex], which describe moving speed of 1 item (just like 1 water molecule in Yangtze River).

The velocity [latex]v(\rho)[/latex] is defined by:  [latex]v(\rho)=\frac{v_{max}}{1+\int_0^{1}\rho (s, t) ds}[/latex]

or by [latex]v(\rho)=v_{max}(1-\frac{\int_0^{1}\rho (s,t) ds}{L_{max}})[/latex].  Here [latex]L_{max}[/latex] is the max load.

I haven’t looking up the article where these two definition come from.  But it can be concluded from experience I think.

In this model, the changing chain: influx–WIP–velocity–outflux