the question that you have brought forward touches the heart of the SD-Methodolody: Why should we use models in continuous time as opposed to models in discrete time (e.g. just setting dt to 1)? The answer is that the sometimes akward 'inaccuracy' will not matter or rather will be offset by the grip we will get upon complex reality. If in a system there are 'lots' of accumulation and draining processes and many interdependencies (e.g. feedback) the benefit of continuous time models with their ease of catching these system features should prove much greater than being off from an exact value that often never exists for the (unknown) model that produces reality.
That said I would in the example you have given be careful in what is meant with 'time to drain'. The term 'solution' with regard to a system of differential equations is mathematically speaking a function
. In your case it is the exponential function and one could give an exact function that gives the result for what should be in the stock so one could then build this as an instantaneous process using just auxiliaries with no need to integrate. So the exponential decay process
you are describing has a time constant
(time to drain) that for a first order differential equation gives the mean residence time
Your model is continuously compounding
and the fractional rate of decay is a constant. This is the first thing that has to be explained to a customer not familiar with SD - SD-models show whats in the stocks for any multiple of DT approaching continuos time dt = 1 IMHO is a very dangerous time step to use all to easily the model will not work properly at say dt = 0.5 or less. So the proper question for setting up an SD-model is what do you know about the behaviour of the Stock
- this amounts to the question for the order of the delay with the pipeline delay as an extreme - and what do you know about the value of the Stock after one year. If you know that the stock should be zero after one year and you know it is about a third order delay, then you might fit the time to drain accordingly
so that the behavior approaches what you know about reality (you might use a threshold with a IF-THEN-ELSE - function to force the stock value to zero while this is not neat of course).
As you can see the time to drain all of a sudden takes on a different meaning for physical, real world processes that behave like an exponential decay. If what you are trying to do is to build a model that represents linear depreciation in a continuous model (the business economics way of looking at depreciation). Then here is a good paper to have a look at:http://www.systemdynamics.org/conferences/2004/SDS_2004/PAPERS/246SCHWA.pdf
If you take a look at figre 4 on page 8 (left hand side) you can see that the time constant (e.g. your time to drain) is adjusted every DT so that the scrap value (e.g. zero) is reached at the eand of the economic life.
Hope that helps.
Guido W. Reichert
The comparison that you have given does illustrate the above quite nicely. The options you have given (100% probability of death vs. 2 times 50% probablity of death) take the probability (e.g. the time constant) as a constant. But it must be variable for the options to be equivalen: The probability for the first half year should be 50% and that for the second half year accordingly 100% (it is of course a conditional probability affecting those that have survived up to that point in time). In the end all are dead - as Keynes has also noted for the long run.
In a proper SD-model the choice of DT should not affect the simulation outcome. It should be a purely technical construct that determines accuracy and speed for a given numerical integration technique.