Dealing with Time in System Dynamics Models

Sourced from Geoff Coyle

Importance of Time in Simulation

For discrete event: “Time moves forward when something happens”.For system dynamics: “Time is a fundamental issue”. Outputs and inputs are plotted against time.

Aspects of Time in Dynamo

DelaysThe time unit usedTime-dependent exogenous influencesLENGTH (reserved word)TIME (reserved word)DT (reserved word)Output control

DelaysPhysical Delays Delay1 – a distribution delay (similar to

erlang) Delay2 – a pipeline delay contains a number

of internal levels (seldom happens in actual systems)

Information Delays Smooth or DLINF3

Delays produced by decision rules Ex. PR.KL=(DESIRED.K-ACTUAL.K)/STAT Where STAT is stock adjustment time (used

as a first order delay of STAT in the system)

Note about Delays

Delays should be measured in the same time units.A model with delays in days and delays in years are confusing.Modeling Production Policy uses delays in daysModeling Capacity planning and acquisition uses delays in years

Time UnitTime units must have the same dimensions as delays.Follow the natural choice of the system, when selecting time.If there is a conflict in choosing time, then the purpose of the model is not clear (both the modeler and user)Time magnitude can be 1 week, 1 year, 1 month, etc. The important things is that it resides in the modeler’s mind and understanding of the problem.

Time dependent Exogenous Inputs

May or may not happen in your model.It includes simulated demand patterns such as STEP and PULSE (there are others).STEP is used to test model’s stability and to detect errors.PULSE should be handled carefully. It differs on the software packages.

LENGTHIt is a built in feature of dynamoIt connotes the duration of the model run.The dimension of LENGTH should be the same as the Time Unit.

How long is LENGTH?The LENGTH should be at least 10 times as long as the longest delay in the model.It must be long enough to allow any exogenous input to have sufficient time to show their effects.For STEP and PULSE, LENGTH must be long enough until no further changes are seen in the model’s behavior.It must allow for at least 4 cycles.It must be sufficient to allow policies in the model to reveal their consequences.

How long is LENGTH?It must be credible to the client/user.If confusion arises on the length, the longest is not necessarily better.Recall the fundamental aspect of the system (modeling a business for 100 years maybe absurd)A simple test is to run it with LENGTH doubled and LENGTH half and see if there is a difference. (Watch out for absurd realities such as negative values)It should be a sensible round number.

A Special command “TIME”

A counter which represents the passage of time in the modelAutomatically creates a level equation of the form: L TIME.K=TIME.J+DTN TIME=1900 (Time can be initialized, but extra care)LENGTH should be adjusted when time is initialized.Initializing time does not mean that you are predicting the future instead of showing policy effects.

DT

It is the time step in the model that progressively moves time towards its termination at LENGTH.DT for Dynamo and TIMESTEP for other packagesDT is an artefact of the modeller’s mind and not the system.

DT

It should have the same dimension as TIME.It is never placed on the right hand side of a rate or auxiliary.It has nothing to do with the problem.If DT is too large it becomes numerically unstable. If DT is too small it will take a longer run.

DT

Observe this: DT<= DEL/4NWhere DEL is the magnitude or duration of delay in the model and N is the order.In practice, the lower possible value from the limit shown above is chosen (binary fraction).Most common values of DT are 0.5, 0.25, and 0.125If there are multiple delays inside the model, choose DT that is a common multiple of the delays’ magnitude.

Output ControlNot a key determinant of time.It is good practice to print, on paper, or tables of all variables in a model while the model is being debugged.An equivalent set of calculations can be made using a spreadsheet (such as EXCEL) to be able to see clearly the numerical changes that happens in the model.The same equations from dynamo can be used in printing or constructing the table for the variables.