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Math and Statistical Distributions 709
Integration vs. summation in the Holding Tank block
Integration vs. summation in the Holding Tank block
As discussed on page 91, the Holding Tank block (Value library) has integration capabilities. The block’s dialog gives the option to either sum or integrate its input. In general, a good rule of thumb is:
• IntegratewhenthevaluegoingintotheHoldingTankisbasedonsimulationtimeunits,such as a rate. For example, you would integrate when the Holding Tank block’s input represents dollars per year or gallons per hour.
• Sum when the value is to be added to the block at each step or dt calculation. For example, you would sum when the Holding Tank block’s input is orders or people.
Summing adds the given input to the total at each step, regardless of the time units. Integration considers the input to be spread evenly over each time unit; at each step integration adds a por- tion of the input to the total. For example, the following table shows the effect of inputting $2000 to the Holding Tank block when the time units are in years and dt (delta time) is set to 0.25 (for 1/4 of the year).
As seen in the table, if the Holding Tank is set to sum its inputs, it would be the equivalent of adding $2000 to the account every quarter. If the Holding Tank is set to integrate, it would be the equivalent of adding $500 per quarter.
Time Step
0 0 0.25 1 0.50 2 0.75 3 1 4
Summed Integrated (delay)
2000 0 4000 500 6000 1000 8000 1500 10000 2000
Integrated (no delay)
500
1000
1500
2000
2500
• •
•
Summationoccursateachstepsothereisanamountcalculatedattime0.Sincesummation treats its input as an amount, the entire 2000 is added at each step.
Theintegrated(delay)choicetreatsitsinputasarateandcalculatesanewresultatthe next step. Because this backward Euler integration occurs during the interval between steps, there is no amount at time 0.
Theintegrated(nodelay)choicealsotreatsitsinputasarate.However,itcalculatesanew result at the current step. Because its integration occurs at each step, there is an amount at time 0.
It is also important to note that if you subsequently change the delta time to something other than 0.25, the total amount in the summed Holding Tank would be different from the amounts shown above, but the total amount in the integrated Holding Tank would remain the same.
Your choice of integration methods depends on the model:
• Youwouldgenerallyusetheintegrated(delay)choicewhenthereisonlyoneintegrating block in the model, when the integrating blocks are not interdependent or cross-coupled, or when there is no feedback.
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