1. CAP4800/5805 Systems Simulation
System Dynamics 2
CAP4800/5805
Systems Simulation

2. What we covered last time
What is System Dynamics
Causal Loop Diagram
Augmenting Causal CLD
Loop dominance
Labeling link polarity
Determining loop polarity
Exogenous items and delays

3. System Dynamics Modeling
Identify a problem
Develop a dynamic hypothesis explaining the cause of the problem
Create a basic causal graph
Augment the causal graph with more information
Convert the augmented causal graph to a System Dynamics flow graph
Translate a System Dynamics flow graph into DYNAMO programs or equations
Simulate the DYNAMO programs or equations

4. Casual-loops Provide insight into a system's structure
Often difficult to infer the behavior of a system from its casual-loop representation
Need to use computer simulation
Simulation model: flow diagrams, equations, simulation language
DYNAMO (DYNAmic Models):
Not a general-purpose language but special purpose language to aid in building computer models

5. Flow Graph Symbols Level Rate Flow arc Auxiliary Cause-and-effect arc
Source/Sink
Constant

6. Level: AKA stock, accumulation, or state variable
A quantity that accumulates over time
Change its value by accumulating or integrating rates
Change continuously over time even when the rates are changing discontinuously

7. Rate/Flow: AKA flow, activity, movement Change the values of levels
The value of a rate is
Not dependent on previous values of that rate
But dependent on the levels in a system along with exogenous influences

8. Auxiliary:
Arise when the formulation of a level’s influence on a rate involves one or more intermediate calculations
Often useful in formulating complex rate equations
Used for ease of communication and clarity
Value changes immediately in response to changes in levels or exogenous influences

9. Source and Sink:
Source represents systems of levels and rates outside the boundary of the model
Sink is where flows terminate outside the system

10. Example 1 (Population and birth)
Births
Population

11. Example 2 (Children and adults)- +
Children maturing
Births
children
Adults

12. DYNAMO Originally developed by Jack Pugh at MIT
First system dynamics simulation language
For a long time the language and the field were considered synonymous
Provides an equation based development environment for system dynamics models
DYNAMO today runs on PC compatibles under Dos/Windows.

13. Time in DYNAMO LEVEL.K: a level calculated at the present time
LEVEL.J: a level calculated one time interval earlier
DT: the length of the time interval between J and K dt dt
L: future
J: past
K: present

14. DYNAMO Program (Population and birth model)
Births
Population
Star statement
* Population Growth
L POP.K = POP.J + DT*BIRTH.JK
N POP = 10
R BIRTH.KL = (POP.K)(PAR)
C PAR = 0.1
SPEC DT = 1.0
Level statement
present
one time interval earlier
between J and K
Initial value statement
Rate statement
Constant statement
SPEC statement

15. Integral/Differential Equations
Diagran R1 R2 L
Integral Equation
L(t) = ∫ [R1(s) – R2(s) ] ds + L(t0) t t0
Differential Equation
dL/dt = Net Change in L = R1(t) - R2(t)

16. System Dynamics Algorithm
Program Main
We are given a concept graph with modes and arcs
The arcs require sign (+,-) labeling
The nodes require labeling: source, rate, level, constant, auxiliary
For each level node (L) with an input rate node (R1) and and output rate node (R2) write:
dL/dt = k1 * R1 – k2 * R2 ; k1 and k2 are rate constants
End for
For all other nodes (N) write:
N(t) = a linear function of all inset members of this node
End Main

17. From Causal Loop Diagram To Simulation Models 1
Causal Graph
Flow Graph R L
Equations
dL/dt = k1*R(t)
R(t) = k2*L(t)
 dL/dt = k1*k2*L(t)
Block Model L’ L ∫
k1*k2

18. From Causal Loop Diagram To Simulation Models 2
Flow Graph
Equations
dL/dt = R1 – R2
R2 = k2*L
R1 = k1
 dL/dt = k1 - k2*L R1 R2 L
Block Model
L1’ L1 ∫ k2 k1 -

19. From Causal Loop Diagram To Simulation Models 3
Flow Graph
Equations
dL1/dt = R1 – R2
dL2/dt = R2 – R3
R1 = k1
R2 = K2 * L1
R3 = K3 * L2
 dL1/dt = k1 – k2*L1
 dL2/dt = k2*L1 – K3*L2 R1 R2 R3 L1 L2
Block Model
L1’ L1
L2’ L2 ∫ - ∫ k2 k3 k1 -

20. Building construction
Problem statement
Fixed area of available land for construction
New buildings are constructed while old buildings are demolished
Primary state variable will be the total number of buildings over time
Causal Graph - - - -

21. Simulation models Flow Graph Equations dBl/dt = Cr – Dr
Cr = f1(CF, Bl)
Dr = f2(AL,Bl)
CF = f3(FLO)
FLO = f4(LA,AA,Bl)
Construction (C)
Demolition (D)
Industrial
Buildings (B)
Average lifetime for buildings (AL)
Construction
fraction
(CF)
Fraction of
land occupied
(FLO)
Land available for industrial buildings (LA)
Average area per building (AA)

22. Next Class VenSim System Dynamics Simulation Tool

23. References
Simulation Model Design and Execution, Fishwick, Prentice-Hall, 1995 (Textbook)
Introduction to Computer Simulation: A system dynamics modeling approach, Nancy Roberts et al, Addison-wesley, 1983
Business Dynamics: Systems thinking and modeling for a complex world, John D. Sterman, McGraw-Hill,2000