1. Simulation with Arena, 4 th ed.Chapter 11 – Continuous & Combined Discrete/Continuous ModelsSlide 1 of 11 Continuous and Combined Discrete/ Continuous Models Chapter 11 Last revision August 19, 2006

2. Simulation with Arena, 4 th ed.Chapter 11 – Continuous & Combined Discrete/Continuous ModelsSlide 2 of 11 What We’ll Do... What is a continuous system? Simple linear continuous systems Combined discrete/continuous systems Non-linear and complex systems

3. Simulation with Arena, 4 th ed.Chapter 11 – Continuous & Combined Discrete/Continuous ModelsSlide 3 of 11 Continuous Systems Discrete systems – State changes occur at isolated points in time called events Continuous systems – State changes may occur continuously over time  Flow of fluids and fluid-like materials  Temperature changes  Chemical operations  Biological processes

4. Simulation with Arena, 4 th ed.Chapter 11 – Continuous & Combined Discrete/Continuous ModelsSlide 4 of 11 Continuous Systems Simple systems (linear)  Rate of change is constant between events  Future value can be calculated from starting value and rate  Can step directly to calculated event Complex systems (non-linear)  Rate of change may depend on other continuous processes  Specialized approaches used to capture change  Approximates continuous change by making a series of small steps between the usual discrete events

5. Simulation with Arena, 4 th ed.Chapter 11 – Continuous & Combined Discrete/Continuous ModelsSlide 5 of 11 Continuous Systems Example of simple continuous system filling a tank smoothly over time (Model 11-1)

6. Simulation with Arena, 4 th ed.Chapter 11 – Continuous & Combined Discrete/Continuous ModelsSlide 6 of 11 Continuous Systems Basic constructs: Levels & Rates from Elements Panel A Level is the value that is changing over time A Rate determines the rate of change of the level Both are similar to Variables in that they can be assigned a new value at any time. Levels may also change as time advances if the value of the associated Rate is non-zero. A Level and a Rate should be used as a pair (e.g. If you have 4 Levels you should have 4 Rates)

7. Simulation with Arena, 4 th ed.Chapter 11 – Continuous & Combined Discrete/Continuous ModelsSlide 7 of 11 Simple Continuous Systems Continuous Element specifies integration parameters:  Number of Dif Equations – In simple systems, leave at default of number of Rate/Level pairs.  Number of State Equations – Ignore in simple systems.  Minimum step size – The minimum time advance between integration steps. Use 0.0 in simple systems.  Maximum step size – The maximum time advance between integration steps. Use a high value (100) in simple systems.  Save Point Interval – The maximum time between save points for recording continuous statistics (CSTATS).  Method – Use Euler linear algorithm for simple systems.

8. Simulation with Arena, 4 th ed.Chapter 11 – Continuous & Combined Discrete/Continuous ModelsSlide 8 of 11 Simple Continuous Systems Discrete control loop to empty and refill a tank (Model 11-2a)

9. Simulation with Arena, 4 th ed.Chapter 11 – Continuous & Combined Discrete/Continuous ModelsSlide 9 of 11 Combined Discrete/Continuous Modify Model 11-2a to Model 11-2b to detect when Tank Level rises to 100 or falls to 0 Detect Module from Blocks panel “watches” for and helps predict events. Watches for value of a variable to cross a threshold value (e.g. a tank level reaching its maximum value) Similar to Create Module in that an entity is created when crossing occurs.

10. Simulation with Arena, 4 th ed.Chapter 11 – Continuous & Combined Discrete/Continuous ModelsSlide 10 of 11 Combined Discrete/Continuous Fill and empty logic using Detect modules Further examples (details in text)  Model 11-3: Coal-loading operation with above approach  Model 11-4: Coal-loading operation with Flow Process panel – special-purpose modules for such models

11. Simulation with Arena, 4 th ed.Chapter 11 – Continuous & Combined Discrete/Continuous ModelsSlide 11 of 11 Complex Systems Non-linear systems require special differential- equation algorithms (Runge-Kutta-Fehlberg)  Step sizes must be set carefully  Smaller step size will generate more accurate results because Arena will calculate continuous-change variables more often  Larger step size will run faster, but your error tolerances will need to be set higher Many situations (like a gravity fed tank) are actually non-linear, but can be accurately approximated with faster, linear methods Model 11-5: Soaking-pit furnace (details in text)  Differential equations defined in VBA