Stochastic Simulation The modelling process Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby – Denmark Email: bfni@dtu.dk
Explanation: What is the problem with the Explanation: What is the problem with the Pareto distribution Pareto distribution • Moment distributions • For nonnegative valued random variables � x � x 0 t j f ( t ) dt 0 t j f ( t ) dt G j ( x ) = 0 t j f ( t ) dt = � ∞ E ( X j ) The contribution to the j ’th moment from values ≤ x . � t � t � x � x � x � − k − 1 � − k tk t 1 f ( t ) dt = dt = k dt β β β β β 0 � t � x � � − k +1 � � − k � x k − 1 k βk = β dt = 1 − k − 1 k − 1 β β β β DTU 02443 – lecture 11 2
Explanation: What is the problem with the Explanation: What is the problem with the Pareto distribution Pareto distribution Pareto distribution Pareto distribution Pareto distribution 1 1 1 1 - exp(-k*log(t)) 1 - exp(-k*log(t)) 1 - exp(-k*log(t)) 1 - exp(-(k-1)*log(t)) 1 - exp(-(k-1)*log(t)) 1 - exp(-(k-2)*log(t)) 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 5 5 5 10 10 10 15 15 15 20 20 20 25 25 25 30 30 30 • The first moment distribution for the Pareto distribution (green) • The second moment distribution for the Pareto distribution (blue) DTU 02443 – lecture 11 3
Some numbers β = 1 Some numbers β = 1 F ( t ) = 1 − t − k f ( t ) = kt − k − 1 G 1 ( t ) = 1 − t − k +1 G 2 ( t ) = 1 − t − k +2 For k = 2 . 05 F ( t ) G 1 ( t ) G 2 ( t ) t 2 0.7585 0.5170 0.0341 10 0.9911 0.9109 0.1190 100 0.9999 0.9921 0.2057 1 − 10 − 6 844.5 0.9992 0.2860 • Even when if we simulate 10 6 values we can not expect to get a decent estimate of the variance! DTU 02443 – lecture 11 4
What to learn: What to learn: • Care is needed when using simulation • Especially if one wants to study strange or rare phenomena. • Always use your practical, theoretical and intuitive understanding of the system to support the analysis by simulation. DTU 02443 – lecture 11 5
The modelling process The modelling process • Problem identification • Goal/purpose • System analysis and data gathering • System syntesis - model formulation • Estimation of model parameters • Preliminary model validation • Program development • Final validation • Experimental design • Statistical analysis DTU 02443 – lecture 11 6
Goal/purpose Goal/purpose • Describe the objective of the problem • e.g. to design an inventory control system with stable production DTU 02443 – lecture 11 7
Problem identification Problem identification • Define which part of reality which is to be modelled • A company producing one product with an ingoing and outgoing inventory wants an adequate inventory control system DTU 02443 – lecture 11 8
System analysis and data gathering System analysis and data gathering • Investigate the system identifying parts with direct impact on the goal. Provide data on these parts. • Demand, relations between order, production and inventory. Insignificant: unemployment, weather. DTU 02443 – lecture 11 9
System syntesis System syntesis • Define state variables. Describe dynamics and relations, determine distributions. • Example: Inventory at time t , determine the type and form of the demand function. DTU 02443 – lecture 11 10
Estimation of model parameters Estimation of model parameters • Determine parameter values, values for model constants/parameters. • Example: Estimate mean and variance of the demand function. DTU 02443 – lecture 11 11
Preliminary model evaluation Preliminary model evaluation • Control of fundamental logical structures. Common sense control of parameters • Does the model imply inventory sizes exceeding system capacity DTU 02443 – lecture 11 12
Program development Program development • Translate state definitions into data structures. Formalise logical and physical relations. Prepare the program for debugging. Make a modular program which is easy to extend. • Example: Inventory size is defined as an integer variable. Time is define integer or continuous depending on the context. DTU 02443 – lecture 11 13
Final validation Final validation • Run the program for input combination with known analytical solution. Run the program with extreme values of the parameters. Common sense control of output. Study animations. Compare with real world data if possible (existing system). • Example: Choose exponential distribution. Reduce/simplify relations. DTU 02443 – lecture 11 14
Experimental design Experimental design • Planning of sensitivity analyses. • Realistic/interesting combinations of parameter values. • Example: Periods of constant demand. Periods with highly varying demand. DTU 02443 – lecture 11 15
Statistical analysis Statistical analysis • Estimation of systemparameters. Confidence intervals. Variance reduction techniques. Time series analysis. • Example: Variance of number of orders. DTU 02443 – lecture 11 16
Verification and validation of simulation models Verification and validation of simulation models - L&K chapter 5 - L&K chapter 5 • Validation is the process of determining whether a simulation model (as opposed to the computer program) is an accurate representation for the particular objectives in study. • Verification is concerned with determining whether the conceptual simulation model (model assumptions) has been correctly translated into a computer “program”. • A simulation model and its results have Credibility if the manager and other key project personnel accept them as “correct”. DTU 02443 – lecture 11 17
Validiation Validiation • Conceptually, if a simulation model is “valid” then it can be used to make decisions about the system similar to those that would be made if it were feasible and cost-effective to experiment with the system itself. DTU 02443 – lecture 11 18
Credibility Credibility • The managers understanding and agreement with the models assumptions. • Demonstration that the model has been validated and verified. • The managers ownership of and involvement with the project • Reputation of the model developers DTU 02443 – lecture 11 19
What is simulation? What is simulation? • Computer experiments with mathematical model • General engineering technique • Analytical/numerical solutions DTU 02443 – lecture 11 20
Course goal Course goal • Topics related to scientific computer experimentation • Specialised techniques ⋄ Variance reduction methods ⋄ Random number generation ⋄ Random variable generation ⋄ The event-by-event principle • Simulation based statistical techniques ⋄ Markov chain Monte Carlo simulated annealing ⋄ Bootstrap • Validition and verification of models • Model building DTU 02443 – lecture 11 21
Project types Project types • Model a system (e.g., like the ferry example) in order to assess performance, under varying designs. • Study a mathematical model, that is impossible or hard to analyze • Study any one of the techniques we have been through, more closely. For example, generating random numbers with the Mersenne Twister. • Come up with your own project type. DTU 02443 – lecture 11 22
Simulation projects Simulation projects • Post office (discrete event) • Simulation and estimation in a Markov model of breast cancer (discrete event) • Simulation of queueing system with input generated by interrupted Poisson processes (discrete event) • Simulation of Levy processes (discrete event) • Simulation of queues with Brownian motion input (discrete event, can be developed to something more advanced) • Microemulsions (MCMC) • Your own suggestion to discuss
General guidelines General guidelines • Do not make model too complicated • Clear objective • Time to experiment with model • Apply variance reduction techniques if possible DTU 02443 – lecture 11 24
Deliverable 2: Project report Deliverable 2: Project report • Precision of objective • Model validation • Program verification • Experimental design Standard report, the important issue though is to have some time to experiment with your model. • Deadline Thursday June 27th (Monday June 30th if you like) DTU 02443 – lecture 11 25
Registering Registering • Register online. I have created an exercise for group hand in on Campusnet. • Register with a group name. Use the title/content of your project as (part of) the group name DTU 02443 – lecture 11 26
Plan for the next two weeks Plan for the next two weeks • Nicolai, Jakob and I will be available on a consultancy basis. The availability will be communicated through Campusnet and/or the web page. • I will be unavailable from noon monday (today) until noon tuesday (tomorrow) DTU 02443 – lecture 11 27
And remember And remember • Course evaluation (Campus net) • Comments and suggestions, at any level of detail DTU 02443 – lecture 11 28
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