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Chapter 5

Chapter 5. Simulation Modeling. Introduction. In many situations a modeler is unable to construct an analytic (symbolic) model adequately explaining the behavior being observed because of its complexity or the intractability of the proposed explicative model .

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Chapter 5

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  1. Chapter 5 Simulation Modeling

  2. Introduction • In many situations a modeler is unable to construct an analytic (symbolic) model adequately explaining the behavior being observed because of its complexity or the intractability of the proposed explicative model. • In some circumstances, it may not be feasible either to observe the behavior directly or to conduct experiments. • Examples • Elevator systems • Communications network • Location of machines in a new industrial plant • Evacuation routes • Drag forces on new designs

  3. Monte Carlo Simulation • A Monte Carlo Simulation is a probabilistic technique (stochastic simulation) where a computer is used to draw samples of random numbers in order estimate the behavior of physical phenomena that may otherwise be untractable. • Examples

  4. 5.1 Simulating Deterministic Behavior: Area Under a Curve • Example: • Approximation of 

  5. Volume Under a Surface • Volume of part of part of the sphere x2 + y2 + z2 ≤ 1 that lies in the first octant (the actual volume is  /6  0.5236)

  6. 5.2 Generating Random Numbers • Middle-Square Method • Start with a four-digit number x0, called the seed. • Square it to obtain an eight-digit number (add a leading zero if necessary). • Take the middle four digits as the next random number. • Example • Major drawback: tendency to degenerate to zero.

  7. Generating Random Numbers • Linear Congruence • Modulus c • Multiplier a • Increment b • Cycling: At some point all pseudorandom number generators begin to cycle, i.e. the sequence repeats itself. • Example • With a=1, b=7, and c=10

  8. 5.3 Simulating Probabilistic Behavior • Relative frequency approach to Probability: • Over the long run, the probability of an event can be thought of as the ratio • Example • Law of Large numbers for a fair coin

  9. Simulating Probabilistic Behavior • Example • Monte Carlo simulation of the roll of a fair die. • Monte Carlo simulation of the roll of an unfair die. • Non-uniform random numbers. • Normally distributed random numbers

  10. Simulating Probabilistic Behavior • Project #5, page 193

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