Discrete Event Simulation

1. Discrete Event Simulation • How to generate RV according to a specified distribution? • geometric • Poisson etc. • Example of a DEVS: • repair problem

2. Outline What is discrete event simulation? Events Probability Probability distributions Sample application

3. What is discrete event simulation? “Simulation is the process of designing a model of a real system and conducting experiments with this model for the purpose either of understanding the behavior of the system or of evaluating various strategies (within the limits imposed by a criterion or set of criteria) for the operation of a system.” -Robert E Shannon 1975 “Simulation is the process of designing a dynamic model of an actual dynamic system for the purpose either of understanding the behavior of the system or of evaluating various strategies (within the limits imposed by a criterion or set of criteria) for the operation of a system.” -Ricki G Ingalls 2002 Definitions from http://staff.unak.is/not/andy/Year%203%2Simulation/Lectures/SIMLec2.pdf

4. What is discrete event simulation? • A simulation is a dynamic model that replicates the behavior a real system. • Simulations may be deterministic or stochastic, static or dynamic, continuous or discrete. • Discrete event simulation (DEVS) is stochastic, dynamic, and discrete. • DEVS is not necessarily spatial – it usually isn’t, but the ideas are applicable to many spatial simulations

5. What is discrete event simulation? • DEVS has been around for decades, and is supported by a large set of supporting tools, programming languages, conventional practices, etc. • Like other kinds of simulation, offers an alternative, often simple way of solving a problem – simulate system and observe results, instead of coming up with analytical model. • Many other benefits like repeatability, ability to use multiple parameter sets, experimental treatment of what may be unique system, cheap compared to physical experiment, etc. • Usually involves posing a question. The simulation estimates an answer.

6. What is discrete event simulation? • Stochastic = probabilistic • Dynamic = changes over time • Discrete = instantaneous events are separated by intervals of time Time may be modeled in a variety of ways within the simulation.

7. Alternate treatments of time Time divided into equal increments: Unequal increments: Cyclical:

8. Termination • Simulation may terminate when a terminating condition is met. • May also be periodic. • Can also be conceptually endless, like weather, terminated at some arbitrary time. • Usually converges or stabilizes on particular result.

9. Events • May change the state of the system. • Have no duration. • If time is event-based, events happen when time advances. • Canonical example: flipping a coin. • Events usually have a value associated with them (e.g. coin: true/false). • The definition of an event depends on the subject of the model.

10. Random Variables • A random variable (“X”) is numerical value associated with a random event. • X can have different values: • X1, X2, X3…Xn • In a stochastic DEVS, the value associated with event is probablistic, that is it occurs with a given probability. • That probability is what allows us to use DEVS to estimate results when we don’t know the precise value of input parameters to our model.

11. Pseudorandom Number Generators • A stochastic simulation depends on a pseudorandom number generator to generate usable random numbers. • These generators can vary significantly in quality. • For practical purposes, it may be important to check the random number generator you are using to make sure it behaves as expected. Example: Using matlab’s rand(), let us simulate a coin toss and test how good the random number generator is.

12. Probability distributions • In a DEVS, you need to decide what probability distribution functions best model the events. • Pseudorandom number generators generate numbers in a uniform distribution • One basic trick is to transform that uniform distribution into other distributions. • There are many basic probability distributions are convenient to represent mathematically. • They may or may not represent reality, but can be useful simplifications.

13. Normal or Gaussian distribution • Ubiquitous in statistics • Many phenomena follow this distribution • When an experiment is repeated, the results tend to be normally distributed.

14. Simulating a Probability distribution Sampling values from an observational distribution with a given set of probabilities (“discrete inverse transform method”). Generate a random number U If U < p0 return X1 If U < p0 + p1 return X2 If U < p0 + p1 + p2 return X3 etc. This can be speeded up by sorting p so that the larger intervals are processed first, reducing the number of steps.

15. Poisson distribution • Example of algorithm to sample from a distribution. • X follows a Poisson distribution if: An algorithm for sampling from a Poisson distribution: 1. Generate a random number U 2. If i=0, p=e-l, F=p 3. If U < F, return I 4. P = l * p / (i + 1), F = F + p, i = i + 1 5. Go to 3 There are similar tricks to sampling from other probability distributions. Some tools like Matlab have these built in.

16. A repair problem n machines are needed to keep an operation. There are s spare machines. The machines in operation fail according to some known distribution (e.g. exponential, Poisson, uniform etc. with a known mean). When a machine fails, it is sent to repair shop and the time to fix is a random variable that follows a known distribution. Question: What is the expected time for the system to crash? System crashes when fewer than n machines are available.

17. Plotting outcome of experiments • The expected value (mean) will converge after a number of repetitions. • We can demonstrate the process of flipping coins…with a simulation. Here are two runs of a simulated sequences of 1000 coin flips (see code), displaying the average result (sum of Xi / n): Note that the variance is high initially and diminishes with time.