Simulation. Version 1.1 c 2010, 2009, 2001 José Fernando Oliveira Maria Antónia Carravilla FEUP

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1 Simulation Version 1.1 c 2010, 2009, 2001 José Fernando Oliveira Maria Antónia Carravilla FEUP

2 Systems - why simulate? System any object or entity on which a particular study is run. Model representation of the object or entity on which the study will be actually performed. Experiments on the system itself, stimulating and recording the observed changes in the parameters that define its state more reliable results. Systems too large, not directly accessible or ethically experiential experimentation on a model: simulation. The best thing you can do, after watch the real system in operation, is to make a simulation of that system.

3 Model types Scale models the system is replaced by a scaled version of the system itself. Conceptual models the system is replaced by a conceptual version of its operation: a certain set of mathematical equations or cause-effect relationships. Modelos de escala Sistema Modelos conceptuais Contínuos Estatísticos Discretos

4 Conceptual models Better than scale modelos because: more transportable; much more versatile, allowing for easier changes in its structure; implementable with a computer! Types of conceptual models: Deterministic: Continuous the system states vary continuously with the independent variable. Discrete the system can only be at a finite number of states, changing instantly from one state to another. Statistical the system state does not depend deterministically on the input variables but its response is random and characterized by a set of probability distributions.

5 Simulation with continuous models (continuous simulation) In a continuous simulation model the state Y of a system depends continuously on the independent variable x, being this relationship governed by equations (possibly differential equations) knowledge of Y = E(x). Example: simulation of the speed of a bus, since it starts until it stops again. System bus Vel System state velocity fm(t) Independent variable time 0, t 0 < t < t 1 fa(t) MOVIMENTO fa(t), t 1 t < t 2 E(t) = PARADO fm(t), t 2 t < t 3 fp(t), t 3 t < t t0 t1 t2 t3 t4 4 fp(t) PARADO t

6 Simulation with statistical models (Statistical simulation or Stochastic simulation or Monte Carlo simulation) We don t deal with a model describing the system complex relationships. Instead, we deal with a set of probability functions that characterize the system behavior. Example: System = Dice Deterministic simulation writing the equations governing the movement of a solid, having as parameters the initial velocity, the angle with which it is thrown, the friction offered by the table, etc.. Statistical simulation sampling the probability function that gives the probability of, at any trial, the result being a given face.

7 Monte Carlo simulation Physicists at Los Alamos Scientific Laboratory were investigating radiation shielding and the distance that neutrons would likely travel through various materials. Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus or how much energy the neutron was likely to give off following a collision, the problem could not be solved with theoretical calculations. John von Neumann and Stanislaw Ulam suggested that the problem be solved by modeling the experiment on a computer using chance. Being secret, their work required a code name. Von Neumann chose the name Monte Carlo. The name is a reference to the Monte Carlo Casino in Monaco where Ulam s uncle would borrow money to gamble. in http: // en. wikipedia. org/ wiki/ Monte_ Carlo_ method

8 Estimation of the area of a circle Estimate the area of the circle given by the following equation: (x 1) 2 + (y 2) 2 = 25 i.e, a circle with radius 5 e and center (x, y) = (1, 2). Procedure: fit the circle in a square of side equal to the diameter of the circle; build a generator of random points of the square, which should occur with equal probability; then, if in a random sample of n points, m are inside the circle: Area of the circle m n Area of the square = m n (10 10)

9 Estimation of the area of a circle We can get points inside the square with equal probability by representing the coordinates x and y of any point of the square by the following distributions: f(x) = 1 10, 4 x 6 g(y) = 1 10, 3 y 7 Let R 1 and R 2 be two random numbers between 0 and 1. Then, a point (x, y) inside the square will be given by: x = 4 + [6 ( 4)]R 1 = R 1 y = 3 + [7 ( 3)]R 2 = R 2 Finally, a random point (x, y ) is inside the circle if: (x 1) 2 + (y 2) 2 25

10 Random numbers A number is said to be random if its occurrence in a sequence of numbers is unpredictable from any past occurrences and if it has the same probability of occurring over a predefined range of values. Truly random phenomena: radioactive noise, background noise in electronic components, etc.. (all physical systems). With computers we can only generate pseudo-random sequences: Use of an iterative process, performed on a mathematical expression, that based on the previous number computes the following one n i+1 = f(n i ) so that the relationship between the two is as unpredictable as possible within a large range of values.

11 Pseudo-random numbers n i+1 = f(n i ) Disadvantage Once one of the numbers is repeated all the sequence repeats itself. Advantage It is possible to exactly duplicate and replicate a sequence of random numbers: important for the detection and correction of mistakes and bugs, and the validation and verification of a simulation model. Different functions f(n i ) Different methods to generate the pseudo-random numbers.

12 Pseudo-random numbers generation Most common method: Multiplicative congruential method n i+1 = (b n i + c)mod(m) em que: mod stands for the remainder of the integer division; m the largest number that can be generated (exclusive); b e c formula parameters; the first value used in the sequence, n 0, is called the seed of the generator. Concrete values for the generation of 16-bits and 32-bits integers, respectively: f(n i ) = (3993n i + 1)mod(32767) f(n i ) = (16807n i + 0)mod( ) To generate a value between 0 e 1 it is enough to divide n i+1 by m: R i+1 = n i+1 m

13 Why is this a statistical simulation model? Monte-Carlo simulation model Inputs Circle radius and center Random points (x0,y0) Computation In/Out Outcome Outputs Estimate of the circle area For the same input we may (will) have different outputs. And if the random points have to follow a non-uniform probability function? histograms (discrete) Probability functions sampling distributions (continuous)

14 Construction of histograms from experimental data Observed variable: number of cars that enter in a minute on a given street. 11 samples were taken: 17, 20, 12, 17, 17, 10, 17, 20, 11, 12, 17 Data classification: N. of cars Total Percentage Cumulative (per min) cars H(x) percentage F (x) = = = = = H(x) F(x) 0,5 1,0 0,4 0,8 0,3 0,6 0,2 0,4 0,1 0,2 0, ,

15 Histogram sampling From a random number u i, belonging to the interval [0, 1], get one of the x values, according to the probabilities induced by the histogram. F(x) 1,0 u i 0,8 0,6 0,4 0,2 0, Assumption: the sequence of (pseudo-)random numbers reflects a constant probability distribution.

16 Production line maintenance A factory has a production line that works 24 hours a day, 360 days a year. This production line generates an added value of 500 euros per hour. Sometimes the line has faults needing repair. Detailed historical data showed that the number of operating hours between failures has the following probability distribution: N. of hours Probability N. of hours Probability The average operating time (between failures) is therefore of hours.

17 Production line maintenance After a breakdown the time required for repair is also variable. If the repair time exceeds 3 days, then a replacement unit can be obtained under the guarantee contract, to restore the factory in operation. As this takes one day, in the worst case the repair time will be 4 days. The probability distribution for repair time is given by the following table: Repair time (hours) Probability The average repair time is 60 hours. We are assuming a repair time multiple of 24 hours.

18 Production line maintenance histogram sampling Cumulative probability histograms Operating time between failures F(x) Repair time F(x) 1,0 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0, Nº de horas ,0 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0, Tempo de reparação

19 Production line maintenance results from a simulation run of size 10 Trial Operating Repair Total % of wasted Failures Total cost ( ) time time time time (TP) 8640h TT ( ( ) (hours) (TR) (TT) TR TT % TP) % % % % % % % % % % Médias %

20 Inverse transform sampling When the probability distribution is given in an analytical form P (x) and x varies continuously in the given range. P (x) origin: fitness to experimental values or a priori knowledge of the stochastic parameters of the model e.g. they follow a normal distribution, or a negative exponential distribution, etc. Fitting a continuous distribution to the experimental distribution of the number of cars: Probability function (approximated) P(x) Cumulative probability function F(x) 0,5 0,5 0,4 0,4 0,3 0,3 0,2 0,2 0,1 0,1 0, Nº de carros 1,0 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0, Nº de carros Being known the analytic expression of F (x), we can find a value of x for a given u i [0, 1] by using F (x) inverse: x = F 1 (u i ).

21 Typical distributions sampling 1. Uniform distribution 1/(b-a) P(x) F (x) = = x 1 a b a dx 1 (x a) b a F 1 (x) = (b a)x + a a b x P (x) = 1 b a, a x b x = (b a)u i + a (with u i uniformly distributed over the interval [0,1])

22 Typical distributions sampling 2. Negative exponential distribution P(x) P (x) = λe λx, x > 0 x F (x) = x 0 λe λx dx = 1 e λx F 1 ln(1 x) (x) = λ x = ln(1 u i) λ (with u i uniformly distributed over the interval [0,1])

23 Typical distributions sampling 3. Normal distribution F (x) = 1 2π x e (x µ)2 2σ 2 dx 1 2 P(x) P (x) = 1 (x µ) 2 2π e 2σ 2 x there is no analytical way of computing this integral computing the integral, and necessarily F 1 (x), using numeric methods Box and Muller method x = 2 ln u i cos(2πv i ) (with u i and v i uniformly distributed over the interval [0,1] and x N(0, 1)) To generate a variable z N(µ, σ 2 ): z = µ + σx

24 Distribution sampling convolution method Idea: express the desired sample as a statistical sum (convolution) of samples from other distributions that are easier to sample.

25 Poisson distribution X Number of occurrences per time unit or period t. P (X = k) = e λ t (λ t) k k! com k = 0, 1,..., n Discrete distribution that describes the occurrence of random phenomena along time. t has to fulfill a few conditions (check Probability Theory bibliography). λ is the average number of occurrences per time unit t.

26 Sampling a Poisson distribution It is known that the time t between successive occurrences of a Poisson process with parameter λ (number of occurrences per unit time) follows a negative exponential distribution, also with parameter λ. From the negative exponential distribution we generate a sequence t i and count how many occurrences fit in successive intervals t: t t t t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10 t 11 X=3 X=2 X=5 t

27 Getting results with a simulation model the circle area estimation case How many times should I repeat the random sampling to have a good estimate of the circle area? For sample sizes 78 varying between Área n = 100 and n = 10000, the experience was replicated 10 times 70 for each n, in each replicate being used a different sequence Tamanho da amostra (n) of random numbers. Área exacta Réplica 1 Réplica 2 Média (10 réplicas) Desvio-padrão (10 réplicas) Results of 2 replicas and mean and standard deviation of the 10 replicates.

28 The circle area estimation case conclusions The estimate of the circle area improves as the sample size increases. The average of the 10 replicates for each sample size n gives a better estimate than each of the individual replicas. The precision of the average value of the 10 replicas increases with the sample size n, as is clearly shown by the decrease of the standard deviation.

29 Getting results with a simulation model the production line maintenance case Results for different simulations (average values): Simulation size % of wasted time N. of breakdowns per year Annual cost % % What confidence can I have on the results of a simulation? How large must my sample be (number of attempts and/or replicas) so that the simulation results are reliable? As simulation runs deliver results with random variations, it is essential to express them as confidence intervals.

30 Confidence intervals Parameter estimation Point estimator a function of sample values that gives an approximate value for a parameter of the population distribution. Different samples will lead to different estimates, even using the same estimator. So which of these values match the value of the parameter being estimated? Probably none! Interval estimation Confidence interval = point estimator + interval + probability of having a hit

31 Confidence on the interval We denote by α the percentage of times that, on average, the range does not include (not hits ) the value of the parameter that we want to estimate. To build a confidence interval is then necessary: - An estimator for the parameter under analysis - A particular value of this estimator - The distribution of the estimator

32 Example Confidence intervals for the expected value of a population The most natural estimator for the expected value of a population is the sample mean. In the case of a large sample of a population, regardless of the probability distribution, with expected value µ and standard-deviation σ, the sample mean X follows a normal distribution, which can be converted into a reduced normal distribution: X N(µ, σ2 n ) Z = X µ σ N(0, 1) n

33 In general σ is unknown but it can be replaced with negligible error (because the sample is large), by the value given by the estimator sample standard-deviation : s = 1 n (X i X) n 1 2 Thus, assuming that σ s: i=1 Z = X µ s X µ σ N(0, 1) n n This distribution allows us to calculate the confidence interval for µ at (1 α) 100%, ie, an interval around the sample mean X that, with a probability of (1 α) 100%, contains the true expected value of the population µ.

34 Let z( α ) be a value so that P [Z > 2 z( α )] = α and let z( α ) be the symmetric value for which P [Z < z( α 2 )] = α 2 : Then: P [ z( α 2 ) < Z < z( α 2 )] = 1 α P [ z( α ) < X µ 2 σ < z( α )] = 1 α 2 n P [X z( α 2 ) σ n < µ < X + z( α 2 ) σ n ] = 1 α i.e, the interval [X z( α2 ) σ n, X + z( α2 ) σ n ] is the confidence interval we were looking for. Considering that s approximates well σ: [ X z( α 2 ) s, X + z( α n 2 ) s ] n

35 Sizing samples Associated with a confidence interval there are always two parameters, which depend on sample size: the extent of the confidence interval, which, for the same degree of confidence, will be smaller, the larger the sample; the degree of confidence of the interval that, for the same amplitude, will be greater, the larger the sample. In simulation applications the most frequent situation is that we want to size the sample in order to get a confidence interval with a previously set degree of confidence and an interval amplitude not greater than a given value.

36 Let us take, for example, a confidence interval for the expected value of the population: [ X z( α 2 ) s, X + z( α n 2 ) s ] n Let us suppose that we want a confidence interval at 95%. In this case α/2 = and we would get from a reduced normal distribution table the following value of z: z(0.025) = Problem: we want a value for n, to know how many simulation trials or replicas we should run (the sample size), but to calculate it we need the value of X and s (i.e. the sample results). Solution: run the simulation model with a first small sample, compute the parameters X and s and use them to calculate the size of the sample for the final experiments.

37 Example: from a first sample we got X = 1.70 and s = The confidence interval is then the following: [ , ] n n Finally, imposing a maximum amplitude of, let us say, 0.04, we would be able to compute the sample size that is necessary to fulfill those requirements: n n n 0.04 n 25

38 Bibliography Hillier and Lierberman (2001). Introduction to Operations Research. Mc Graw-Hill. Joseph G. Ecker and Michael Kupferschmid (1988). Introduction to Operations Research. Wiley. Handy A. Taha (1987). Operations Research: an introduction. Prentice Hall. Ravindram, Philips and Solberg (1987). Operations Research: principles and practice. Wiley.