1. MA3264 Mathematical Modelling Lecture 6 Monte Carlo Simulation

2. Motivation assign elevators to even / odd floors? Elevator Service : Should we Experimental Method : May be disruptive since one way streets? use express elevators? Traffic Control : Where should we locate upset customers may take stairs! traffic lights? frustrated drivers may take sidewalks! Therefore decision makers require simulation.

3. Motivation Analogue : wind tunnels Simulation can be Stochastic : Markov Chains, etc. Mathematical Models Digital : computes using mathematical models Computations Deterministic : formuli such as Analytic : directly compute variables or probabilities Monte Carlo : uses random number generator Problem : very often intractible

4. Random Number Generation  gives m x n matrix whose entries are independent random numbers uniformly distributed over the interval [0,1] >> m=4; n = 2; >> rand(m,n) ans = 0.9501 0.8913 0.2311 0.7621 0.6068 0.4565 0.4860 0.0185 >> x = rand(10,1); >> y = zeros(10,1); >> plot(x,y,'r*') >> hold on >> x = rand(10,1); >> plot(x,y,'bo')

5. Random Number Generation >> x = rand(10,1); >> y = rand(10,1); >> plot(x,y,'r*') >> hold on >> x = rand(10,1); >> y = rand(10,1); >> plot(x,y,'bo') >> x = rand(100,1); >> y = rand(100,1); >> plot(x,y,'r*')

6. Random Number Generation

8. Area Under a Curve >> s = 0:.001:1; >> fs = s.^2; >> plot(s,fs,'ko') >> title('f(s) = s^2') >> count = 0; >> for j = 1:10000 if y(j) < x(j)^2 count = count + 1; end >> area_estimate = count/10000 area_estimate = 0.3387 Question What is the area under the curve f(s) = s^2 ?

9. Area Under a Curve >> z = rand(1000000,2); >> count = sum(z(:,2) < z(:,1).^2); >> area_estimate = count/1000000 area_estimate = 0.3329 >> z = rand(1000000,2); >> count = sum(z(:,2) < z(:,1).^2); >> area_estimate = count/1000000 area_estimate = 0.3330 >> z = rand(1000000,2); >> count = sum(z(:,2) < z(:,1).^2); >> area_estimate = count/1000000 area_estimate = 0.3328 Question How does accuracy depend on # points used ? >> for m = 1:100 z = rand(1000000,2); count = sum(z(:,2) < z(:,1).^2); area_estimate(m) = count/1000000; end

10. Volume Inside of a Solid Object Problem Compute the volume of a solid that is formed by intersecting a ball of radius 1 and a cube of diameter 3/2 that has the same center as the ball? Solution Clearly the volume of the cube = 27/8 = 3.750 Therefore is suffices to compute the fraction of random points chosen inside the cube that are inside the sphere. The following MATLAB commands do the computation >> N = 10000000; >> count = 0; >> xyz = 1.5*rand(N,3)-0.75; >> radii = sqrt(xyz(:,1).^2 + xyz(:,2).^2 + xyz(:,3).^2); >> count = sum(radii < 1); >> fraction = count/10000000 fraction = 0.9212 >> volume_estimate = (27/8)*fraction volume_estimate = 3.1092

11. Generating Random Numbers Middle Square Method, invented by Ulam, Metropolis + 1. Start with a four digit number, called the seed. http://en.wikipedia.org/wiki/John_von_Neumann 2. Square it to obtain an eight digit number (add leading zeros if necessary). 3. Take the middle four digits as the next random number. >> x(1) = 3084; >> for k = 1:17 s = x(k)^2; s6 = mod(s,1000000); r6 = s6 - mod(s6,100); x(k+1) = r6/100; end 3084 5110 1121 2566 5843 1406 9768 4138 1230 5129 3066 4003 240 576 3317 24 5 0 Question Whats wrong? Question Where to go?

12. Generating Random Numbers >> x(1) = 962947; >> for k = 1:100000 s = x(k)^2; s9 = mod(s,1000000000); r9 = s9 - mod(s9,1000); x(k+1) = r9/1000; end hist(x/1000000) >> x = rand(100000,1); >> hist(x) Question What did we do? Question What method is better?

13. Generating Random Numbers Linear Congruence Method, invented by 1. Start with a four digit number, called the seed. http://en.wikipedia.org/wiki/Derrick_Henry_Lehmer 2. Apply the transformation Here a, b, c are three required positive integers. The random integers are between 0 and c-1. >> a = 15625; >> b = 22221; >> c = 2^31; >> x(1) = 389274; >> for k = 1:100000 x(k+1)=mod(a*x(k)+b,c); end x = x/(c-1); hist(x)

14. Generating Random Numbers It is often required to compute random numbers that are not uniformly distributed over an interval. Example When a dice is rolled it produces a random number in the set {1,2,3,4,5,6}, it is uniformly distributed over that (discrete) set of points. The following MATLAB commands simulates 100 throws of a dice and computes how many times each value occured. >> for k = 1:100 >> y = rand; >> x(k) = round(6*y+1/2); >> end >> for val = 1:6 >> numbers(val) = sum(x == val); >> end >> numbers numbers = 17 18 19 18 17 11

15. Generating Random Numbers Queueing theory models elevator (lift) and bus service. http://en.wikipedia.org/wiki/Queueing_theory Question How can we generate this random variable using (only) a uniform random number generator? The time between people consecutively arriving at an elevator or a bus stop is a random variable that is exponentially distributed over the interval This means that where is the expected value of

16. Generating Random Numbers Theorem If c > 0 and y is uniformly distributed over [0,1] then x = - c ln(y) is exponentially distributed over Proof The MATLAB code generates 100000 random samples >> c = 1; >> for k = 1:100000 y(k) = rand; x(k) = -c*log(y(k)); end hist(x,100) histogram of

17. Generating Random Numbers Theorem If is uniformly distributed over [0,1] and is independent of and is exponentially distributed with expected value 2 then and and independent and normally distributed with mean 0 and variance 1 Proof The conditions on x and y are satisfied iff

18. Generating Random Numbers The MATLAB commands generate 100000 samples >> for k = 1:50000 u(k) = rand; r(k) = -2*log(rand); x(2*k-1) = sqrt(r(k))*cos(2*pi*u(k)); x(2*k) = sqrt(r(k))*sin(2*pi*u(k)); end hist(x,100)

19. Simulating Choosing m from n People The MATLAB commands below do this % people are called 1, 2, …,n % c = vector containing chosen people % r = vector of people not yet chosen r = 1:n; for k = 1:m % compute random integer between 1 and n-k+1 j = round((n-k+1)*rand+0.5); c (k) = r(j); r(j) = r(n-k+1); r = r(1:n-k); end

20. Suggested Reading&Problems in Textbook 5.1 Area Under a Curve, p 177-181, Prob 5.1 Recommended Websites 5.2 Generating Random Numbers, p 177-181, Prob 5.2 5.3 Simulating Prob. Behavior, p 186-190, Prob 5.3 http://en.wikipedia.org/wiki/Monte_Carlo_method http://en.wikipedia.org/wiki/Random_number_generator http://en.wikipedia.org/wiki/Queueing_theory

21. Tutorial 6 Due Week 6-10 October Page 181. Problem 1. Imagine that every week you purchase lottery tickets until you win a prize. Design and run a simulation program to compute the average number of lottery tickets that you purchase per week. Page 181. Problem 3. Choose a number of random points so that your estimated value of pi is accurate to about 4 significant digits Page 190. Problem 1 (not project 1)

22. Homework 2 Due Friday 10 October 1. Write a computer program to simulate waiting times at a bus station during rush hours using the assumptions and steps below. Run the program 200 times to generate a histogram of the waiting times that people experience and a histogram of the total number of bus arrivals. Explain in detail your logic and algorithms. a. People start to arrive at the bus station at 5:00 at the average rate of 8 per minute. They stop arriving at 7:00. The times between consecutive arrivals is exponentially distributed. For each simulation, first compute an array containing random samples of the times (in order) that people arrive between 5-7PM. b. Buses arrive promptly every 10 minutes starting at 5:10 and continue until the last passenger is picked up. Each bus arrives empty and picks up exactly 60 people. c. Each time a bus arrives the people waiting scramble to board the bus. Simulate this by choosing 60 people randomly, from among those waiting, during each bus arrival.