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Outline terminating and non-terminating systems analysis of terminating systems generation of random numbers simulation by Excel a terminating system a non-terminating system basic operations in Arena 1 Two Types of Systems Terminating and Non-Terminating 2 Two Types of Systems chess piece starts at vertex F moves equally likely to adjacent vertices to estimate E(# of moves) to reach the upper boundary B A GI/G/ 1 queue infinite buffer service times ~ unif[6, 10] interarrival times ~ unif[8, 12] to estimate the E[# of customers in system] C … N(t) D E t, time F 3 Two Types of Systems chess piece initial condition defined by problem termination of a simulation run defined by the system estimation of the mean or probability of a random variable run length defined by number of replications 4 GI/G/ 1 queue initial condition unclear termination of a simulation run defined by ourselves estimation of the mean or probability of the limit of a sequence of random variables run length defined by run time Two Types of Systems Terminating and Non-Terminating chess piece: a terminating systems analysis: Strong Law of Large Numbers (SLLN) and Central Limit Theorem (CLT) GI/G/ 1 queue: a nonterminating system analysis: probability theory and statistics related to but not exactly SLLN, nor CLT 5 Analysis of Terminating Systems 6 Strong Law of Large Numbers - Basis to Analyze Terminating Systems i.i.d. random variables X1, X2, … finite mean and variance 2 define X n X1 ... X n n P {} | lim X n () 1 n 7 Strong Law of Large Numbers - Basis to Analyze Terminating Systems a fair die thrown continuously Xi = the number shown on the ith throw lim X n ? n n Y 1, if X n {3,4}, What Yn Why should lim i 1 n n 0, otherwise. 8 i 1 be?? 3 Strong Law of Large Numbers - Basis to Analyze Terminating Systems in terminating systems, each replication is an independent draw of X Xi E(X) are i.i.d. (X1 + … + Xn)/n 9 Central Limit Theorem - Basis to Analyze Terminating Systems interval estimate & hypothesis testing of normal random variables t, 2, and F i.i.d. random variables X1, X2, … of finite mean and variance 2 Xn d standard normal / n CLT: approximately normal for “large enough” n can use t, 2, and F for 10 Generation of Random Numbers & Random Variates 11 To Generate Random Variates in Excel for uniform [0, 1]: rand() function for other distributions: use Random Number Generator in Data Analysis Tools uniform, discrete, Poisson, Bernoulli, Binomial, Normal tricks to transform uniform [-3.5, 7.6]? normal (4, 9) (where 4 is the mean and 9 is the variance)? 12 To Generate the Random Mechanism general overview, with details discussed later this semester everything based on random variates from uniform (0, 1) each stream of uniform (0, 1) random variates being a deterministic sequence of numbers on a round robin “first” number in the robin to use: SEED many simple, handy generators 13 Simulation by Excel for Terminating Systems 14 Examples Example 1: Generate 1000 samples of X ~ uniform(0,1) Example 2: Generate 1000 samples of Y ~ normal(5,1) Example 3: Generate 1000 samples of Z ~ z: 5 10 15 20 25 30 p: 0.1 0.15 0.3 0.2 0.14 0.11 Example 4. Use simulation to estimate (a) P(X > 0.5) (b) P(2 < Y < 8) (c) E(Z) Using 10 replications, 50 replications, 500 replications, 5000 replications. Which is more accurate? 15 Examples: Probability and Expectation of Functions of Random Variables X ~ x: p(x): Y = Find 100 150 200 250 300 0.1 0.3 0.3 0.2 0.1 2 X 2 50 E(Y) and P(Y 30) 16 Examples: Probability and Expectation of Functions of Random Variables X ~ N(10, 4), Y ~ N(9,1), independent estimate P(X < Y) Cov(X, Y) = E(XY) - E(X)E(Y) 17 Example: Newsboy Problem Pieces of “Newspapers” to Order order 2012 calendars in Sept 2011 cost: $2 each; selling price: $4.50 each salvage value of unsold items at Jan 1 2012: $0.75 each from historical data: demand for new calendars Demand: Prob. : 100 150 0.3 0.2 objective: profit maximization questions 200 250 0.3 0.15 how many calendars to order with the optimal order quantity, P(profit 400) 18 300 0.05 Example: Newsboy Problem Pieces of “Newspapers” to Order D = the demand of the 2012 calendar D follows the given distribution Q = the order quantity {100, 150, 200, 250, 300} V = the profit in ordering Q pieces = 4.5 min (Q, D) + 0.75 max (0, Q - D) - 2Q objective: find Q* to maximize E(V) 19 Example: Newsboy Problem Pieces of “Newspapers” to Order two-step 1 solution procedure estimate E(profit) for a given Q generate demands find the profit for each demand sample find the (sample) mean profit of all demand samples look for Q*, which gives the largest mean profit 2 20 Example: Newsboy Problem Pieces of “Newspapers” to Order our simulation of 1000 samples, Q = 100: E(V) = 250 Q = 150: E(V) = 316.31 Q = 200: E(V) = 348.31 Q = 250: E(V) = 328.75 Q = 300: E(V) = 277.17 Q* = 200 is optimal remarks: many papers on this issue 21 Simulation by Excel for a Non-Terminating System 22 Simulation a GI/G/1 Queue by its Special Properties Dn = delay time of the nth customer; D1 = 0 Sn = service time of the nth customer Tn = inter-arrival time between the nst and the (n+1)st customer Dn+1 = [Dn + Sn - Tn]+, where []+ = max(, 0) N average delay = Dn / N n 1 23 Arena Model 03-1, Model 03-02, Model 03-03 24 Model 03-01 a drill press processing one type of product interarrival times ~ i.i.d. exp(5) service times ~ i.i.d. triangular (1,3,6) all random quantities are independent one type of parts; parts come in and are processed one by one 25 a drill press Model 03-02 and Model 03-03 Model 03-02: sequential servers Alfie checks credit Betty prepares covenant Chuck prices loan Doris disburses funds Model 03-03: parallel servers Each employee can do any tasks 26