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1 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

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2 Two Types of Systems Terminating and Non-Terminating

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3 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 GI/G/ 1 queue infinite buffer service times ~ unif[6, 10] interarrival times ~ unif[8, 12] to estimate the E[# of customers in system] F E D C B A N(t)N(t) t, time …

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4 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 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

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5 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 non- terminating system analysis: probability theory and statistics related to but not exactly SLLN, nor CLT

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6 Analysis of Terminating Systems

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7 Strong Law of Large Numbers - Basis to Analyze Terminating Systems i.i.d. random variables X 1, X 2, … finite mean and variance 2 define

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8 Strong Law of Large Numbers - Basis to Analyze Terminating Systems a fair die thrown continuously X i = the number shown on the ith throw be? What

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9 Strong Law of Large Numbers - Basis to Analyze Terminating Systems in terminating systems, each replication is an independent draw of X X i are i.i.d. E(X) (X 1 + … + X n )/n

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10 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 X 1, X 2, … of finite mean and variance 2 CLT: approximately normal for “ large enough ” n can use t, 2, and F for

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11 Generation of Random Numbers & Random Variates

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12 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) ?

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13 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

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14 Simulation by Excel for Terminating Systems

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15 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 101520 25 30 p: 0.10.150.30.20.140.11 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?

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16 Examples: Probability and Expectation of Functions of Random Variables X ~ x: 100 150 200 250 300 p(x): 0.1 0.3 0.3 0.2 0.1 Y = Find E(Y) and P(Y 30)

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17 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)

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18 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: 100 150 200 250 300 Prob. : 0.3 0.2 0.3 0.15 0.05 objective: profit maximization questions how many calendars to order with the optimal order quantity, P(profit 400)

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19 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)

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20 Example: Newsboy Problem Pieces of “ Newspapers ” to Order two-step solution procedure 1 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 2 look for Q *, which gives the largest mean profit

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21 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

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22 Simulation by Excel for a Non-Terminating System

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23 Simulation a GI/G/1 Queue by its Special Properties D n = delay time of the nth customer; D 1 = 0 S n = service time of the nth customer T n = inter-arrival time between the nst and the (n+1)st customer D n+1 = [D n + S n - T n ] +, where [ ] + = max( , 0) average delay =

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24 Arena Model 03-1, Model 03-02, Model 03-03

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25 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 a drill press one type of parts; parts come in and are processed one by one

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26 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

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