 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

 2  Two Types of Systems  Terminating and Non-Terminating

 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 …

 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

 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

 6  Analysis of Terminating Systems

 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

 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

 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

 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

 11  Generation of Random Numbers & Random Variates

 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) ?

 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

 14  Simulation by Excel for Terminating Systems

 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: p:  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?

 16  Examples: Probability and Expectation of Functions of Random Variables  X ~ x: p(x):  Y =  Find E(Y) and P(Y  30)

 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)

 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 : $0.75 each  from historical data: demand for new calendars Demand: Prob. :  objective: profit maximization  questions  how many calendars to order  with the optimal order quantity, P(profit  400)

 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) max (0, Q - D) - 2Q  objective: find Q * to maximize E(V)

 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

 21  Example: Newsboy Problem  Pieces of “ Newspapers ” to Order  our simulation of 1000 samples,  Q = 100: E(V) = 250  Q = 150: E(V) =  Q = 200: E(V) =  Q = 250: E(V) =  Q = 300: E(V) =  Q * = 200 is optimal  remarks: many papers on this issue

 22  Simulation by Excel for a Non-Terminating System

 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 =

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

 25  Model  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

 26  Model and Model  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