 1  Outline  terminating and non-terminating systems  theories for output analysis  Strong Law of Large Numbers  Central Limit Theorem  Regenerative Processes  random variate generation from Excel  simulating with Excel  simple random variables  functions of random variables  newsboy problem

 2  Two Book Stores  Bookstore A  from 8 am to 8 pm  inventory count at 8 pm  order up to B  leadtime: 12 hours  customer arrivals: Poisson process of rate, each for one book  loss sales  Bookstore B  24 hours every day  continuous review  order up to B  leadtime: 12 hours  customer arrivals: Poisson process of rate, each for one book  loss sales

 3  Book Store A B books I 1 (12) I 2 (12)  beginning of Day 1 and Day 2: similar  I 1 (12) and I 2 (12): i.i.d. Day 1Day 2

 4  Book Store B B books X books I 1 (12) I 2 (12)  X and B have different distributions  dependence: X & I 1 (12); X & I 2 (12)  I 1 (12) & I 2 (12) Day 1Day 2

 5  Two Book Stores  first bookstore  from 8 am to 8 pm  I i (12) ~ i.i.d.  ~ i.i.d.  terminating  second bookstore  24 hrs every day  I i (12) dependent, different distributions  dependent, different distributions  non-terminating

 6  Two Book Stores  differences  termination condition  run length  quantities of interest  initial condition

 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

 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  Output Analysis – Terminating Systems  n replications; sample space for ith replication  i   =  1  n ; sample space of the whole experiment   = (  1, ,  n ), where  i is outcome of the ith replication  sampled values: X 1 (  1 ), …, X n (  n )  estimate  = E(X)  estimate by  estimate  by= g(X 1 (  1 ), …, X n (  n ))

 12   unbiased estimator of  ?  variance of estimator  efficient estimator of  ?  confidence on the range estimator  # of simulation runs (replications) required? Output Analysis – Terminating System  statistical tests for

 13  Output Analysis – Non-Terminating System  similar questions as terminating systems  non-terminating  possibly with dependent random variables  the mean and probability of quantities that follow the stationary (limiting) distribution

 14  Theory of Regenerative Processes - Basis to Analyze Non-Terminating Systems  Cho-Free low-fat chocolate milk in a 24-hour store  order 20 bottles when out of stock  order lead-time = four hours  shelf life of each bottle = one week  expired milk: thrown away immediately  customer arrivals: Poisson process of rate  unsatisfied customers never return  what probability & expectation are we talking about?  P(store is out of stock of Cho-Free)  E(number of bottles in the store)

 15  Theory of Regenerative Processes - Basis to Analyze Non-Terminating Systems  the actual story: X t = number of bottles in the store at t  distribution of X t  a function of t  converges to that of X   questions of interest: with respect to X   P(X  = k), E(X  )

 16  Theory of Regenerative Processes - Basis to Analyze Non-Terminating Systems  {B(t)} is a regenerative process if there exists a non-negative random variable T  {B(t)| 0  t < T} is independent of {B(t)| T  t}  {B(t)| 0  t} and {B(t)| t  T} are stochastically equivalent  most practical systems are regenerative

 17  Theory of Regenerative Processes - Basis to Analyze Non-Terminating Systems  for a regenerative process {B(t)}, under mild conditions  for the 24-hour store

 18  Theory of Regenerative Processes - Basis to Analyze Non-Terminating Systems  in practice, simulate one long run  questions  unbiased estimator, variance of estimator, confidence interval, interval estimator  untouched questions  dependent random variables  imprecise result for finite simulation time  biased by initial condition

 19  To Generate the Random Mechanism  will discuss later, general overview  everything based on random variates from uniform (0, 1)  often each stream of uniform (0, 1) random variates is a deterministic sequence of numbers on a round robin  “ first ” number in the robin to use: SEED  many simple, handy generators

 20  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) ?

 21  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:

 22  Examples: Probability and Expectation of Random Variables  Example 4. Use simulation to estimate  (a) P(X > 0.5)  (b) P(2 < Y < 8)  (c) E(Z)  questions before solving  terminating or non-terminating?  which theorem to base on?  state?

 23  Examples: Probability and Expectation of Random Variables  use  10 replications  50 replications  500 replications  5000 replications  accuracy?

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

 25  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)

 26  Example: Newsboy Problem - # of “ Newspapers ” to Order  order 2010 calendars in Sept 2009  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)

 27  Example: Newsboy Problem - # of “ Newspapers ” to Order  D = the demand of the 2007 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)

 28  Example: Newsboy Problem - # 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 largest mean profit

 29  Example: Newsboy Problem - # 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

 30  Exercise  situation similar to the example  salvage value  = 0 for the first 50 pieces  = $0.75 / piece from the 51 st piece onwards  questions  find Q *  P(profit  400)