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OutlineOutline

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

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Two Types of SystemsTwo 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

ED

CB

A

N(t)

t, time

…

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Two Types of SystemsTwo 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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Two Types of SystemsTwo Types of Systems Terminating and Non-Terminating 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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Analysis of Terminating SystemsAnalysis of Terminating Systems

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Strong Law of Large Numbers Strong Law of Large Numbers - - Basis to Analyze Terminating SystemsBasis to Analyze Terminating Systems

i.i.d. random variables X1, X2, …

finite mean and variance 2

1)(lim|}{ n

nXP

nXX

nnX ...1 define

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Strong Law of Large Numbers Strong Law of Large Numbers - - Basis to Analyze Terminating SystemsBasis to Analyze Terminating Systems

a fair die thrown continuously Xi = the number shown on the ith throw

?lim n

nX

?3

1lim should Why

otherwise.

},4,3{ if

,0

,11

n

YX

Y

n

ii

n

nn be?What

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Strong Law of Large Numbers Strong Law of Large Numbers - - Basis to Analyze Terminating SystemsBasis to Analyze Terminating Systems

in terminating systems, each replication is an independent draw of X Xi are i.i.d.

E(X) (X1 + … + Xn)/n

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Central Limit Theorem Central Limit Theorem - - Basis to Analyze Terminating SystemsBasis 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

normal standard/

dn

n

X

CLT: approximately normal for “large enough” n can use t, 2, and F for

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Generation of Generation of Random Numbers & Random VariatesRandom Numbers & Random Variates

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To Generate To Generate Random Variates in ExcelRandom 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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To Generate the To Generate the Random MechanismRandom 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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Simulation by Excel Simulation by Excel

for Terminating Systemsfor Terminating Systems

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ExamplesExamples

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?

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Examples: Examples: Probability and Expectation Probability and Expectation of Functions of Random Variablesof Functions of Random Variables

X ~

x: 100 150 200 250 300

p(x): 0.1 0.3 0.3 0.2 0.1

502 2 X Y =

Find E(Y) and P(Y 30)

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Examples: Examples: Probability and Expectation Probability and Expectation of Functions of Random Variablesof 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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Example: Newsboy Problem Example: Newsboy Problem Pieces of “Newspapers” to Order 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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Example: Newsboy Problem Example: Newsboy Problem Pieces of “Newspapers” to Order 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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Example: Newsboy Problem Example: Newsboy Problem Pieces of “Newspapers” to OrderPieces 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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Example: Newsboy Problem Example: Newsboy Problem Pieces of “Newspapers” to Order 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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Simulation by Excel Simulation by Excel

for a Non-Terminating Systemfor a Non-Terminating System

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Simulation a Simulation a GIGI//GG/1 Queue /1 Queue by its Special Propertiesby 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)

average delay = 1/

N

nn

D N

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

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Model 03-01Model 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 pressone type of parts; parts come in and are processed one by one

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Model 03-02 and Model 03-03Model 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