1 outline terminating and non-terminating systems analysis of terminating systems generation of...
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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