Transcript Simulating by Excel

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
 1 
Two Types of Systems
 Terminating and Non-Terminating
 2 
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
B
A
GI/G/ 1 queue

infinite buffer
service times ~ unif[6, 10]
interarrival times ~ unif[8, 12]
to estimate the
E[# of customers in system]
C
…
N(t)
D
E
t, time
F
 3 
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
 4 
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

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 nonterminating system
 analysis: probability
theory and statistics
related to but not
exactly SLLN, nor CLT

 5 
Analysis of Terminating Systems
 6 
Strong Law of Large Numbers
- Basis to Analyze Terminating Systems
i.i.d. random variables X1, X2, …
 finite mean  and variance 2


define X n 

X1 ...  X n
n

P {} | lim X n ()    1
n
 7 
Strong Law of Large Numbers
- Basis to Analyze Terminating Systems
a fair die thrown continuously
 Xi = the number shown on the ith throw

lim X n  ?
n 
n
Y
1, if X n {3,4},
What
Yn  
Why should lim i 1
n 
n
0, otherwise.
 8 
i
1
be??
3
Strong Law of Large Numbers
- Basis to Analyze Terminating Systems

in terminating systems, each replication is an
independent draw of X
 Xi
 E(X)
are i.i.d.
 (X1 + … + Xn)/n
 9 
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 X1, X2, … of finite mean  and
variance 2
Xn  d

 standard normal
/ n

CLT: approximately normal for “large enough” n
 can use t, 2, and F for
 10 
Generation of
Random Numbers & Random Variates
 11 
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)?
 12 
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

 13 
Simulation by Excel
for Terminating Systems
 14 
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
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?
 15 
Examples: Probability and Expectation
of Functions of Random Variables
X
~
x:
p(x):
Y
=
 Find
100 150 200 250 300
0.1 0.3 0.3 0.2 0.1
2 X 2  50
E(Y) and P(Y  30)
 16 
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)
 17 
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:
Prob.
:
100
150
0.3
0.2

objective: profit maximization

questions
200
250
0.3 0.15

how many calendars to order

with the optimal order quantity, P(profit  400)
 18 
300
0.05
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)
 19 
Example: Newsboy Problem
 Pieces of “Newspapers” to Order
 two-step
 1
solution procedure
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
look for Q*, which gives the largest
mean profit
 2
 20 
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
 21 
Simulation by Excel
for a Non-Terminating System
 22 
Simulation a GI/G/1 Queue
by 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)
N

average delay =  Dn / N
n 1
 23 
Arena Model 03-1,
Model 03-02, Model 03-03
 24 
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
one type of parts; parts come in
and are processed one by one
 25 
a drill press
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
 26 