Transcript Ch.1 Basic Simulation Modeling

Ch. 1 Basic Simulation
Modeling
1
Notas
http://delta.cs.cinvestav.mx/~lixo/teaching
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Basic Simulation Modeling
•
•
•
•
The nature of simulation
Systems, Models and Simulation
Discrete-Event Simulation
Simulation of a single-server queuing
system
• Steps in a sound simulation study
• Advantages, disadvantages and pitfalls of
simulation
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The Nature of Simulation
•
•
•
•
Conceptions
Application areas
Academic level
Impediments
4
Conceptions
• Simulation course is about techniques for using computers to
imitate or simulate the operations of various kinds of real
world facilities or processes
• System: the facility or process of interest
• Model: a set of assumptions about how the system works,
which usually take the form of mathematical or logical
relationships, constitute a model that is used to try to gain
more understanding of how the corresponding system
behaves.
• Analytic soulution: to obtain exact information on questions of
intresets.
• Simulation: use a computer to evaluate a model numerically,
and data are gathered in order to estimate the desired true
characteristics of the model.
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Example
• a manufacturing company contemplates
building a large extension onto one of its
plants, but is not sure if the potential gain
in
productivity
would
justify
the
construction cost.
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Application areas
• Designing and analyzing manufacturing systems
• evaluating military weapons systems or their logistics
requirements
• determining hardware requirements or protocols for
communication networks
 Determining hardware and software requirements for a
computer system
 Designing and operating transportation systems such as
airports, freeways, ports and subways
 Evaluating designs for service organizations such as call
centers, fast-food restaurants, hospitals, and post offices
 Reengineering of business processes
 Determining ordering polices for an inventory system
 Analyzing financial or economic systems
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Academic level
 Winter Simulation Conference (600-700 people
every year)
 Is one of the three important operationsresearch techniques (in serveys related to the
use of operations research techniques: math
programming, statistics, simulation)
 The second only to “math programming” among
13 techniques considered (in 1294 papers from
the journal Interfaces from 1970 through 1992)
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Impediments
 Models used to study large-scale systems tend
to be very complex, and writing computer
programs to execute them can be an arduous
task indeed. (excellent software products)
 Large amount of computer time is sometimes
required. (cheaper and faster computer)
 An unfortunate impression that simulation is just
an exercise in computer programming, albeit a
complicated one. (attitude, simulation
methodology)
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Systems, Models and Simulation
• System is defined to be a collection of entities, e.g.,
people or machines, which act and interact together
toward the accomplishment of some logical end.
• System depends on the objectives of a particular
study.
• State of a system: collection of variables necessary
to describe a system at a particular time, relative to
the objectives of a study. (the number of busy tellers,
the number of customers in the bank, the time of
arrival of each customer in the bank)
– discrete system: the state variables change instantaneously at
separated points in time. (a bank, e.g., the number of customers in the
bank)
– continuous system: the state variables change continuously with
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respect to time. (an airplane moving through the air, e.g., position and
Continue...
• Study on a system: try to gain some insight
into the relationships among various
components, or to predict performance under
some new conditions being considered.
• Ways to study a system
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System
Experiment
with a model
of the system
Experiment
with the
actual system
Mathematical
model
Physical
model
Analytical
solution
Simulation
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Example
 One study on a bank to determine the
number of tellers needed to provide
adequate service for customers who want
just to cash a check or make a savings
deposite, the system can be defined to be
that portion of the bank consisting of the
tellers and the customers waiting in line or
being served.
 If the loan officer and the safety deposite
boxes are to be included, the definition of
the system must be expanded in an
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obvious way.
Discrete Event Simulation
• Definition
• Time-Advance Mechanisms
• Components and Organization of a
Discrete -Event Simulation Model
– Components
– Logic Organization
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Definitions
• Discrete-event simulation concerns the
modeling of a system as it evolves over
time by a representation in which the state
variables change instantaneously at
separate points in time. Or the system
can change at only a countable number of
points in time.
• Event is defined as an instantaneous
occurrence that may change the state of
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the system.
Example 1.1
• Single-server queuing system: a
barbershop, to estimate the (expected)
average delay in queue (line) of arriving
customers
– State variables: the status of the server (busy or
idle), the number of customers waiting in queue
to be served, the time of arrival of each person
waiting in queue.
– Events: the arrival of a customer and the
completion of service for a customer, which
results in the customer’s departure.
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Time-Advance Mechanism
• Simulation clock: the variable in a simulation model
that gives the current value of simulated time.
– to keep track of the current value of simulated time as the
simulation proceeds
– to advance simulated time from one value to another
• Advancing the simulation clock
– next-event time advance (mostly used)
– fixed-increment time advance (a special case of the first)
• Next-event time-advance approach
– simulation clock is initialized to zero
– the times of occurrence of future events are determined.
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Example 1.2
• Notations:
– : time of arrival of the ith customer ()
– : interarrival time between (i-1)st and ith arrivals
of customers
– : time that server actually spends serving ith
customer (exclusive of customer’s delay in
queue)
– : delay in queue of ith customer
– : time that ith customer completes service and
departs
– : time of occurrence of ith event of any type (ith
value the simulation clock takes on, excluding 18
e1
e0
0
e2 e3
t1
A1
t2
A2
S1
c1
e4
e5
t3
c2
Time
A3
S2
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Components and Organization of a
Discrete -Event Simulation Model
• Components (10)
 Systems state: The collection of state variables
necessary to describe the system at a particular
time
 Simulation clock: A variable giving the current
value of simulated time
 Event list: A list containing the next time when
each type of event will occur
 Statistical counters: Variables used for storing
statistical information about system performance
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Continue...
 Initialization routine: A subprogram to initialize the
simulation model at time 0
 Timing routine: A subprogram that determines the
next event from the event list and then advances
the simulation clock to the time when that event is
to occur
 Event routine: A subprogram that updates the
system state when a particular type of event occurs
(there is one event routine for each event type)
 Library routines: A set of subprograms used to
generate random observations from probability
distributions that were determined as part of the
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simulation model
Continue...
 Report generator: A subprogram that computes
estimates (from the statistical counters) of the
desired measures of performance and produces a
report when the simulation ends
 Main program: A subprogram that invokes the
timing routine to determine the next event and then
transfers control to the corresponding event routine
to update the system state appropriately. The main
program may also check for termination and invoke
the report generator when the simulation is over.
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Start
Initialization routine
1. Set simulation
0
clock=0
2. Initialize system state
and statistical counters
3. Initialize event list
Main program
Time routine
0. Invoke the initialization routine
1. Invoke the timing routine Repeatedly
2. Invoke event routine
1
i
1. Determine the next
event type, say, i
2. Advance the
simulation clock
2
Event routine i
1.Update system state
2.Update statistical counters
3.Generate future events and add to
event list
Is
simulation
over?
Library routines
Generate random
variates
No
Report generator
Yes
1. Compute estimates of interest
2. Write report
Stop
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Simulation of A Single-Server
Queueing System
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Components of a queueing system
1.
Arrival process:
•
•
•
2.
Service mechanism
•
•
•
•
3.
Interarrivals: Ai
Mean interarrival time: E(A)
Arrival rate:   1 / E ( A)
Service time: Si
Number of servers: s
Mean service time: E(S)
Service rate:   1 / E ( S )
Queue discipline
•
•
•
FIFO
LIFO
Priority
* D. Gross, and C.M. Harris, Fundamentals of Queueing Theory, 3d ed., John Wiley, New York,
1998
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Notation for queueing systems
1.
2.
3.
4.
s servers in parallel and one FIFO queue feeding all servers
A1, A2, are IID random variables
S1, S2,…are IID random variables
The Ai’s and Si’s are independent
1
2
. . .
s
FIGURE 1.71
A GI/G/s queue
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GI/G/s queue
• GI (general independent): the distribution of the Ai’s
• G (general): the distribution of the Si’s
• Symbols: M (exponential), Ek (k-Erlang), D
(deterministic times)
• M/M/1: a single-server queueing system with
exponential interarrival and service times and a FIFO
discipline
•    /( s) : the utilization factor of the queueing
system
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A departure customer
Server
Customer in service
Customers in queue
An arrival customer
FIGURE 1.4
A single-server queueing system.
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Measures of Performance for Queueing Systems
Di : delay in queue of ith customer
Wi  Di  Si : waiting time in system of ith customer
Q (t ) : number of customers in queue at time t
L(t ) : number of customers in system at time t [Q(t) plus number of customers being served at time t]
n
d  lim
D
i
i 1
n 
w.p. 1
n
: steady-state average delay
n
w  lim
W
n 
Q  lim
T 
L  lim
T 
i
i 1
w.p. 1 : steady-state average waiting time
n

T
0

T
0
Q (t ) dt
T
L(t ) dt
T
w.p.1
w.p.1
Q  d and L  w
: steady-state time-average number in queue
: steady-state time-average number in system
w  d  E (S )
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e1, e2, …, e13 : 13 successive events
n=6 : delays in queue
Interarrival times:
A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2,
A6=1.6, A7=0.2, A8=1.4, A9=1.9, …
Service times:
S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6, …
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Measures of Performance for the example
Di : delay in queue of ith customer
pi : the expected proportion of the time that Q(t) is equal to i
Q (t ) : number of customers in queue at time t
The expected average delay in queue
n
 Di
dˆ ( n ) 
i 1
n
The expected average number of customers in queue

q( n )   ipi
i 0

qˆ ( n )   ipˆ i
i 0

 iT 
qˆ (n ) 

i 0
T(n)
i
T (n)
0
Q(t)dt
T (n)
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Q(t)
3
2
1
1
0
e1=0.4
Arrivals
2
3
4
e2=1.6
e8=4.0
e3=2.1
e7=3.8
5
6
e11=5.8
e10=5.6
7
8
9
t
e12=7.2
e6=3.3
Departures
e4=2.4 e5=3.1
e9=4.9
e13=8.6=T(6)
FIGURE 1.5
Q(t), arrival times, and departure times for a realization of a single-server queueing system.
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Measures of Performance for the example
1 if theserver is busy at timet
B( t )  
0 if theserver is idle at timet
The expected utilization of the server

uˆ (n) 
T (n)
0
B(t )dt
T (n)
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B(t)
1
1
0
e1=0.4
Arrivals
2
3
4
e2=1.6
e8=4.0
e3=2.1
e7=3.8
5
6
e11=5.8
e10=5.6
7
8
9
t
e12=7.2
e6=3.3
Departures
e4=2.4 e5=3.1
e9=4.9
e13=8.6=T(6)
FIGURE 1.6
B(t), arrival times, and departure times for a realization of a single-server queueing system.
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Intuitive Explanation
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Initialization
time = 0
System state
A 0.4
0
D
Clock
0
Sever
status
System
0
Event list
0
Number
in
Times
queue
of
arrival
Time
of last
event

Statistical counters
0
0
0
0
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(a)
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Arrival
time = 0.4
System state
A 1.6
0.4
0.4
Clock
1
Sever
status
System
D 2.4
0
Event list
0.4
Number
in
Times
queue
of
arrival
Time
of last
event
Statistical counters
1
0
0
0
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(b)
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Arrival
time = 1.6
System state
A 2.1
1.6
1.6
0.4
1
1.6
System
Sever
status
D 2.4
Clock
1
Event list
1.6
Number
in
Times
queue
of
arrival
Time
of last
event
Statistical counters
1
0
0
1.2
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(c)
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Arrival
time = 2.1
System state
A 3.8
2.1
0.4
1
1.6
2.1
System
Sever
status
2
1.6
2.1
D 2.4
Clock
Event list
2.1
Number
in
Times
queue
of
arrival
Time
of last
event
Statistical counters
1
0
0.5
1.7
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(d)
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Departure
time = 2.4
System state
A 3.8
2.4
2.1
1.6
1
2.1
System
Sever
status
D 3.1
Clock
1
2.4
Number
in
Times
queue
of
arrival
Time
of last
event
Event list
Statistical counters
2
0.8
1.1
2.0
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(e)
40
Departure
time = 3.1
System state
A 3.8
3.1
2.1
Clock
1
Sever
status
System
D 3.3
0
3.1
Number
in
Times
queue
of
arrival
Time
of last
event
Event list
Statistical counters
3
1.8
1.8
2.7
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(f)
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Departure
time = 3.3
System state
A 3.8
3.3
D
Clock
0
Sever
status
System
0
3.3
Number
in
Times
queue
of
arrival
Time
of last
event

Event list
Statistical counters
3
1.8
1.8
2.9
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(g)
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Arrival
time = 3.8
System state
A 4.0
3.8
3.8
Clock
1
Sever
status
System
D 4.9
0
3.8
Number
in
Times
queue
of
arrival
Time
of last
event
Event list
Statistical counters
4
1.8
1.8
2.9
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(h)
43
Departure
time = 4.0
System state
A 5.6
4.0
4.0
3.8
1
4.0
System
Sever
status
D 4.9
Clock
1
4.0
Number
in
Times
queue
of
arrival
Time
of last
event
Event list
Statistical counters
4
1.8
1.8
3.1
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(i)
44
Departure
time = 4.9
System state
A 5.6
4.9
4.0
Clock
1
Sever
status
System
D 8.6
0
4.9
Number
in
Times
queue
of
arrival
Time
of last
event
Event list
Statistical counters
5
2.7
2.7
4.0
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(j)
45
Arrival
time = 5.6
System state
A 5.8
5.6
5.6
4.0
1
5.6
System
Sever
status
D 8.6
Clock
1
5.6
Number
in
Times
queue
of
arrival
Time
of last
event
Event list
Statistical counters
5
2.7
2.7
4.7
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(k)
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Arrival
time = 5.8
System state
A 7.2
5.8
4.0
1
5.6
5.8
System
Sever
status
2
5.6
5.8
D 8.6
Clock
5.8
Number
in
Times
queue
of
arrival
Time
of last
event
Event list
Statistical counters
5
2.7
2.9
4.9
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(l)
47
Arrival
time = 7.2
System state
A 9.1
7.2
4.0
1
5.6
5.8
7.2
System
Sever
status
3
5.6
5.8
7.2
D 8.6
Clock
7.2
Number
in
Times
queue
of
arrival
Time
of last
event
Event list
Statistical counters
5
2.7
5.7
6.3
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(m)
48
Departure
time = 8.6
System state
A 9.1
8.6
5.6
1
5.8
7.2
System
Sever
status
2
5.8
7.2
D 9.2
Clock
8.6
Number
in
Times
queue
of
arrival
Time
of last
event
Event list
Statistical counters
6
5.7
9.9
7.7
Number Total
Area
Area
delayed delay under Q(t) under B(t)
Computer Representation
(n)
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Comments
• Interaction between the simulation clock and
the event list
• Process updates of the state variables and
statistical counters
• Contingencies
– A departing customer could leave behind an empty
queue
– Termination conditions
• Time ties
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1.4.3 Program Organization and Logic
• Why general-purpose language
• Model being simulated: M/M/1 queue
– n=1000
– Interarrival and service times
• Interarrival: exponential distribution (mean=1 min)
• Service: exponential distribution (mean=0.5 min)
f ( x) 
•
1

e
x

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Arrival
Event
FIGURE 1.8
Flowchart for arrival routine, queueing model
Schedule the next
arrival event
Yes
Is
the server
busy?
Set delay=0
for this customer
and gather statistics
Add 1 to the
number in queue
Write error
Yes
message and stop
simulation
No
Add 1 to the
number of
customers delayed
Is
the queue
full?
No
Make the
server busy
Store time of
arrival of this
customer
Schedule a
departure event
for this customer
Return
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Departure
Event
Yes
Is
the queue
empty?
FIGURE 1.9
Flowchart for arrival routine, queueing model
No
Make the server idle
Subtract 1 from
the number
in queue
Eliminate departure
event from
consideration
Compute delay of
customer entering service
and gather statistics
Add 1 to the
number of customers
delayed
Schedule a
departure event
for this customer
Move each customer
in queue (if any) up
one place
Return
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C Program for Queueing Model
(Fig 1.19 – Fig. 1.27)
•
•
•
•
•
•
•
•
•
External definitions
Main functions
Function initialize
Function timing
Function arrive
Function departure
Function report
Function update_avg_stats
Function expon
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Simulation Output and Discussion
(Fig. 1.28 Output report, queueing model)
Single-server queueing system
Mean interarrival time
Mean service time
Number of customers
1.000 mins
0.500 mins
1000
Average delay in queue
Average number in queue
Server utilization
Time simulation ended
0.430 mins
0.418
0.460
1027.915 mins
55
Alternative Stopping Rules
(Fig. 1.33 –Fig. 1.37)
Stop: fixed run time 480 mins
•
•
•
•
•
External definitions
Main functions
Function initialize
Function report
Output report
56
Determining the Events and Variables
Event-graph
Arrival
Departure
FIGURE 1.38
Event graph, queueing model
57
Determining the Events and Variables
Event-graph
Arrival
Enter
service
Departure
FIGURE 1.39
Event graph, queueing model with separate “enter-service” event
58
Determining the Events and Variables
Event-graph
Arrival
Departure
End
simulation
FIGURE 1.40
Event graph, queueing model with fixed run length
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STEPS IN A SOUND SIMULATION STUDY
1. Formulate the problem and plan the study
a. Problem of interest is stated by manager.
b. One or more kickoff meeting for the study are conducted,
with the project manager, the simulation analysts, and
subject-matter experts (SMEs) in attendance. The
following issues are discussed:
• Overall objectives of the study
• Specific questions to be answered by the study
• Performance measures that will be used to evaluate the effeciency
of different system configurations
• Scope of the model
• System configurations to be modeled
• Software to be used
• Time frame for the study and the required resources
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STEPS…
2. Collect data and define a model.
a. Collect information on the system layout and operating procedures.
•
•
•
No single person or document is sufficient.
Some people may have inaccurate information---make sure that ture SMEs are identified.
Operating procedures may not be formalized.
b. Collect data (if possible) to specify model parameters and input probability distributions
c. Delineate the above information and data in an “assumptions document,” which is the
conceptual model
d. Collect data (if possible) on the performance of the existing systsem (for validation purpose
in Step 6)
e. The level of model detail should depend on the following:
•
•
•
•
•
•
•
Project objectives
Performance measures
Data availability
Credibility concerns
Computer constraints
Opinions of SMEs
Time and money constraints
f.
There need not be a one-to-one correspondence between each element of the model and
the corresponding element of the system.
g. Interact with the manager (and other key project personnel) on a regular basis
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STEPS…
3. Is the conceptual model valid?
a. Perform a structured walk-through of the conceptual model using the
assumptions document before an audience of managers, analysts, and
SMEs
•
•
•
Helps ensure that the model’s assumptions are correct and complete
Promotes ownership of the model
Take place before programming begins to avoid significant
reprogramming later
4. Construct a computer program and verify.
a. Program the model in a programming language or in simulation
software. Benefits of using a programming language are that one is
often known, they have a low purchase cost, and they may result in a
smaller model execution time. The use of simulation software, on the
other hand, reduces programming time and results in a lower project
cost.
b. Verify (debug) the simulation computer program.
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STEPS…
5. Make pilot runs.
a. Make pilot runs for validation purposes in Step 6.
6. Is the programed model valid?
a. If ther is an existing system, then compare model and system (from
Step 2) performance measures for existing system
b. Regardless of whether there is an existing system, the simulation
analysts and SMEs should review the model results for correctness.
c. Use sensitivity analyses to determine what model factors have a
significant impact on performance measures and, thus, have to be
modeled carefully.
7. Design experiments.
a. Specify the following for each system configuration of interest:
• Length of each run
• Length of the warmup period, if one is appropriate
• Number of independent simulation runs using different random numbers--facilitates construction of confidence intervals
63
STEPS…
8. Make production runs.
a. Production runs are made for use in Step 9.
9. Analyze output data.
a. Two major objectives in analyzing output data are:
• Determining the absolute performance of certain system configurations
• Comparing alternative system configurations in a relative sense
10. Document, present, and use results.
a. Document assumptions (see Step 2), computer program, and study’s
results for use in the current and future projects.
b. Present study’s results.
• Use animation to communicate model to managers and other people who
are not familiar with all of the model details.
• Discuss model building and validation process to promote credibility.
c. Results are used in decision-making process if they are both valid and
credible.
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Formulate problem
and plan the study
Collect data and
define a model
Conceptual
model valid?
No
Yes
Construct a computer
program and verify
Make pilot runs
Programmed
Model valid?
No
Yes
Design experiments
Make production runs
FIGURE 1.68
Steps in a simulation study.
Analyze output data
Document, present,
and use results
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ADVANTAGES OF SIMULATION
• Most complex, real-world systems with stochastic elements cannot be
accurately described by a mathematical model that can be evaluated
analytically. Thus, a simulation is often the only type of investigation
possible.
• Simulation allows one to estimate the performance of an existing system
under some projected set of operating conditions.
• Alternative proposed system designs (or alternative operating policies for a
single system) can be compared via simulation to see which best meets a
specified requirement.
• In a simulation we can maintain much better control over experimental
conditions than would generally be possible when experimenting with the
system itself.
• Simulation allows us to study a system with a long time frame---e.g., an
economic system---in compressed time, or alternatively to study the
detailed workings of a system in expanded time.
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DISADVANTAGES OF SIMULATION
• Each run of a stochastic simulation model produces only
estimates of a model’s true characteristics for a particular set
of input parameters. If a “valid” analytic model is available or
can be easily de developed, it will generally be preferable to a
simulation model.
• Simulation models are often expensive and time-consuming to
develop.
• If a model is not a “valid” representation of a system under
study, the simulation results, no matter how impressive they
appear, will provide little useful information about the actual
system.
•
In some studies both simulation and analytic models might
be useful. In particular, simulation can be used to check the
validity of assumptions needed in an analytic mode. On the
other hand, an analytic model can suggest reasonable
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alternatives to investigate in a simulation study.
Pitfalls to the successful completion of
a simulation study
• Failure to have a well-defined set of objectives at the beginning of
the simulation study
• Inappropriate level of model detail
• Failure to communicate with management throughout the course of
the simulation study
• Misunderstanding of simulation by management
• Treating a simulation study as if it were primarily an exercise in
computer programming
• Failure to have people with a knowledge of simulation methodology
and statistics on the modeling team
• Failure to collect good system data
• Inappropriate simulation software
• Obliviously using simulation software products whose complex
marco statement may not be well documented and may not
implement the desired modeling logic
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Pitfalls to the successful completion of a
simulation study
• Belief that easy-to-use simulation packages, which require little or
no programming, require a significantly lower level of technical
competence
• Misuse of animation
• Failure to account correctly for sources of randomness in the actual
system
• Using arbitrary distributions (e.g., normal, uniform, or trianglar) as
input to the simualation
• Analyzing the output data from one simulation run (replication) using
formulas that assume independence
• Making a single replication of a particular system design and treating
the output statistics as the “true answers”
• Comparing alternative system design on the basis of one replication
for each design
• Using the wrong performance measures
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Notes
• In some studies both simulation and
analytic models might be useful. In
particular, simulation can be used to
check the validity of assumptions needed
in an analytic model. On the other hand,
an analytic model can suggest reasonable
alternatives to investigate in a simulation
study.
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Tarea 1: Simulation of a Single server queueing system
Modify the code for the single-server queue in Chapter 1 to
compute and write in addition the following measures of
performance:
1.
The time-average number in the system
2.
The average total time in the system
3.
The maximum queue length
4.
The maximum delay in quque
5.
The maximum time in the system
6.
The proportion of customers having a delay in queue in excess of 1 minute
Entegar: report on the simulation including simulation model description, design of
the experiments, results of above measures after the simulation, conclusion on
the simulation, and program code.
Fecha limite: 18 de Febrero de 2005
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