1. Ch. 1 Basic Simulation Modeling

2. 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

3. 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.

5. 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.

6. 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

7. Academic level
Winter Simulation Conference ( people every year)
Is one of the three important operations-research 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)

8. 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)

9. 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 respect to time. (an airplane moving through the air, e.g., position and velocity )

10. 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

11. System
Experiment
with a model
of the system
Experiment
with the
actual system
Physical
model
Mathematical
model
Analytical
solution
Simulation

12. 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 obvious way.

13. Discrete Event Simulation
Definition
Time-Advance Mechanisms
Components and Organization of a Discrete -Event Simulation Model
Components
Logic Organization

14. 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 the system.

15. 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.

16. 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.

17. 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 the value )

18. e0 e1 e2 e3 e4 e5
Time t1 t2 c1 t3 c2 A1 A2 A3 S1 S2

19. Components and Organization of a Discrete -Event Simulation Model
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

20. 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 simulation model

21. 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.

22. Start
Initialization routine
Main program
Time routine
1. Set simulation
clock=0
2. Initialize system state
and statistical counters
3. Initialize event list
0. Invoke the initialization routine
1. Invoke the timing routine
2. Invoke event routine
1. Determine the next
event type, say, i
2. Advance the
simulation clock 1
Repeatedly i 2
Event routine i
Library routines
1.Update system state
2.Update statistical counters
3.Generate future events and add to
event list
Generate random
variates Is
simulation
over? No
Report generator
Yes
1. Compute estimates of interest
2. Write report
Stop

23. Simulation of A Single-Server Queueing System

24. Components of a queueing system
Arrival process:
Interarrivals: Ai
Mean interarrival time: E(A)
Arrival rate:
Service mechanism
Service time: Si
Number of servers: s
Mean service time: E(S)
Service rate:
Queue discipline
FIFO
LIFO
Priority
* D. Gross, and C.M. Harris, Fundamentals of Queueing Theory, 3d ed., John Wiley, New York, 1998

25. Notation for queueing systems
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

26. 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
: the utilization factor of the queueing system

27. A departure customer
Server
Customer in service
Customers in queue
An arrival customer
FIGURE 1.4
A single-server queueing system.

28. Measures of Performance for Queueing Systems
: delay in queue of ith customer
: waiting time in system of ith customer
: number of customers in queue at time t
: number of customers in system at time t [Q(t) plus number of customers being served at time t]
: steady-state average delay
: steady-state average waiting time
: steady-state time-average number in queue
: steady-state time-average number in system

29. 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, …

30. Measures of Performance for the example
: delay in queue of ith customer
: the expected proportion of the time that Q(t) is equal to i
: number of customers in queue at time t
The expected average delay in queue
The expected average number of customers in queue

31. Q(t) 3 2 1 1 2 3 4 5 6 7 8 9 t Arrivals Departures FIGURE 1.5
e1=0.4
e2=1.6
e8=4.0
e11=5.8
Arrivals
e10=5.6
e12=7.2
e3=2.1
e7=3.8
e6=3.3
Departures
e13=8.6=T(6)
e4=2.4
e5=3.1
e9=4.9
FIGURE 1.5
Q(t), arrival times, and departure times for a realization of a single-server queueing system.

32. Measures of Performance for the example
The expected utilization of the server

33. B(t) 1 1 2 3 4 5 6 7 8 9 t Arrivals Departures FIGURE 1.6
e1=0.4
e2=1.6
e8=4.0
e11=5.8
Arrivals
e10=5.6
e12=7.2
e3=2.1
e7=3.8
e6=3.3
Departures
e13=8.6=T(6)
e4=2.4
e5=3.1
e9=4.9
FIGURE 1.6
B(t), arrival times, and departure times for a realization of a single-server queueing system.

34. Intuitive Explanation

35. Initialization
time = 0
System state A
0.4 D
Clock
Event list
Time
of last
event
Sever
status
Number in
queue
Statistical counters
Times of
arrival
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
System
Computer Representation
(a)

36. Arrival
time = 0.4
System state A
1.6
0.4 D
2.4
0.4
Clock
Event list 1
0.4
Time
of last
event
Sever
status
Number in
queue
Statistical counters
Times of
arrival 1
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
System
Computer Representation
(b)

37. Arrival
time = 1.6
System state A
2.1
1.6 D
2.4
1.6
0.4
Clock
Event list 1 1
1.6
Time
of last
event
Sever
status
Number in
queue
Statistical counters
1.6
Times of
arrival 1
1.2
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
System
Computer Representation
(c)

38. Arrival
time = 2.1
System state A
3.8
2.1 D
2.4
1.6
0.4
Clock
Event list 1 2
2.1
2.1
Time
of last
event
Sever
status
Number in
queue
Statistical counters
1.6
Times of
arrival 1
0.5
1.7
2.1
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
System
Computer Representation
(d)

39. Departure
time = 2.4
System state A
3.8
2.4 D
3.1
2.1
1.6
Clock
Event list 1 1
2.4
Time
of last
event
Sever
status
Number in
queue
Statistical counters
2.1
Times of
arrival 2
0.8
1.1
2.0
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
System
Computer Representation
(e)

40. Departure
time = 3.1
System state A
3.8
3.1 D
3.3
2.1
Clock
Event list 1
3.1
Time
of last
event
Sever
status
Number in
queue
Statistical counters
Times of
arrival 3
1.8
1.8
2.7
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
System
Computer Representation
(f)

41. Departure
time = 3.3
System state A
3.8
3.3 D
Clock
Event list
3.3
Time
of last
event
Sever
status
Number in
queue
Statistical counters
Times of
arrival 3
1.8
1.8
2.9
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
System
Computer Representation
(g)

42. Arrival
time = 3.8
System state A
4.0
3.8 D
4.9
3.8
Clock
Event list 1
3.8
Time
of last
event
Sever
status
Number in
queue
Statistical counters
Times of
arrival 4
1.8
1.8
2.9
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
System
Computer Representation
(h)

43. Departure
time = 4.0
System state A
5.6
4.0 D
4.9
4.0
3.8
Clock
Event list 1 1
4.0
Time
of last
event
Sever
status
Number in
queue
Statistical counters
4.0
Times of
arrival 4
1.8
1.8
3.1
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
System
Computer Representation
(i)

44. Departure
time = 4.9
System state A
5.6
4.9 D
8.6
4.0
Clock
Event list 1
4.9
Time
of last
event
Sever
status
Number in
queue
Statistical counters
Times of
arrival 5
2.7
2.7
4.0
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
System
Computer Representation
(j)

45. Arrival
time = 5.6
System state A
5.8
5.6 D
8.6
5.6
4.0
Clock
Event list 1 1
5.6
Time
of last
event
Sever
status
Number in
queue
Statistical counters
5.6
Times of
arrival 5
2.7
2.7
4.7
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
System
Computer Representation
(k)

46. Arrival
time = 5.8
System state A
7.2
5.8 D
8.6
5.6
4.0
Clock
Event list 1 2
5.8
5.8
Time
of last
event
Sever
status
Number in
queue
Statistical counters
5.6
Times of
arrival 5
2.7
2.9
4.9
5.8
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
System
Computer Representation
(l)

47. Arrival
time = 7.2
System state A
9.1
7.2 D
8.6
5.6
4.0
Clock
Event list 1 3
5.8
7.2
7.2
Time
of last
event
Sever
status
Number in
queue
Statistical counters
5.6
Times of
arrival 5
2.7
5.7
6.3
5.8
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
7.2
System
Computer Representation
(m)

48. Departure
time = 8.6
System state A
9.1
8.6 D
9.2
5.8
5.6
Clock
Event list 1 2
7.2
8.6
Time
of last
event
Sever
status
Number in
queue
Statistical counters
5.8
Times of
arrival 6
5.7
9.9
7.7
7.2
Number
delayed
Total
delay
Area
under Q(t)
Area
under B(t)
System
Computer Representation
(n)

49. 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

50. 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)

51. Arrival Event Schedule the next arrival event Is the server busy? Yes
FIGURE 1.8
Flowchart for arrival routine, queueing model
Schedule the next
arrival event Is
the server
busy?
Yes No
Add 1 to the
number in queue
Set delay=0
for this customer
and gather statistics
Add 1 to the
number of
customers delayed Is
the queue
full?
Write error
message and stop
simulation
Yes No
Make the
server busy
Store time of
arrival of this
customer
Schedule a
departure event
for this customer
Return

52. customer entering service
Departure
Event
FIGURE 1.9
Flowchart for arrival routine, queueing model Is
the queue
empty?
Yes 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

53. 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

54. Simulation Output and Discussion (Fig. 1
Simulation Output and Discussion (Fig Output report, queueing model)
Single-server queueing system
Mean interarrival time mins
Mean service time mins
Number of customers
Average delay in queue mins
Average number in queue
Server utilization
Time simulation ended 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

59. STEPS IN A SOUND SIMULATION STUDY
Formulate the problem and plan the study
Problem of interest is stated by manager.
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

60. STEPS… 2. Collect data and define a model.
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.
Collect data (if possible) to specify model parameters and input probability distributions
Delineate the above information and data in an “assumptions document,” which is the conceptual model
Collect data (if possible) on the performance of the existing systsem (for validation purpose in Step 6)
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
There need not be a one-to-one correspondence between each element of the model and the corresponding element of the system.
Interact with the manager (and other key project personnel) on a regular basis

61. STEPS… 3. Is the conceptual model valid?
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.
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.
Verify (debug) the simulation computer program.

62. STEPS… 5. Make pilot runs. 6. Is the programed model valid?
Make pilot runs for validation purposes in Step 6.
6. Is the programed model valid?
If ther is an existing system, then compare model and system (from Step 2) performance measures for existing system
Regardless of whether there is an existing system, the simulation analysts and SMEs should review the model results for correctness.
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.
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. 9. Analyze output data.
Production runs are made for use in Step 9.
9. Analyze output data.
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.
Document assumptions (see Step 2), computer program, and study’s results for use in the current and future projects.
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.
Results are used in decision-making process if they are both valid and credible.

64. Steps in a simulation study.
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
FIGURE 1.68
Steps in a simulation study.
Make production runs
Analyze output data
Document, present,
and use results

65. 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.

66. 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 alternatives to investigate in a simulation study.

67. 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

68. 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

69. 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.

70. 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:
The time-average number in the system
The average total time in the system
The maximum queue length
The maximum delay in quque
The maximum time in the system
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

71. Ch. 3 Simulation Software

72. Programming of discrete-event simulation models
Generating random numbers, that is, observations from a U(0,1) probability distribution
Generating random variates from a specified probability distribution (e.g., exponential)
Advancing simulated time
Determining the next event from the event list and passing control to the appropriate block of code
Adding records to, or deleting records from, a list
Collecting output statistics and reporting the results
Detecting error conditions

73. Advantages of Using a Simulation Package
Simulation packages automatically provide most of the features needed to build a simulation model, resulting in a significant decrease in “programming” time and a reduction in overall project cost.
They provide a natural framework for simulation modeling. Their basic modeling constructs are more closely akin to simulation than are those in a general-purpose programming language like C.
Simulation models are generally easier to modify and maintain when written in a simulation package.
They provide better error detection because many potential types of errors are checked for automatically. Since fewer modeling constructs need to be included in a model, the chance of making an error will be probably smaller.

74. Advantages of Using General-Purpose Programming Language
Most modelers already know a programming language, but this is often not the case with a simulation package.
A simulation model efficiently written in C or C++ may require less execution time than a model developed in a simulation package. This is so because a simulation package is designed to address a wide variety of systems with one set of modeling constructs, whereas a C program can be more closely tailored to a particular application. This consideration has, however, become less important with the availability of inexpensive, high-speed PCs.
Programming languages may allow greater programming flexibility than certain simulation packages.
Software cost is generally lower, but total project cost may not be.

75. 3.3 Classification of Simulation Software
General-purpose Versus Application-Oriented Simulation Packages
Modeling approaches
Common Modeling Elements

76. 3.3.1 General-purpose Versus Application-Oriented Simulation Packages
Historically,
simulation languages
application-oriented simulators.
Advances:
simulation languages: employing graphical model-building approach such as icons, dialog boxes
Simulators: allowing programming in certain model locations using an internal pseudo-language.
Modify existing modeling constructs
Create new constructs
Recently,
general-purpose simulation packages: Arena, Extend
application-oriented simulation packages: Manufacturing, Communications Networks, Process Reengineering and Services, Health Care, Call Centers, Animation (standalone)

77. 3.3.2 Modeling Approaches Event-scheduling approach Process approach
Identify characteristic events
Write a set of event routines
Execute events (no simulated time passes during its execution)
Process approach
A process is a time-ordered sequence of interrelated events separated by intervals of time, which describes the entire experience of an “entity” as it flows through a “system”.

78. Process Approach x x x Entity arrives Entity begins service
Entity completes
service
Time x x x
Event
“Event”
Event
Possible
passage of time
in queue
Possible of time
for serving
of entity
FIGURE 3.1
Process describing the flow of an entity through a system

79. Process Approach Routine customer Arrival event 1 Schedule an arrival
event for the next
customer entity 3 4 Is
the server
idle ? 2 No
Place this
customer entity
in the queue
Wait until selected
for service
Passage of
simulation time
Yes 5
Remove this
customer entity
from the queue 6
Make the server busy
Departure event 7
Schedule a departure
event for the next
customer entity 9
Make the server idle 8
Wait until service
is completed 10
Termination of
this customer entity
Return
Passage of
simulation time
FIGURE 3.2
Prototype customer-process routine for a single-server queueing system.

80. 3.3.3 Common Modeling Elements
Type of system
Entities
Attributes
Resources
Queues
Manufacturing
Part
Part number, due date
Machines, workers
Queues or buffers
Communications
Message
Destination, message length
Nodes, links
Buffers
Airport
Airplane
Flight number, weight
Runways, gates
Insurance agency
Application, claim
Name, policy number, amount
Agents, clerks

81. 3.4 Desirable Software Features
General capabilities (including modeling flexibility and ease of use)
Hardware and software considerations
Animation
Statistical features
Customer support and documentation
Output reports and plots

82. 3.4.1 General capabilities modeling flexibility
Ability to define and change attributes for entities and also global variables, and to use both in decision logic (e.g., if-then-else constructs)
Ability to use mathematical expressions and mathematical functions (logarithms, exponentiation, etc.)
Ability to create new modeling constructs and to modify existing ones, and to be able to use the new or modified modeling constructs in current and future models

83. 3.4.1 General capabilities ease of use
Graphical user interface
Modeling constructs (e.g. icons or blocks) that are neither too “primitive” nor too “macro”
Tabs in dialog boxes can help manage a large number of options
Hierarchy allows a user to combine several basic modeling constructs into a new higher-level construct. (reuse)
Debugging aids
Follow a single entity through the model to see if it is processed correctly
See the state of the model every time a particular event occurs (e.g., a machine breakdown)
Set the value of certain attributes or variables to “force” an entity down a logical path that occurs with small probability

84. 3.4.1 General capabilities ease of use
Fast model execution speed
User-friendly model “front ends”
Run-time version
Running new or existing scenarios for a model built by an internal or external consultant
Sales tool for equipment suppliers or system integrators
Training
Import data from (and export data to) other applications (e.g., an Excel spreadsheet or a database)
Automatically simulate different scenarios that iterate on some model parameter (e.g., the number of machines in a work station for a factory)
Combined discrete-continuous simulation
External routines
Easily initialized in a nonempty and idle state
The state of a simulation can be saved at the end of a run
Cost
From $500 to $50,000
Other costs: maintenance fees, upgrade fees, additional hardware and software, etc.

85. 3.4.2 Hardware and Software Requirements
Computer platforms: Windows based PCs, UNIX workstations, Apple computers
Required RAM: 128Mb
Operating systems: Windows 98, Windows NT, UNIX

86. 3.4.3 Animation and Dynamic Graphics
Uses of animation
Communicating the essence of a simulation model (or simulation itself) to a manager or to other people who may not be aware of (or care about) the technical details of the model
Debugging the simulation program
Showing that a simulation model is not valid
Suggesting improved operational procedures for a system (some things may not be apparent from looking at just the simulation’s numerical results)
Training operational personnel
Promoting communication among the project team

87. 3.4.3 Animation and Dynamic Graphics
Types of animations
Concurrent animation
Post-processed
Desirable features for animation
Default animation
Library of standard icons
Smooth movement of icons
Control to speed up or slow down the animation
Zoom and pan, named animation views
Vector-based graphics (pictures are drawn with lines, arcs, and fills) rather than pixel-based graphics
3-D animation
Import CAD drawings and clip art
Dynamic graphics and statistics: clocks, dials, level meters, dynamically update a histograms, time plots, etc.

88. 3.4.4 Statistical capabilities
Good random number generator
At least 100 different streams that can be assigned to different sources of randomness
Same results on different executions
Theoretical distributions
Continuous: exponential, gamma, Weibull, lognormal, normal, uniform, triangular, beta
Discrete: binomial, geometric, negative binomial, Poisson, discrete uniform
Empirical distribution
Independent replications (or runs)
Each runs uses separate sets of different random numbers
Each run uses the same initial conditions
Each run resets the statistical counters

89. 3.4.4 Statistical capabilities
A statistically sound method available for constructing a confidence interval for a mean
Warmup period
Allow the user to specify what performance measures to collect output data on
Optimization

90. 3.4.5 Customer Support and Documentation
Public and customized training at the client’s site
Good technical support for questions (tel. help)
Good documentation
User’s guide or reference manual, numerous detailed examples, context-dependent online help, library of mini examples
Detailed description of how each modeling construct works
Free demo disk
Free trail
Newsletter, yearly user’s conference
Regular updates of the software

91. 3.4.6 Output Reports and Graphics
Standard reports for the estimated performance measures
Customize reports
Histogram
Time plot
Database
Correlation plot
Export individual model output observations to other software packages for further analysis and display. (spreadsheets, databases, statistics packages, and graphical packages)

92. 3.5 General-purpose Simulation Packages
ARENA, Extend, MODSIM III,

93. 3.6 Object-oriented Simulation
Advantages :
It promotes code reusability
It helps manage complexity
It makes model changes easier
It facilitates large projects with several programmers.
Disadvantages:
Some O-O simulation packages may have a steep learning curve.
One must do many projects and reuse objects to achieve its full benefits.

94. Ch.4 Review of Basic Probability and Statistics

95. simulation output data
4.1 Introduction
Perform statistic
analyses of the
simulation output data
Design the
simulation
experiments
Probability
and
statistics
Model a
probabilistic
system
Generate random
samples from the
input distribution
Choose the
input probabilistic
distribution
Validate the
simulation
model

96. 4.2 Random variables and their properties
Experiment is a process whose outcome is not known with certainty.
Sample space (S) is the set of all possible outcome of an experiment.
Sample points are the outcomes themselves.
Random variable (X, Y, Z) is a function that assigns a real number to each point in the sample space S.
Values x, y, z

97. Examples flipping a coin S={H, T} tossing a die S={1,2,…,6}
flipping two coins
S={(H,H), (H,T), (T,H), (T,T)}
X: the number of heads that occurs
rolling a pair of dice
S={(1,1), (1,2), …, (6,6)}
X: the sum of the two dice

98. Distribution (cumulative) function
: the probability associated with the event
Properties:
F(x) is nondecreasing [i.e., if ]. .

99. Discrete random variable
A random variable X is said to be discrete if it can take on at most a countable number of values.
The probability that X takes on the value
Then
Probability mass function on I=[a,b]

100. Examples p(x) x p(x) for the demand-size random variable X. 1 2 3 41 2 3 4 x
p(x)
1/6
1/3
1/2
2/3
5/6
p(x) for the demand-size random variable X.

101. F(x) for the demand-size random variable X.1 2 3 4 x
F(x)
1/6
1/3
1/2
2/3
5/6
F(x) for the demand-size random variable X.

102. Continuous random variables
A random variable is said to be continuous if there exists a nonnegative function f(x) such that for any set of real number B
and
f(x) is called the probability density function.
for all

103. f(x) x
Interpretation of the probability density function

104. Uniform random variable on the interval [0,1]
, then

105. x
F(x) 1 x
f(x) 1
f(x) for a uniform random variable on [0,1]
F(x) for a uniform random variable on [0,1]
where

106. Exponential random variable
F(x) x
f(x) 1 x
f(x) for an exponential random
variable with mean
F(x) for an exponential random
variable with mean

107. Joint probability mass function
If X and Y are discrete random variables, then let
for all x, y
where p(x,y) is called the joint probability mass function of X and Y.
X and Y are independent if
for all x, y
where
are the (marginal) probability mass functions of X and Y.

108. Example 4.9
Suppose that X and Y are jointly discrete random variables with
Then
for x=1,2
for y=2,3,4
Since
For all x, y, the random variables X and Y
are independent.

109. Joint probability density function
The random variables X and Y are jointly continuous if there exists a nonnegative
function f(x,y), such that for all sets of real numbers A and B,
X and Y are independent if
for all x and y
where
are the (marginal) probability density functions of X and Y, respectively.

110. Example 4.11
Suppose that X and Y are jointly continuous random variables with
Then
for
for
Since
X and Y are not independent.

111. Mean or expected value
The mean is one measure of central tendency in the sense that
it is the center of gravity

112. Examples
For the demand-size random variable, the mean is given by
For the uniform random variable, the mean is given by

113. Properties of means 1. 2.
Even if the ‘s are dependent.

114. Median The median of the random variable is defined to be the
smallest value of x such that x
area=0.5
The median for a continuous random variable

115. Example 4.14
1. Consider a discrete random variable X that takes on each of the
values, 1, 2, 3, 4, and 5 with probability 0.2. Clearly, the mean
And the median of X are 3.
2. Now consider random variable Y that takes on each of the
values, 1, 2, 3, 4, and 100 with probability 0.2. The mean
and the median of X are 22 and 3, respectively.
Note that the median is insensitive to this change in the distribution.
The median may be a better measure of central tendency than the mean.

116. Variance For the demand-size random variable,
For the uniform random variable on [0,1],

117. large
small
Density functions for continuous random variables with
large and small variances.

118. Properties of the variance1. 2. 3.
if the ‘s are
independent (or uncorrelated).

119. Standard deviation
The probability that is between and is 0.95.

120. Covariance The covariance between the random variables and
is a measure of their dependence.
if i=j,

121. Example 4.17
For the jointly continuous random variables X and Y in Example 4.11

122. If and are independent random variables
and are uncorrelated.
Generally, the converse is not true.

123. Correlated If , then and are said to be positively correlated.
and tend to occur together
and tend to occur together
If , then and are said to be negatively correlated.
and tend to occur together
and tend to occur together

124. Correlation
If is close to +1, then and are highly positively correlated.
If is close to -1, then and are highly negatively correlated.
For the random variable in Example 4.11

125. 4.3 Simulation output data and stochastic processes
Stochastic process is a collection of "similar" random variables
ordered over time, which are all defined on a common
sample space.
State space is the set of all possible values that these random
variables can take on.
Discrete-time stochastic process:
Continuous-time stochastic process:

126. Example 4.19 M/M/1 queue with IID interarrival times IID service times
FIFO service
Define the discrete-time stochastic process of delays in queue
and
are positively correlated.
simulation
input random variables
output stochastic process
The state space: the set of nonnegative real numbers

127. Example 4.20 For the queueing system of Example 4.19,
Let be the number of customers in the queue at time t .
Then is a continuous-time stochastic process with state space

128. Covariance-stationary
Assumptions about the stochastic process are necessary to draw inferences in practice.
A discrete-time stochastic process
is said to be
covariance-stationary, if
and is independent of i for

129. Covariance-stationary process
For a covariance-stationary process, the mean and variance are stationary over time, and the covariance between two observations and depends only on the separation j and not actual time value i and i+j.
We denote the covariance and correlation between and by
and respectively, where

130. Example 4.22
Consider the output process for a covariance-stationary M/M/1 queue with

133. Warmup period
In general, output processes for queueing systems are positively correlated.
If is a stochastic process beginning at time 0 in a simulation, then it is quite likely not to be covariance-stationary.
However, for some simulation will be approximately covariance-stationary if k is large enough, where k is the length of the warmup period.

134. 4.4 Estimation of means, variance, and correlations
Suppose are IID random variables with finite population mean and finite population variance
Unbiased estimators:
Sample mean
Sample variance

135. How close is to ?

136. How close is to
to construct a confidence interval
Unbiased estimator

137. Density function
for
Density function
for
n large
n small
Distributions of for small and large n.

138. Estimate the variance of the sample mean .
´s are independent
´s are uncorrelated
However, the simulation output data are almost always correlated.
are from a covariance-stationary stochastic process,
Then, is an unbiased estimator of , however, is no
longer an unbiased estimator of Since

139. However, simulation output data are always correlated. Since
(1)
For a covariance-stationary process:
(2)

140. If one estimates from (correct in the IID case)
there are two errors:
the bias in as an estimator of
the negligence of the correlation terms in Eq. (2).
Solution: combine Eq. (1) and Eq. (2)
(3)
If , then and

141. Example 4.24
from the process of delays for a covariance-stationary M/M/1 queue with Eq.(1) and (3)
Thus, is a gross underestimate of , and we are likely to be overly optimistic about the closeness of to

142. Estimate .
In general "good" estimates of the 's will be difficult to obtain unless n is very large and j is small relative to n.

144. 4.5.1 Confidence Intervals

145. 4.5.1 Confidence Intervals Central Limit Theorem: as , where ,
the distribution function of a normal random variable with
and , is given by
If n is "sufficiently large", the random variable will be approximately distributed as a standard normal random variable, regardless of the underlying distribution of the 's. For large n, the sample mean is approximately distributed as a normal random variable with mean and variance

146. where ( ) is the upper critical point for
a standard normal random variable.

147. f(x )
Shaded area = a - 1 2 / z x
If n is sufficiently large, an approximate percent confidence interval for is given by

148. Interpretation I: If one constructs a very large number of independent percent confidence intervals, each based on n observations, where n is sufficiently large, the proportion of these confidence intervals that contains (cover)
should be
Interpretation II: If the 's are normal random variables, the random variable has a t distribution with n-1 degree of freedom (df), and an exact (for any )
percent confidence interval for is given by
Where is the upper critical point for the t distribution with n-1 df

149. Standard normal distribution
f(x)
t distribution with 4df x
Figure 4.16
Density function for the t distribution with 4df
and for the standard normal distribution.

150. Example 4.26
10 observations: 1.20, 1.50, 1.68, 1.89, 0.95, 1.49, 1.58, 1.55,
0.50, 1.09 are from a normal distribution,
To construct a 90% confidence interval for .

151. Table 4.1 Estimated coverages based on 500 experiments
Distribution
Skewness v
n=5
n=10
n=20
n=40
Normal
0.00
0.910
0.902
0.898
0.900
Exponential
2.00
0.854
0.878
0.870
0.890
Chi Square
2.83
0.810
0.830
0.848
Lognormal
6.18
0.758
0.768
0.842
0.852
Hyperexponential
6.43
0.584
0.586
0.682
0.774

152. 4.5.2 Hypothesis tests for the mean:
If is large, is not likely to be true.
If is true, the statistic will have a t distribution with n-1 df.

153. Example 4.27
For Example 4.26,
To test the null hypothesis that at level
We reject

154. 4.6 The Strong Law of Large Numbers
Theorem 4.2

155. Example 4.29

156. Tarea II: Teoría de la probabilitad y estatística
A.M. Law and W.D. Kelton, Simulation, Modeling and Analysis, 3rd edition, pp
Problems 4.1, 4.2, 4.4, 4.7, 4.9, 4.10, 4.13, 4.20, 4.21, 4.23, 4.24, 4.25, 4.26

157. Chap. 5 Building Valid, Credible, and Appropriately Detailed Simulation Models

158. 5.1 Introduction and Definitions (1)
Verification is concerned with determining whether the conceptual simulation model (model assumptions) has been correctly translated into a computer program, i.e., debugging the simulation computer program.

159. 5.1 Introduction and Definitions(2)OJ '. I
5.1 Introduction and Definitions(2)
Validation is the process of determining whether a simulation model (as opposed to the computer program) is an accurate representation of the system, for the particular objectives of the study.

160. Validation A valid model can be used to make decisions.
A validation process depends on the complexity of the system and on whether a version of the system currently exists.
A model can only be an approximation.
A model is valid for one purpose.
The measures of performance used to validate the model should include those that the decision maker will actually use for evaluating system design.

161. 5.1 Introduction and Definitions(3)
A simulation model and its results have credibility if the manager and other project personnel accept them as "correct“.
A credible model is not necessarily valid, and vice versa.

162. Establish credibility
Validation
Verification
Validation
Establish credibility
Establish
credibility
Simulation
program
“Correct”
results
available
Results used
in decision-
making
process
System
Conceptual
model
Analysis
and data
1, 2, 3
Programming 4
Make
model runs
5,6,7,8,9
Sell
results to
management 10
Figure 5.1
Timing and relationships of validation, verification, and establishing credibility

163. 5.2 Guidelines for Determining the Level of Model Detail (1)
Carefully define the specific issues to be investigated by the study and the measures of performance that will be used for evaluation.
The entity moving through the simulation model does not always have to be the same as the entity moving through the corresponding system.
Use subject-matter experts (SMEs) and sensitivity analyses.
“Moderately detailed“ model.
Regular interaction.

164. 5.2 Guidelines for Determining the Level of Model Detail (2)
Do not have more detail in the model than is necessary to address the issues of interest, subject to the proviso that the model must have enough detail to be credible.
The level of model detail should be consistent with the type of data available.
In all simulation studies, time and money constraints are a major factor in determining the amount of model detail.
If the number of factors (aspects of interest) for the study is large, then use a "coarse" simulation model or analytic model to identify what factors have a significant impact on system performance.

165. 5.3 Verification of Simulation Computer Program
Tech 1: Write and debug the computer program on modules or subprograms.
Tech 2: More than one person review the computer program (structured walk­ through of the program).
Tech 3: Run the simulation under a variety of settings of input parameters, and check to see that the output is reasonable.
Tech 4: "trace", interactive debugger.
Tech 5: The model should be run under simplifying assumptions for which its true characteristics are known or can easily be computed.
Tech 6: Observe an animation of the simulation output.
Tech 7: Compute the sample mean and variance for each simulation input probability distribution, and compare them with the desired mean and variance.
Tech 8: Use a commercial simulation package to reduce the amount of programming required.

166. 5.4 Techniques for Increasing Model Validity and Credibility (1)
Collect high-quality information and data on the system
Conversation with subject matter experts
in MS, machine operators, engineers, maintenance personnel, schedulers, managers, vendors, …
Observation of the system
Data are not representative of what one really wants to model
Data are not of the appropriate type or format
Data may contain measurement, recording, or rounding errors
Data may be biased because of self interest
Data may be inconsistent
Existing theory IID exponential random variables
Relevant results from similar simulation study
Experience and intuition of the modelers

167. 5.4 Techniques for Increasing Model Validity and Credibility (2)
Interact with the manager on a regular basis
There may not be a clear idea of the problem to be solved at initiation of the study.
The manager’s interest and involvement in the study are maintained.
The manager’s knowledge of the system contributes to the actual validity of the model
The model is credible since the manager understands and accepts the model’s assumptions.
Maintain an assumptions document and perform a structured walk-through
Validate components of the model by using quantitative techniques.
Validate the output from the overall simulation model
Animation

168. 5.5. Management's Role in the Simulation Process
Formulating problem objectives.
Directing personnel to provide informationand data to the simulation modeler and to attend the structured walk-through.
Interacting with the simulation modeler on a regular basis.
Using the simulation results as an aid in the decision-making process.

169. 5.6 Statistical Procedures for Comparing Real-world Observations and Simulation Output Data
Inspection approach.
Confidence-interval approach based on independent data.
Time-series approach

170. 5.6.1 Inspection Approach
Statistics: sample mean, sample variance, the sample correlation function, histograms.
dangerous! for sample size 1.
Correlated inspection approach

171. Table 5.4
Results for three experiments with the inspection approach
Experiment 1
0.90
0.70
0.20 2
0.71
-0.01 3
1.08
0.35
0.73

172. Historical system
input data
Historical system
input data
Actual system
Simulation model
System output data
Compare
Model output data
Figure 5.2
The correlated inspection approach

173. Table 5.5
Results for the first 10 of 500 experiments with the correlated and basic
Inspection approaches, and a summary for all 500
Experiment j 1
3.06
3.81
2.62
-0.75
0.44 2
2.79
3.37
2.05
-0.58
0.74 3
2.21
2.61
4.56
-0.40
-2.35 4
2.54
3.59
1.86
-1.05
0.68 5
9.27
11.02
2.41
-1.75
6.86 6
3.09
3.75
1.85
-0.66
1.24 7
2.50
2.84
1.13
-0.34
1.37 8
0.31
0.71
3.12
-2.81 9
3.17
3.94
5.09
-0.77
-1.92 10
0.98
1.18
1.25
-0.20
-0.27
Sample mean
of all 500
2.10
2.85
2.70
-0.60
Sample variance
2.02
2.28
2.12
0.08
4.08

174. 5.6.2 Confidence-Interval Approach based on Independent Data
Condition: it is possible to collect a potentially large amount of data for both the model and the system.