ch01
TRANSCRIPT
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Ch. 1 Basic Simulation Modeling
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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
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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 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)
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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 respect to time. (an airplane moving through the air, e.g., position and velocity )
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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 the
actual system
Physicalmodel
Analyticalsolution
Simulation
Experiment with a model of the system
Mathematicalmodel
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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 obvious way.
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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 the system.
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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 the value )
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e0 e1 e2 e3 e4 e5
0 t1 t2 c1 t3 c2
A1 A2 A3
S1 S2
Time
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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 simulation model
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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
1. Set simulation clock=02. Initialize system state and statistical counters3. Initialize event list
0. Invoke the initialization routine
1. Invoke the timing routine2. Invoke event routine
1. Determine the next event type, say, i2. Advance the simulation clock
1.Update system state2.Update statistical counters3.Generate future events and add to event list
Repeatedly
Initialization routine Main program Time routine
Event routine i
Generate randomvariates
Library routines
Issimulation
over?
1. Compute estimates of interest2. Write report
Stop
Report generator
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2
i
No
Yes
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Simulation of A Single-Server Queueing System
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Components of a queueing system1. Arrival process:
• Interarrivals: Ai• Mean interarrival time: E(A)• Arrival rate:
2. Service mechanism• Service time: Si• Number of servers: s• Mean service time: E(S)• Service rate:
3. Queue discipline• FIFO• LIFO• Priority
* D. Gross, and C.M. Harris, Fundamentals of Queueing Theory, 3d ed., John Wiley, New York, 1998
)(/1 AE
)(/1 SE
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Notation for queueing systems
1. s servers in parallel and one FIFO queue feeding all servers
2. A1, A2, are IID random variables
3. S1, S2,…are IID random variables
4. The Ai’s and Si’s are independent
1 2 s. . .
FIGURE 1.71A 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
• : the utilization factor of the queueing system
)/( s
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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
iD
1 w.p.lim 1
n
Dd
n
ii
n
wLdQ and
: delay in queue of ith customer
iii SDW : waiting time in system of ith customer
)(tQ : number of customers in queue at time t
)(tL : number of customers in system at time t [Q(t) plus number of customers being served at time t]
1 w.p.lim 1
n
Ww
n
ii
n
: steady-state average delay
: steady-state average waiting time
1 w.p. )(
lim 0
T
dttQQ
T
T
1 w.p. )(
lim 0
T
dttLL
T
T
: steady-state time-average number in queue
: steady-state time-average number in system
)(SEdw
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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
iD
)(ˆ 1
n
Dnd
n
ii
: delay in queue of ith customer
: the expected proportion of the time that Q(t) is equal to i
)(tQ : number of customers in queue at time t
)(
Q(t)
)()(ˆ
T(n)
00
nT
dt
nT
iTnq i
i
0
)(i
iipnq
0
ˆ)(ˆi
ipinq
ip
The expected average delay in queue
The expected average number of customers in queue
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t
Q(t)
0 1 2 3 4 5 6 7 8 9
e1=0.4 e2=1.6
e3=2.1
e4=2.4 e5=3.1
e6=3.3
e7=3.8
e8=4.0
e9=4.9
e10=5.6e11=5.8
e12=7.2
e13=8.6=T(6)
FIGURE 1.5Q(t), arrival times, and departure times for a realization of a single-server queueing system.
Arrivals
Departures
1
2
3
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Measures of Performance for the example
t
ttB
at time idle isserver theif0
at timebusy isserver theif1)(
)(
)()(ˆ
)(
0
nT
dttBnu
nT
The expected utilization of the server
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t
B(t)
0 1 2 3 4 5 6 7 8 9
e1=0.4 e2=1.6
e3=2.1
e4=2.4 e5=3.1
e6=3.3
e7=3.8
e8=4.0
e9=4.9
e10=5.6
e11=5.8
e12=7.2
e13=8.6=T(6)
FIGURE 1.6B(t), arrival times, and departure times for a realization of a single-server queueing system.
Arrivals
Departures
1
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Intuitive Explanation
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0 0 0
0
0 0 0 0
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Initialization time = 0
System
(a)
Number in
queueTimes
of arrival
Time of last event
0.4
A
D
Numberdelayed
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1 0 0.4
0.4
1 0 0 0
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Arrival time = 0.4
System
(b)
Number in
queueTimes
of arrival
Time of last event
1.6A
D
0.4
2.4
Numberdelayed
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1 1 1.6
1.6
1 0 0 1.2
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Arrival time = 1.6
System
(c)
Number in
queueTimes
of arrival
Time of last event
2.1A
D
0.4
2.4
1.6
1.6
Numberdelayed
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1 2 2.1
2.1
1 0 0.5 1.7
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Arrivaltime = 2.1
System
(d)
Number in
queueTimes
of arrival
Time of last event
3.8A
D
0.4
2.4
1.6
1.6
2.1
2.1
Numberdelayed
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1 1 2.4
2.4
2 0.8 1.1 2.0
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Departuretime = 2.4
System
(e)
Number in
queueTimes
of arrival
Time of last event
3.8A
D
1.6
3.1
2.1
2.1
Numberdelayed
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1 0 3.1
3.1
3 1.8 1.8 2.7
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Departuretime = 3.1
System
(f)
Number in
queueTimes
of arrival
Time of last event
3.8A
D
2.1
3.3
Numberdelayed
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0 0 3.3
3.3
3 1.8 1.8 2.9
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Departuretime = 3.3
System
(g)
Number in
queueTimes
of arrival
Time of last event
3.8A
D
Numberdelayed
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1 0 3.8
3.8
4 1.8 1.8 2.9
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Arrivaltime = 3.8
System
(h)
Number in
queueTimes
of arrival
Time of last event
4.0A
D
3.8
4.9
Numberdelayed
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1 1 4.0
4.0
4 1.8 1.8 3.1
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Departuretime = 4.0
System
(i)
Number in
queueTimes
of arrival
Time of last event
5.6A
D
3.8
4.9
4.0
4.0
Numberdelayed
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1 0 4.9
4.9
5 2.7 2.7 4.0
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Departuretime = 4.9
System
(j)
Number in
queueTimes
of arrival
Time of last event
5.6A
D 8.6
4.0
Numberdelayed
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1 1 5.6
5.6
5 2.7 2.7 4.7
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Arrivaltime = 5.6
System
(k)
Number in
queueTimes
of arrival
Time of last event
5.8A
D
4.0
8.6
5.6
5.6
Numberdelayed
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1 2 5.8
5.8
5 2.7 2.9 4.9
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Arrivaltime = 5.8
System
(l)
Number in
queueTimes
of arrival
Time of last event
7.2A
D
4.0
8.6
5.6
5.6
5.8
5.8
Numberdelayed
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1 3 7.2
7.2
5 2.7 5.7 6.3
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Arrivaltime = 7.2
System
(m)
Number in
queueTimes
of arrival
Time of last event
9.1A
D
4.0
8.6
5.6
5.6
5.8
5.8
7.2
7.2
Numberdelayed
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1 2 8.6
8.6
6 5.7 9.9 7.7
Computer Representation
System state
Statistical countersSeverstatus
Totaldelay
Areaunder Q(t)
Areaunder B(t)
Clock Event list
Departuretime = 8.6
System
(n)
Number in
queueTimes
of arrival
Time of last event
9.1A
D
5.6
9.2
5.8
5.8
7.2
7.2
Numberdelayed
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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)
•
x
exf
1
)(
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Arrival Event
Schedule the next arrival event
Isthe server
busy?
Set delay=0for this customer
and gather statistics
Add 1 to the number in queue
Add 1 to the number of
customers delayed
Make the server busy
Schedule a departure event for this customer
Isthe queue
full?
Write errormessage and stop
simulation
Store time of arrival of this
customer
Return
Yes
Yes
No
No
FIGURE 1.8Flowchart for arrival routine, queueing model
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DepartureEvent
Isthe queueempty?
Subtract 1 from the number
in queue
Compute delay of customer entering service
and gather statistics
Add 1 to the number of customers
delayed
Schedule a departure event for this customer
Return
Yes No
FIGURE 1.9Flowchart for arrival routine, queueing model
Move each customer in queue (if any) up
one place
Make the server idle
Eliminate departure event from
consideration
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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 1.000 minsMean service time 0.500 minsNumber of customers 1000
Average delay in queue 0.430 minsAverage number in queue 0.418Server utilization 0.460Time simulation ended 1027.915
mins
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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
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Determining the Events and Variables
Event-graph
Arrival Departure
FIGURE 1.38Event graph, queueing model
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Determining the Events and Variables
Event-graph
ArrivalEnter
service
FIGURE 1.39Event graph, queueing model with separate “enter-service” event
Departure
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Determining the Events and Variables
Event-graph
FIGURE 1.40Event graph, queueing model with fixed run length
Arrival Departure
Endsimulation
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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 distributionsc. Delineate the above information and data in an “assumptions document,” which is the
conceptual modeld. 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 systemb. 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
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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 problemand plan the study
Collect data anddefine a model
Conceptualmodel valid?
Construct a computerprogram and verify
Make pilot runs
ProgrammedModel valid?
Design experiments
Make production runs
Analyze output data
Document, present,and use results
No
Yes
No
Yes
FIGURE 1.68Steps in a simulation study.
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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 alternatives to investigate in a simulation study.
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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