OUTLINE
• Basic Concepts in Modeling and Simulation
• Building Simulation Models
• Verification and Validation
• Designing Experiments
• Output Analysis
• Applications of Simulation Modeling
Simulation with Arena, 3rd ed. Chapter 1 – What Is Simulation?
Slide 1 of 23
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
Slide 2 of 51
SIMULATION
Imitate the operations of a facility or process, usually via computer
What’s being simulated is the system To study system, often make
assumptions/approximations, both logical and mathematical, about how it works
These assumptions form a model of the system If model structure is simple enough, could use
mathematical methods to get exact information on questions of interest — analytical solution
Ways to Study Systems
– Simulation is “method of last resort?” Maybe …
– But with simulation there’s no need (or less need) to “look where the light is”
Slide 4 of 23
Work With the System?
–Advantage — unquestionably looking at the right thing
But it’s often impossible to do so in reality with the actual system–System doesn’t exist–Would be disruptive, expensive, or dangerous
Slide 5 of 23
Computer Simulation
• Methods and applications to imitate or mimic real systems usually via computer.
• No longer regarded as the approach of “last resort”.
• Today, it is viewed as an indispensable problem-solving methodology for engineers, designers, and managers.
• Can be used to study simple models but should not use it if an analytical solution is available
• Real power of simulation is in studying complex models
Slide 6 of 23
Applications of Simulation
• Applies in many fields and industries– Manufacturing facility– Bank operation– Airport operations (passengers, security, planes, crews, baggage)– Transportation/logistics/distribution operation– Hospital facilities (emergency room, operating room, admissions)– Computer network– Freeway system– Business process (insurance office)– Criminal justice system– Chemical plant– Fast-food restaurant– Supermarket– Theme park– Emergency-response system
Slide 7 of 23
Advantages of Simulation
• Flexibility to model things as they are (even if messy and complicated) - Allows uncertainty, nonstationarity in modeling
• New policies, operating procedures can be explored without disrupting ongoing operation of the real system.
• New hardware designs, physical layouts, transportation systems can be tested without committing resources for their acquisition.
• Time can be compressed or expanded to allow for a speed-up or slow-down of the phenomenon
• Advances in simulation software, computing and information technology are all increasing popularity of simulation
Slide 8 of 23
The Bad News
• Don’t get exact answers, only approximations, estimates
• Model building requires special training.
• Simulation modeling and analysis can be time consuming and expensive.
• Simulation results can be difficult to interpret.
• Get random output (RIRO) from stochastic simulations Statistical design, analysis of simulation experiments
SIMULATION vs. OPTIMIZATIONSIMULATION vs. OPTIMIZATIONIn an In an optimization modeloptimization model, the values of the , the values of the decision variables are decision variables are outputsoutputs that will that will maximize (or minimize) the value of the maximize (or minimize) the value of the objective function.objective function. InIn a a simulation modelsimulation model, the values of the , the values of the decision variables decision variables (controllable ones) (controllable ones) are are inputsinputs. The model evaluates the objective . The model evaluates the objective function for a particular set of valuesfunction for a particular set of values and and provides various performance measuresprovides various performance measures.. RIRO (Random input Random Output)RIRO (Random input Random Output)
Simulation Simulation Model TaxonomyModel Taxonomy
Chapter 2 – Fundamental Simulation Concepts
Slide 11 of 46 Simulation with Arena, 3rd ed.
The System:A Simple Processing System
ArrivingBlank Parts
DepartingFinished Parts
Machine(Server)
Queue (FIFO) Part in Service
4567
• General intent: Estimate expected production Waiting time in queue, queue length, proportion of time
machine is busy
• Time units Can use different units in different places … must declare Be careful to check the units when specifying inputs Declare base time units for internal calculations, outputs Be reasonable (interpretation, roundoff error)
Chapter 2 – Fundamental Simulation Concepts
Slide 12 of 46 Simulation with Arena, 3rd ed.
Model Specifics
• Initially (time 0) empty and idle• Base time units: minutes• Input data (assume given for now …), in minutes:
Part Number Arrival Time Interarrival Time Service Time1 0.00 1.73 2.902 1.73 1.35 1.763 3.08 0.71 3.394 3.79 0.62 4.525 4.41 14.28 4.466 18.69 0.70 4.367 19.39 15.52 2.078 34.91 3.15 3.369 38.06 1.76 2.37
10 39.82 1.00 5.3811 40.82 . .
. . . .
. . . .
• Stop when 20 minutes of (simulated) time have passed
Chapter 2 – Fundamental Simulation Concepts
Slide 13 of 46 Simulation with Arena, 3rd ed.
System
Clock
B(t)
Q(t)
Arrival times of custs. in queue
Event calendar
Number of completed waiting times in queue
Total of waiting times in queue
Area under Q(t)
Area under B(t)
Q(t) graph B(t) graph
Time (Minutes) Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:Setup
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
Chapter 2 – Fundamental Simulation Concepts
Slide 14 of 46 Simulation with Arena, 3rd ed.
System
Clock 0.00
B(t) 0
Q(t) 0
Arrival times of custs. in queue
<empty>
Event calendar [1, 0.00, Arr] [–, 20.00, End]
Number of completed waiting times in queue 0
Total of waiting times in queue 0.00
Area under Q(t) 0.00
Area under B(t) 0.00
Q(t) graph B(t) graph
Time (Minutes) Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:t = 0.00, Initialize
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
Chapter 2 – Fundamental Simulation Concepts
Slide 15 of 46 Simulation with Arena, 3rd ed.
System
Clock 0.00
B(t) 1
Q(t) 0
Arrival times of custs. in queue
<empty>
Event calendar [2, 1.73, Arr] [1, 2.90, Dep] [–, 20.00, End]
Number of completed waiting times in queue 1
Total of waiting times in queue 0.00
Area under Q(t) 0.00
Area under B(t) 0.00
Q(t) graph B(t) graph
Time (Minutes) Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand: t = 0.00, Arrival of Part 1
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
1
Chapter 2 – Fundamental Simulation Concepts
Slide 16 of 46 Simulation with Arena, 3rd ed.
System
Clock 1.73
B(t) 1
Q(t) 1
Arrival times of custs. in queue
(1.73)
Event calendar [1, 2.90, Dep] [3, 3.08, Arr] [–, 20.00, End]
Number of completed waiting times in queue 1
Total of waiting times in queue 0.00
Area under Q(t) 0.00
Area under B(t) 1.73
Q(t) graph B(t) graph
Time (Minutes) Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand: t = 1.73, Arrival of Part 2
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
12
Chapter 2 – Fundamental Simulation Concepts
Slide 17 of 46 Simulation with Arena, 3rd ed.
System
Clock 2.90
B(t) 1
Q(t) 0
Arrival times of custs. in queue
<empty>
Event calendar [3, 3.08, Arr] [2, 4.66, Dep] [–, 20.00, End]
Number of completed waiting times in queue 2
Total of waiting times in queue 1.17
Area under Q(t) 1.17
Area under B(t) 2.90
Q(t) graph B(t) graph
Time (Minutes) Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand: t = 2.90, Departure of Part 1
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
2
Chapter 2 – Fundamental Simulation Concepts
Slide 18 of 46 Simulation with Arena, 3rd ed.
System
Clock 3.08
B(t) 1
Q(t) 1
Arrival times of custs. in queue
(3.08)
Event calendar [4, 3.79, Arr] [2, 4.66, Dep] [–, 20.00, End]
Number of completed waiting times in queue 2
Total of waiting times in queue 1.17
Area under Q(t) 1.17
Area under B(t) 3.08
Q(t) graph B(t) graph
Time (Minutes) Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand: t = 3.08, Arrival of Part 3
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
23
Chapter 2 – Fundamental Simulation Concepts
Slide 19 of 46 Simulation with Arena, 3rd ed.
System
Clock 3.79
B(t) 1
Q(t) 2
Arrival times of custs. in queue
(3.79, 3.08)
Event calendar [5, 4.41, Arr] [2, 4.66, Dep] [–, 20.00, End]
Number of completed waiting times in queue 2
Total of waiting times in queue 1.17
Area under Q(t) 1.88
Area under B(t) 3.79
Q(t) graph B(t) graph
Time (Minutes) Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand: t = 3.79, Arrival of Part 4
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
234
Chapter 2 – Fundamental Simulation Concepts
Slide 20 of 46 Simulation with Arena, 3rd ed.
System
Clock 4.41
B(t) 1
Q(t) 3
Arrival times of custs. in queue
(4.41, 3.79, 3.08)
Event calendar [2, 4.66, Dep] [6, 18.69, Arr] [–, 20.00, End]
Number of completed waiting times in queue 2
Total of waiting times in queue 1.17
Area under Q(t) 3.12
Area under B(t) 4.41
Q(t) graph B(t) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand: t = 4.41, Arrival of Part 5
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
2345
Chapter 2 – Fundamental Simulation Concepts
Slide 21 of 46 Simulation with Arena, 3rd ed.
System
Clock 4.66
B(t) 1
Q(t) 2
Arrival times of custs. in queue
(4.41, 3.79)
Event calendar [3, 8.05, Dep] [6, 18.69, Arr] [–, 20.00, End]
Number of completed waiting times in queue 3
Total of waiting times in queue 2.75
Area under Q(t) 3.87
Area under B(t) 4.66
Q(t) graph B(t) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand: t = 4.66, Departure of Part 2
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
345
Chapter 2 – Fundamental Simulation Concepts
Slide 22 of 46 Simulation with Arena, 3rd ed.
System
Clock 12.57
B(t) 1
Q(t) 0
Arrival times of custs. in queue
()
Event calendar [5, 17.03, Dep] [6, 18.69, Arr] [–, 20.00, End]
Number of completed waiting times in queue 5
Total of waiting times in queue 15.17
Area under Q(t) 15.17
Area under B(t) 12.57
Q(t) graph B(t) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand: t = 12.57, Departure of Part 4
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
5
Chapter 2 – Fundamental Simulation Concepts
Slide 23 of 46 Simulation with Arena, 3rd ed.
System
Clock 17.03
B(t) 0
Q(t) 0
Arrival times of custs. in queue ()
Event calendar [6, 18.69, Arr] [–, 20.00, End]
Number of completed waiting times in queue 5
Total of waiting times in queue 15.17
Area under Q(t) 15.17
Area under B(t) 17.03
Q(t) graph B(t) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand: t = 17.03, Departure of Part 5
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
Chapter 2 – Fundamental Simulation Concepts
Slide 24 of 46 Simulation with Arena, 3rd ed.
System
Clock 18.69
B(t) 1
Q(t) 0
Arrival times of custs. in queue ()
Event calendar [7, 19.39, Arr] [–, 20.00, End] [6, 23.05, Dep]
Number of completed waiting times in queue 6
Total of waiting times in queue 15.17
Area under Q(t) 15.17
Area under B(t) 17.03
Q(t) graph B(t) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand: t = 18.69, Arrival of Part 6
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
6
Chapter 2 – Fundamental Simulation Concepts
Slide 25 of 46 Simulation with Arena, 3rd ed.
System
Clock 19.39
B(t) 1
Q(t) 1
Arrival times of custs. in queue
(19.39)
Event calendar [–, 20.00, End] [6, 23.05, Dep] [8, 34.91, Arr]
Number of completed waiting times in queue 6
Total of waiting times in queue 15.17
Area under Q(t) 15.17
Area under B(t) 17.73
Q(t) graph B(t) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand: t = 19.39, Arrival of Part 7
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
67
Chapter 2 – Fundamental Simulation Concepts
Slide 26 of 46 Simulation with Arena, 3rd ed.
Simulation by Hand: t = 20.00, The End
0
1
2
3
4
0 5 10 15 20
012
0 5 10 15 20
67
System
Clock 20.00
B(t) 1
Q(t) 1
Arrival times of custs. in queue
(19.39)
Event calendar [6, 23.05, Dep] [8, 34.91, Arr]
Number of completed waiting times in queue 6
Total of waiting times in queue 15.17
Area under Q(t) 15.78
Area under B(t) 18.34
Q(t) graph B(t) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Chapter 2 – Fundamental Simulation Concepts
Slide 27 of 46 Simulation with Arena, 3rd ed.
Simulation by Hand:Finishing Up
• Average waiting time in queue:
• Time-average number in queue:
• Utilization of drill press:
part per minutes 53261715
queue in times of No.queue in times of Total
..
part 79020
7815value clock Final
curve under Area.
.)( tQ
less)(dimension 92020
3418value clock Final
curve under Area.
.)( tB
Chapter 2 – Fundamental Simulation Concepts
Slide 28 of 46 Simulation with Arena, 3rd ed.
Randomness in Simulation
• The above was just one “replication” — a sample of size one (not worth much)
• Made a total of five replications:
• Confidence intervals for expected values: In general, For expected total production,
nstX n //, 211 )/.)(.(. 56417762803
042803 ..
Notesubstantialvariabilityacrossreplications
Chapter 2 – Fundamental Simulation Concepts
Slide 29 of 46 Simulation with Arena, 3rd ed.
Steps in a Simulation Study
• Understand the system
• Be clear about the goals
• Formulate the model representation
• Translate into modeling software
• Verify “program”
• Validate model
• Design experiments
• Make runs
• Analyze, get insight, document results
Introduction 30
A Simulation Project Requires to Put together a Complete
Mix of Skills on the Team
-Knowledge of the system under investigation
-System analyst skills (model formulation)
-Model building skills (model Programming)
-Data collection skills
-Statistical skills (input data representation,
experimental design, output analysis)
-Management skills (to get everyone pulling in the
same direction)
Steps in a SimulationProject
Data Collection:Input Data Modeling
•Input Analysis activities consist of: data collection data analysis goodness-of-fit testing (Chi-Square and
the Kolmogrov-Smirnov tests).
•The quality of the output is no better than the quality of inputs (GIGO principle).
Introduction 33
Model Translation: Choose The Appropriate Simulation Tools
Assuming Simulation is the appropriate
means, three alternatives exist:1. Build Model in a General Purpose
Language
2. Build Model in a General Simulation Language
3. Use a Special Purpose Simulation Package
Slide 34 of 23
Simulation Languages
• ARENA, Extend, AweSim, Micro Saint, GPSS/SLX, SIMPLE++, SIMUL8 and etc.
Less flexibility Easier to learn More costly
Introduction 35
SPECIAL PURPOSE SIMULATION PACKAGES
NETWORK II.5: Simulator for computer systems
MEDMODEL: Health Care
OPNET: Simulator for communication networks, including wireless networks
SIMFACTORY: Simulator for manufacturing operations
Advantages: Short learning cycle, No programming
Disadvantages: High Cost, Limited Flexibility
Chapter 2 – Fundamental Simulation Concepts
Slide 36 of 46 Simulation with Arena, 3rd ed.
Two Simulation Modeling Approaches
1.Event-Scheduling Approach
2. Process-Interaction Approach
Steps in a SimulationProject
04/19/23 38
Real-World System
Simulation Model(Conceptual Model)
Simulation Program
Validation
Verification
Verfication & Validation
Verification and Validation 39
Calibration and Validationof Models
RealSystem
InitialModel
First revisionof model
Secondrevisionof model
Revise
Revise
Revise
Compare model
to reality
Compare revised model
to reality
Compare 2nd revised model
to reality
<Iterative process of calibrating a model>
04/19/23 40
Example
• Suppose, in our current system, average order-filling time is 16.2 hours for orders received via the web. We hope to reduce this by making changes in our logistics system.
• We can check the validity of our simulation model via a hypothesis test.
• We can set up the following test:
H0: simulation mean fill time = 16.2 H1: simulation mean fill time 16.2
04/19/23 41
Testing
• Run R replications of the simulation model, collecting the average fill time Y1,…,YR on each replication.
• If the data are approximately normally distributed, then we reject
H0 if
1,2//|2.16|
Rt
RSY
04/19/23 42
What can we conclude?
• If we accept, then the model is valid? No! The model and the real system are
not the same; if we make R large enough, we will eventually reject.
• If we reject, then the model is invalid? No! It may be close enough for the
decision we need to make; we might have accepted if R was smaller.
Steps in a SimulationProject
Experimental Design in Simulation
• There is a huge amount of literature on experimental design and most of it is applicable to simulation.
• Experimental design allows us to efficiently explore the relationship between inputs and outputs.
• In experimental design terminology, the input parameters and structural assumptions are called factors (qualitative, quantitative, controllable, uncontrollable) and the output performance measures are called responses.
Experimental I/O ExamplesExperimental I/O Examples
Example Inputs (factors) Outputs (responses)
Chemical reaction PressureTemperatureCatalyst concentration
Yield
Growing tomatoes FertilizerSoil pHSeed hybridWater
YieldHardiness
Simulation of amanufacturingsystem
Job dispatch ruleNumber of machinesMachines’ reliabilityMean downtimes
ThroughputTime in systemUtilizationsQueue sizes
46
What Outputs (Responses) to Collect?
There are typically two types of
output:
Discrete-Time Output Data
Continuous-Time Output Data
47
Discrete-time Output Data
There is a natural “first” observation, “second” observation, etc.—but can only observe them when they “happen”.
If Wi = time in system for the ith part produced (for i = 1, 2, ..., N), and there are N parts produced during the simulation
i1 2 3 N ..................................
Wi
48
Continuous-time Output DataCan jump into system at any point in time (real, continuous time) and take a “snapshot” of something-there is no natural first or second observation.
If Q(t) = number of parts in a particular queue at time t between [0,T] and we run simulation for T units of simulated time
Q(t)
0
1
2
3
t T
49
Simulation Model
Inputs: Cycletimes
Interarrivaltimes
Batchsizes
Outputs: Hourlyproduction
Machineutilization
RIRO
DIDO Vs. RIRO Simulation
Steps in a SimulationProject
OUTPUT ANALYSIS
• Terminating (Transient) Simulations (Starts at time 0 under well-specified initial conditions)
Example: Bank opens at 8:30 am with no customers present and all tellers are available, and closes at 4:30 pm
• Non-terminating (Steady-state) Simulations (Initial conditions are defined by the analyst)
Examples: assembly lines that shut down infrequently, telephone systems, hospital emergency rooms, airport
Whether a simulation is considered to be terminating or non-terminating depends on
both the objectives of the simulation study and the nature of the system.
Simulation with Arena, 3rd ed. Chapter 1 – What Is Simulation?
Slide 51 of 23
52
Analysis for Steady-State Simulations
Objective: Estimate the steady state mean
Basic question: Should you do many short runs or one long run ?????
lim ( )i iE Y
Many short runs
One long run
X1
X2
X3
X4
X5
X1
Simulation with ARENA©
• What is ARENA©?
Arena is a Microsoft Windows based application package for simulation modeling and analysis. It is a product of Rockwell Software, Inc.
Current version: 14.5 (2014)
• ARENA’s User interface: GUI, interactive and menu driven.
Cellular Manufacturing• Cells 1, 2, and 4 each have a single machine, Cell 3
has 2 machines. The two machines in Cell 3 are different: the newer one can process parts in 80% of the time of the older one.
• The system produces 3 parts types, each visiting a different sequence of stations.
• All the process times are triangularly distributed.
• We will collect statistics on resource utilization, time and number in queue, as well as cycle time (time in system, from entry to exit) by part type. Initially, we’ll run the simulation for 2000 minutes.
Exercise 1: Wayne International AirportWayne International Airport primarily serves
domestic air traffic. Occasionally, however,
a chartered plane from abroad will arrive
with passengers bound for Wayne's great amusement
parks.
Whenever an international plane
arrives at the airport the two customs
inspectors on duty set up operations to
process the passengers.
Exercise 1: Wayne International AirportIncoming passengers must first have their
passports and visas checked. This is handled by
one inspector. The time required to check
a passenger's passports and visas can be
described by the following probability distribution:
Time Probability
20 seconds .20
40 seconds .40
60 seconds .30
80 seconds .10
After having their passports and visas checked, the passengers next proceed to the second customs official who does baggage inspections. Passengers form a single waiting line with the official inspecting baggage on a first come, first served basis. The time required for baggage inspection is described by the following probability distribution:
Time Time ProbabilityProbability No Time No Time .25 .25
1 minute 1 minute .60 .60 2 minutes 2 minutes .10 .10 3 minutes 3 minutes .05 .05
Exercise 1: Wayne International Airport
Exercise 1: Wayne International Airport
A chartered plane from abroad lands at Wayne
Airport with 80 passengers. Simulate the processing
of the first 10 passengers through customs. Use the
following random numbers:
For passport control:
93, 63, 26, 16, 21, 26, 70, 55, 72, 89
For baggage inspection:
13, 08, 60, 13, 68, 40, 40, 27, 23, 64
Exercise 1: Wayne International Airport
• Question 1
How long will it take for the first 10 passengers to clear customs?
• Question 2
What is the average length of time a customer waits before having his bags inspected after he clears passport control? How is this estimate biased?
Exercise 1: Wayne International AirportAnswer 1: Passenger 10 clears customs after 9 minutes and
20 seconds.
Answer 2: (Baggage Inspection Begins) - (Passport Control Ends)
= 0+0+0+40+0+20+20+40+40+0 = 120 sec.
Average Wait. Time/passenger=120/10 = 12 sec/passenger
This is a biased estimate because we assume that the
simulation began with the system empty. Thus, the
results tend to underestimate the average waiting time.
EXERCISE 2: Hand Simulation of Ordering Policy
• XYZ company sells CD players (with speakers), which it orders from Fuji Electronics in Japan. Because of shipping and handling costs, each order must be for five CD players. Because of the time it takes to receive an order, the warehouse outlet places an order every time the present stock drops to five CD players. It costs $100 to place an order. It costs the warehouse $400 in lost sales when a customer asks for a CD player and the warehouse is out of stock. It costs $40 to keep each CD player stored in the warehouse. If a customer cannot purchase a CD player when it is requested, the customer will not wait until one comes in but will go to a competitor. The probability distributions for demand and lead time have been determined as follows:
EXERCISE 2: Hand Simulation of Ordering Policy
Demand per Month Probability
0 .04
1 .08
2 .28
3 .40
4 .16
5 .02
6 .02
1.00
EXERCISE 2: Hand Simulation of Ordering Policy
Time to Receive an Order (month) Probability
1 .60
2 .30
3 .10
1.00
EXERCISE 2: Hand Simulation of Ordering Policy
• The warehouse has five CD players in stock. Orders are always received at the beginning of the week. Simulate ordering and sales policy for 10 months using the following random numbers and compute the average monthly cost.
RNs (Demand): 39, 72, 37, 87, 98,99, 93, 21,97, 41
RNs (Lead Time):73,75,15, 62, 47, 69, 95, 78, 16, 25
Exercise 3• George Nanchoff owns a gas station. The cars arrive at the gas station
and they are served by one assistant. Use the following inter-arrival time and service distribution to simulate arrival of five cars.
Using the random number sequence: 92, 44, 15, 97, 21, 80, 38, 64, 74, 08 estimate: – the average customer waiting time ,
– average idle time of the assistant,
– the average time a car spends in the system.
Interarrival time (in minutes)
P(X)Service Time (in minutes)
P (X)
4 .35 2 .30
7 .25 4 .40
10 .30 6 .20
20 .10 8 .10