Chapter 1
Basic Simulation Modeling
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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CONTENTS
1.1 The Nature of Simulation
1.2 Systems, Models, and Simulation
1.3 Discrete-Event Simulation
1.4 Simulation of a Single-Server Queueing System
1.5 Simulation of an Inventory System
1.6 Alternative Approaches to Modeling and Coding
Simulations
1.7 Steps in a Sound Simulation Study
1.8 Other Types of Simulation
1.9 Advantages, Disadvantages, and Pitfalls of Simulation
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1.1 THE NATURE OF SIMULATION
• 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
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1.1 The Nature of Simulation (cont’d.)
• But most complex systems require models that are also
complex (to be valid)
– Must be studied via simulation — evaluate model numerically and
collect data to estimate model characteristics
• Example: Manufacturing company considering extending
its plant
– Build it and see if it works out?
– Simulate current, expanded operations — could also investigate
many other issues along the way, quickly and cheaply
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1.1 The Nature of Simulation (cont’d.)
• Some (not all) application areas
– Designing and analyzing manufacturing systems
– Evaluating military weapons systems or their logistics
requirements
– Determining hardware requirements or protocols for
communications 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 policies for an inventory system
– Analyzing financial or economic systems
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1.1 The Nature of Simulation (cont’d.)
• Use, popularity of simulation
– Several conferences devoted to simulation, notably the Winter
Simulation Conference (www.wintersim.org)
• Surveys of use of OR/MS techniques (examples …)
– Longitudinal study (1973-1988): Simulation consistently ranked
as one of the three most important techniques
– 1294 papers in Interfaces (1997): Simulation was second only to
the broad category of “math programming”
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1.1 The Nature of Simulation (cont’d.)
• Impediments to acceptance, use of simulation
– Models of large systems are usually very complex

But now have better modeling software … more general, flexible, but still
(relatively) easy to use
– Can consume a lot of computer time


But now have faster, bigger, cheaper hardware to allow for much better
studies than just a few years ago … this trend will continue
However, simulation will also continue to push the envelope on computing
power in that we ask more and more of our simulation models
– Impression that simulation is “just programming”


There’s a lot more to a simulation study than just “coding” a model in some
software and running it to get “the answer”
Need careful design and analysis of simulation models – simulation
methodology
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1.2 SYSTEMS, MODELS, AND
SIMULATION
• System: A collection of entities (people, parts, messages,
machines, servers, …) that act and interact together toward
some end (Schmidt and Taylor, 1970)
–
–
–
–
In practice, depends on objectives of study
Might limit the boundaries (physical and logical) of the system
Judgment call: level of detail (e.g., what is an entity?)
Usually assume a time element – dynamic system
• State of a system: Collection of variables and their values
necessary to describe the system at that time
– Might depend on desired objectives, output performance measures
– Bank model: Could include number of busy tellers, time of arrival
of each customer, etc.
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1.2 Systems, Models, and Simulation (cont’d.)
• Types of systems
– Discrete


State variables change instantaneously at separated points in time
Bank model: State changes occur only when a customer arrives or departs
– Continuous


State variables change continuously as a function of time
Airplane flight: State variables like position, velocity change continuously
• Many systems are partly discrete, partly continuous
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1.2 Systems, Models, and Simulation (cont’d.)
• Ways to study a system
– Simulation is “method of
last resort?” Maybe …
– But with simulation there’s
no need (or less need) to
“look where the light is”
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1.2 Systems, Models, and Simulation (cont’d.)
• Classification of simulation models
– Static vs. dynamic
– Deterministic vs. stochastic
– Continuous vs. discrete
• Most operational models are dynamic, stochastic, and
discrete – will be called discrete-event simulation models
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1.3 DISCRETE-EVENT SIMULATION
• Discrete-event simulation: Modeling of a system as it
evolves over time by a representation where the state
variables change instantaneously at separated points in time
– More precisely, state can change at only a countable number of
points in time
– These points in time are when events occur
• Event: Instantaneous occurrence that may change the state
of the system
– Sometimes get creative about what an “event” is … e.g., end of
simulation, make a decision about a system’s operation
• Can in principle be done by hand, but usually done on
computer
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1.3 Discrete-Event Simulation (cont’d.)
• Example: Single-server queue
– Estimate expected average delay in queue (line, not
service)
– State variables



Status of server (idle, busy) – needed to decide what to do
with an arrival
Current length of the queue – to know where to store an
arrival that must wait in line
Time of arrival of each customer now in queue – needed to
compute time in queue when service starts
– Events



Arrival of a new customer
Service completion (and departure) of a customer
Maybe – end-simulation event (a “fake” event) – whether
this is an event depends on how simulation terminates (a
modeling decision)
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1.3.1 Time-Advance Mechanisms
• Simulation clock: Variable that keeps the current value of
(simulated) time in the model
– Must decide on, be consistent about, time units
– Usually no relation between simulated time and (real) time needed
to run a model on a computer
• Two approaches for time advance
– Next-event time advance (usually used) … described in detail
below
– Fixed-increment time advance (seldom used) … Described in
Appendix 1A


Generally introduces some amount of modeling error in terms of when
events should occur vs. do occur
Forces a tradeoff between model accuracy and computational efficiency
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1.3.1 Time-Advance Mechanisms (cont’d.)
• More on next-event time advance
– Initialize simulation clock to 0
– Determine times of occurrence of future events – event list
– Clock advances to next (most imminent) event, which is executed

Event execution may involve updating event list
– Continue until stopping rule is satisfied (must be explicitly stated)
– Clock “jumps” from one event time to the next, and doesn’t “exist”
for times between successive events … periods of inactivity are
ignored
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1.3.1 Time-Advance Mechanisms (cont’d.)
• Next-event time advance for the single-server queue
ti = time of arrival of ith customer (t0 = 0)
Ai = ti – ti-1 = interarrival time between (i-1)st and ith customers (usually
assumed to be a random variable from some probability distribution)
Si = service-time requirement of ith customer (another random variable)
Di = delay in queue of ith customer
Ci = ti + Di + Si = time ith customer completes service and departs
ej = time of occurrence of the jth event (of any type), j = 1, 2, 3, …
– Possible trace of events (detailed narrative in text)
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1.3.2 Components and Organization of a
Discrete-Event Simulation Model
• Each simulation model must be customized to target system
• But there are several common components, general
organization
–
–
–
–
–
–
–
–
–
–
System state – variables to describe state
Simulation clock – current value of simulated time
Event list – times of future events (as needed)
Statistical counters – to accumulate quantities for output
Initialization routine – initialize model at time 0
Timing routine – determine next event time, type; advance clock
Event routines – carry out logic for each event type
Library routines – utility routines to generate random variates, etc.
Report generator – to summarize, report results at end
Main program – ties routines together, executes them in right order
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1.3.2 Components and Organization of a
Discrete-Event Simulation Model (cont’d.)
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1.3.2 Components and Organization of a
Discrete-Event Simulation Model (cont’d.)
• More on entities
– Objects that compose a simulation model
– Usually include customers, parts, messages, etc. … may include
resources like servers
– Characterized by data values called attributes
– For each entity resident in the model there’s a record (row) in a
list, with the attributes being the columns
• Approaches to modeling
– Event-scheduling – as described above, coded in general-purpose
language
– Process – focuses on entities and their “experience,” usually
requires special-purpose simulation software
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1.4 SIMULATION OF A SINGLE-SERVER
QUEUEING SYSTEM
• Will show how to simulate a specific version of the singleserver queueing system
• Book contains code in FORTRAN and C … slides will
focus only on C version
• Though simple, it contains many features found in all
simulation models
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1.4.1 Problem Statement
• Recall single-server queueing model
• Assume interarrival times are independent and
identically distributed (IID) random variables
• Assume service times are IID, and are
independent of interarrival times
• Queue discipline is FIFO
• Start empty and idle at time 0
• First customer arrives after an interarrival
time, not at time 0
• Stopping rule: When nth customer has
completed delay in queue (i.e., enters service)
… n will be specified as input
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1.4.1 Problem Statement (cont’d.)
• Quantities to be estimated
– Expected average delay in queue (excluding service time) of the n
customers completing their delays

Why “expected?”
– Expected average number of customers in queue (excluding any in
service)


A continuous-time average
Area under Q(t) = queue length at time t, divided by T(n) = time simulation
ends … see book for justification and details
– Expected utilization (proportion of time busy) of the server


Another continuous-time average
Area under B(t) = server-busy function (1 if busy, 0 if idle at time t), divided
by T(n) … justification and details in book
– Many others are possible (maxima, minima, time or number in
system, proportions, quantiles, variances …)
• Important: Discrete-time vs. continuous-time statistics
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1.4.2 Intuitive Explanation
• Given (for now) interarrival times (all times are in minutes):
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
• Given service times:
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
• n = 6 delays in queue desired
• “Hand” simulation:
– Display system, state variables, clock, event list, statistical
counters … all after execution of each event
– Use above lists of interarrival, service times to “drive” simulation
– Stop when number of delays hits n = 6, compute output
performance measures
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1.4.2 Intuitive Explanation (cont’d)
Status
shown is
after all
changes
have
been
made in
each
case …
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
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1.4.2 Intuitive Explanation (cont’d)
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
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1.4.2 Intuitive Explanation (cont’d)
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
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1.4.2 Intuitive Explanation (cont’d)
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
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1.4.2 Intuitive Explanation (cont’d)
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
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1.4.2 Intuitive Explanation (cont’d)
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
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1.4.2 Intuitive Explanation (cont’d)
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
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1.4.2 Intuitive Explanation (cont’d)
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
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1.4.2 Intuitive Explanation (cont’d)
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
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1.4.2 Intuitive Explanation (cont’d)
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
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1.4.2 Intuitive Explanation (cont’d)
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
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1.4.2 Intuitive Explanation (cont’d)
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
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1.4.2 Intuitive Explanation (cont’d)
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
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1.4.2 Intuitive Explanation (cont’d)
Interarrival times:
Service times:
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
Final output performance measures:
Average delay in queue = 5.7/6 = 0.95 min./cust.
Time-average number in queue = 9.9/8.6 = 1.15 custs.
Server utilization = 7.7/8.6 = 0.90 (dimensionless)
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1.4.3 Program Organization and Logic
• C program to do this model (FORTRAN as well is in book)
– Event types: 1 for arrival, 2 for departure
– Modularize for initialization, timing, events, library, report, main
• Changes from hand simulation:
– Stopping rule: n = 1000 (rather than 6)
– Interarrival and service times “drawn” from an exponential
distribution (mean b = 1 for interarrivals, 0.5 for service times)

Density function

Cumulative distribution function
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1.4.3 Program Organization and Logic (cont’d.)
• How to “draw” (or generate) an observation (variate) from
an exponential distribution?
• Proposal:
– Assume a perfect random-number generator that generates IID
variates from a continuous uniform distribution on [0, 1] …
denoted the U(0, 1) distribution … see Chap. 7
– Algorithm:
1. Generate a random number U
2. Return X = – b ln U
– Proof that algorithm is correct:
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1.4.5 C Program;
1.4.6 Simulation Output and Discussion
• Refer to pp. 30, 31, 42-48 in the book (Figures 1.8, 1.9,
1.19-1.27) and the file mm1.c
–
–
–
–
–
–
–
–
–
–
Figure 1.19 – external definitions (at top of file)
Figure 1.20 – function main
Figure 1.21 – function initialize
Figure 1.22 – function timing
Figure 1.23 – function arrive (flowchart: Figure 1.8)
Figure 1.24 – function depart (flowchart: Figure 1.9)
Figure 1.25 – function report
Figure 1.26 – function update_time_avg_stats
Figure 1.27 – function expon
Figure 1.28 – output report mm1.out



Are these “the” answers?
Steady-state vs. terminating?
What about time in queue vs. just time in system?
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1.4.7 Alternative Stopping Rules
• Stop simulation at (exactly) time 8 hours (= 480 minutes),
rather than whenever n delays in queue are completed
– Before, final value of simulation clock was a random variable
– Now, number of delays completed will be a random variable
• Introduce an artificial “end-simulation” event (type 3)
– Schedule it on initialization
– Event routine is report generator
– Be sure to update continuous-time statistics to end
• Changes in C code (everything else is the same)
–
–
–
–
–
Figure 1.33 – external definitions
Figure 1.34 – function main
Figure 1.35 – function initialize
Figure 1.36 – function report
Figure 1.37 – output report mm1alt.out
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1.4.8 Determining the Events and Variables
• For complex models, it might not be obvious what the
events are
• Event-graph method (Schruben 1983, and subsequent
papers) gives formal graph-theoretic method of analyzing
event structure
• Can analyze what needs to be initialized, possibility of
combining events to simplify model
• Software package (SIGMA) to build, execute a simulation
model via event-graph representation
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1.5 SIMULATION OF AN INVENTORY
SYSTEM;
1.5.1 Problem Statement
• Single-product inventory
• Decide how many items to have in inventory for the next
n = 120 months; initially (time 0) have 60 items on hand
• Demands against inventory
– Occur with inter-demand time ~ exponential with mean 0.1 month
– Demand size = 1, 2, 3, 4 with resp. probabilities 1/6, 1/3, 1/3, 1/6
• Inventory review, reorder – stationary (s, S) policy … at
beginning of each month, review inventory level = I
– If I  s, don’t order (s is an input constant); no ordering cost
– If I < s, order Z = S – I items (S is an input constant, order “up to”
S); ordering cost = 32 + 3Z; delivery lag ~ U(0.5, 1) month
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1.5.1 Problem Statement (cont’d.)
• Demand in excess of current (physical) inventory is
backlogged … so (accounting) inventory could be < 0
• Let I(t) be (accounting) inventory level at time t (+, 0, –)
I+(t) = max {I(t), 0} = number of items physically on hand at time t
I –(t) = max {–I(t), 0} = number of items in backlog at time t
• Holding cost: Incur $1 per item per month in (positive)
inventory
Time-average (per month) holding cost =
• Shortage cost: Incur $5 per item per month in backlog
Time-average (per month) backlog cost =
• Average total cost per month: Add ordering, holding,
shortage costs per month
– Try different (s, S) combinations to try to reduce total cost
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1.5.2 Program Organization and Logic
•
•
State variables: Inventory level, amount of an outstanding
order, time of the last (most recent) event
Events:
1.
2.
3.
4.
•
Arrival of an order from the supplier
Demand for the product
Why the ordering
of event types
End of the simulation after n = 120 months
3 and 4?
Inventory evaluation (maybe ordering) at beginning of a month
Random variates needed
–
–
–
Interdemand times: exponential, as in queueing model
Delivery lags ~ U(0.5, 1): 0.5 + (1 – 0.5)U, where U ~ U(0, 1)
Demand sizes: Split [0, 1] into subintervals of width 1/6, 1/3,
1/3, 1/6; generate U ~ U(0, 1); see which subinterval U falls in;
return X = 1, 2, 3, or 4, respectively
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1.5.4 C Program;
1.5.5 Simulation Output and Discussion
• Refer to pp. 64-66, 73-79 in the book (Figures 1.43-1.46,
1.57-1.67) and the file inv.c
–
–
–
–
–
–
–
–
Figure 1.57 – external definitions (at top of file)
Figure 1.58 – function main
Figure 1.59 – function initialize
Figure 1.60 – function order_arrival (flowchart: Figure 1.43)
Figure 1.61 – function demand (flowchart: Figure 1.44)
Figure 1.62 – function evaluate (flowchart: Figure 1.45)
Figure 1.63 – function report
Figure 1.64 – function update_time_avg_stats (flowchart:
Figure 1.46)
– Figure 1.65 – function random_integer
– Figure 1.66 – function uniform
– Figure 1.67 – output report inv.out


Reaction of individual cost components to changes in s and S … overall?
Uncertainty in output results (this was just one run)?
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1.6 ALTERNATIVE APPROACHES TO
MODELING AND CODING SIMULATIONS
• Parallel and distributed simulation
– Various kinds of parallel and distributed architectures
– Break up a simulation model in some way, run the different parts
simultaneously on different parallel processors
– Different ways to break up model


By support functions – random-number generation, variate generation,
event-list management, event routines, etc.
Decompose the model itself; assign different parts of model to different
processors – message-passing to maintain synchronization, or forget
synchronization and do “rollbacks” if necessary … “virtual time”
• Web-based simulation
– Central simulation engine, submit “jobs” over the web
– Wide-scope parallel/distributed simulation
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1.7 STEPS IN A SOUND SIMULATION
STUDY
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STEPS IN A SOUND SIMULATION STUDY
Now that we have looked in some detail at the
inner workings of a discrete-event simulation.
We need to step back and realize that model
programming is just part of the overall effort to
design or analyze a complex system by
simulation. Attention must be paid to a variety
of other concerns such as modeling system
randomness, validation, statistical analysis of
simulation
output
data
and
project
management. Figure 1.46 shows the steps
that will compose a typical, sound simulation
study [see also Banks et al. (2005, pp. 14-18)
and Law (2003)].
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The number beside the symbol representing
each step refers to the more detailed
description of that step below. Note that a
simulation study is not a simple sequential
process. As one proceeds with the study, it
may be necessary to go back to a previous
step.
1. Formulate the problem and plan the
study.
a. Problem of interest is stated by
manager.
 Problem may not be stated correctly
or in quantitative terms.
 An interactive process is often
necessary
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1.
Formulate problem and plan the study
2.
Collect data and define a model
3.
4.
5.
6.
Assumptions
document valid?
Yes
Construct a computer programme and verify
Make pilot runs
Programmed
Model valid?
7.
Yes
Design experiments
8.
Make production runs
9.
Analyze output data
10.
No
No
Document, present, and use results
FIGURE 1.46
a simulation
study
Simulation Modeling and Analysis – Steps
Chapter 1 in
– Basic
Simulation Modeling
Slide 51 of 51
b. One or more kickoff meetings 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 (required to decide level of model
detail)
 Performance measures that will be used to
evaluate the efficacy of different system
configurations
 Scope of the model
 System configurations to be modeled
(required to decide generality of simulation
programme)
Simulation Modeling
and Analysis – Chapter 1 – Basic Simulation Modeling
Slide 52 of 51
c. Select the software for the model (see
Chap.3)
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 true SMEs
are identified.
 Operating procedures may not be
formalized.
b.Collect data (if possible) to specify model
parameters
and
input
probability
distributions
(see
Chap.
6).
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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c. Delineate the above information and data
in an "assumptions document," which is the
conceptual model (see Sec. 5.4.3).
d. Collect data (if possible) on the
performance of the existing system (for
validation purposes in Step 6).
e.Choosing the level of model detail (see Sec.
5.2), which is an art, should depend on the
following:
 Project objectives
 Performance measures
 Data availability
 Credibility concerns
 Computer constraints
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
Slide 54 of 51
f.
There need not be a one-to-one
correspondence between each element of
the model and the corresponding element of
the system.
g. Start with a “simple” model and embellish it
as needed. Modeling each aspect of the
system will seldom be required to make
effective decisions, and might result in
excessive model execution time, in missed
deadlines, or in obscuring important system
factors.
h. Interact with the manager (and other key
project personnel) on a regular basis (see
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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3. Is the assumptions model valid?
a. Perform a structured walk-through of the
assumptions document before an audience
of managers, analysts, and SMEs (see Sec.
5.4.3). This will
 Help ensure that the model's assumptions
are correct and complete
 Promotes interaction among the project
members.
 Promote ownership of the model.
 Take place before programming begins, to
avoid significant reprogramming later
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
Slide 56 of 51
4.Construct a computer program and
verify.
a.Program the model in a programming
language (e.g., C or C++) or in simulation
software (e.g., Arena, Extend, Flexsim
and ProModel). Benefits of using a
programming language are that one is
often known, they offer greater program
control, they have a low purchase cost,
and they may result in a smaller model
execution time. The use of simulation
software (see Chap. 3), on the other
hand, reduces programming time and
results in a lower project cost.
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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b. Verify (debug) the simulation computer
program (see Sec. 5.3).
5. Make pilot runs.
a. Make pilot runs for validation purposes in
Step 6.
6. Is the programmed model valid?
a. If there is an existing system, then
compare model and system (from Step 2)
performance measures for the existing
system (see Sec. 5,4.5).
b. Regardless of whether there is an existing
system, the simulation analysts and SMEs
should review the model results for
correctness.
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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c. Use sensitivity analyses (see Sec. 5.4.4) 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 simulation run
 Length of the warmup period, if one is
appropriate
 Number of independent simulation runs
using different random numbers (see
Chap.
7)—facilitates
construction
of
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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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 (see Chap.
9)
 Comparing
alternative
system
configurations in a relative sense (see
Chap. 10 and Sec. 11.2)
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.
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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b. Present study's results.
 Use animation (see Chap. 3) 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
c. Results
are used in decision-making
credibility.
process if they are both valid and
credible.
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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1.8 OTHER TYPES OF SIMULATION
• Continuous simulation
– Typically, solve sets of differential equations numerically over time
– May involve stochastic elements
– Some specialized software available; some discrete-event
simulation software will do continuous simulation as well
• Combined discrete-continuous simulation
– Continuous variables described by differential equations
– Discrete events can occur that affect the continuously-changing
variables
– Some discrete-event simulation software will do combined
discrete-continuous simulation as well
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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1.8 Other Types of Simulation (cont’d.)
• Monte Carlo simulation
– No time element (usually)
– Wide variety of mathematical problems
– Example: Evaluate a “difficult” integral



Let X ~ U(a, b), and let Y = (b – a) g(X)
Then
Algorithm: Generate X ~ U(a, b), let Y = (b – a) g(X); repeat; average the
Y’s … this average will be an unbiased estimator of I
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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1.9 ADVANTAGES, DISADVANTAGES,
AND PITFALLS OF SIMULATION
• Advantages
– Simulation allows great flexibility in modeling complex systems,
so simulation models can be highly valid
– Easy to compare alternatives
– Control experimental conditions
– Can study system with a very long time frame
• Disadvantages
– Stochastic simulations produce only estimates – with noise
– Simulation models can be expensive to develop
– Simulations usually produce large volumes of output – need to
summarize, statistically analyze appropriately
• Pitfalls
–
–
–
–
Failure to identify objectives clearly up front
In appropriate level of detail (both ways)
Inadequate design and analysis of simulation experiments
Inadequate education, training
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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APPENDIX 1B A PRIMER ON QUEUEING
SYSTEMS
A queueing system consists of one or more
servers that provide service of some kind to
arriving customers. Customers who arrive to
find all servers busy (generally) join one or
more queues (or lines) in front of the servers,
hence the name "queueing" system.
Historically, a large proportion of all discreteevent simulation studies have involved the
modeling of a real-world queueing system, or
at least some component of the system being
simulated
was
a 1queueing
system.
Simulation
Modeling and Analysis
– Chapter
– Basic Simulation Modeling
Slide 65 of 51
Thus, we believe that it is important for the
student of simulation to have at least a basic
understanding of the components of a queueing
system, standard notation for queueing systems,
and measures of performance that are often
used to indicate the quality of service being
provided by a queueing system. Some examples
of real-world queueing systems that have often
been simulated are given in Table 1.4. For
additional information on queueing systems in
general, see Gross and Harris (1998). Bertsekas
and Gallager (1992) is recommended for those
interested
in
queueing
models
of
communications networks. Finally, Shanthikumar
and Buzacott (1993) discuss stochastic models
of manufacturing systems.
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
Slide 66 of 51
TABLE 1.4
Examples of queueing system
System
Servers
Customers
Bank
Tellers
Customers
Hospital
Doctors, nurses, beds
Patients
Computer System
Central processing
unit, input/output
devices
Machines, workers
Jobs
Runways, gates,
security check-instations
Nodes, links
Airplanes, travelers
Manufacturing System
Airport
Communications
network
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
Parts
Messages, Packets
Slide 67 of 51
1B.1 COMPONENTS OF A QUEUEING
SYSTEM
A queueing system is characterized by three
components:
arrival
process,
service
mechanism, and queue discipline. Specifying
the arrival process for a queueing system
consists of describing how customers arrive to
the system. Let Ai be the inter arrival time
between the arrivals of the (i — l)st and ith
customers (see Sec. 1.3). If A1, A2,... are
assumed to be IID random variables, we shall
denote the mean (or expected) inter arrival
time by E(A) and call λ == 1 /E(A) the arrival
rate of customers.
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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The service mechanism for a queueing system
is articulated by specifying the number of
servers (denoted by s), whether each server
has its own queue or there is one queue
feeding all servers, and the probability
distribution of customers' service times. Let 5;
be the service time of the ith arriving customer.
If S1, S2. . .are IID random variables, we shall
denote the mean service time of a customer by
E(S) and call w = 1/E(S) the service rate of a
server.
The queue discipline of a queueing system
refers to the rule that a server uses to choose
the next customer from the queue (if any)
when the server completes the service of the
current
customer.
Simulation
Modeling and
Analysis – Chapter 1 – Basic Simulation Modeling
Slide 69 of 51
Commonly used queue disciplines include
FIFO: Customers are served in a first-in, first-out
manner.
LIFO: Customers are served in a last-in, first-out
manner (see Prob. 2.17).
Priority: Customers are served in order of their
importance (see Prob. 2.22) or on the basis of
their service requirements (see Probs. 1.24, 2.20,
and 2.21).
1B.2 NOTATION FOR QUEUEING SYSTEMS
Certain queueing systems occur so often in
practice that standard notations have been
developed for them. In particular, consider the
queueing system shown in Fig. 1.71, which has
the following characteristics:
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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.
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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We call such a system a GI/G/s queue, where GI
(general independent) refers to the ' distribution of the
Ai's and G (general) refers to the distribution of the
Si’s. If specific distributions are given for the Ai’s and
the Si’s (as is always the case for simulation),
symbols denoting these distributions are used in
place of GI and G. The symbol M is used for the
exponential distribution because of the Markovian,
i.e., memory less, property of the exponential
distribution (see Prob. 4.26), the symbol EK for a kErlang distribution (if X is a k-Erlang random
variable,
X  
Y
then
, where the V/s are IID exponential random
variables), and D-for deterministic (or constant) times.
Thus, a single-server queueing system with
exponential inter arrival times and service times and a
FIFO queue discipline is called an M/M/l queue.
k
i 1
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
i
Slide 72 of 51
For any GI/G/s queue, we shall call the quantity p = λ
/ (sw) the utilization factor of the queueing system
(sw is the service rate of the system when all servers
are busy). It is a measure of how heavily the
resources of a queueing system are utilized.
1B.3 MEASURES OF PERFORMANCE FOR
QUEUEING SYSTEMS
There are many possible measures of performance
for queueing systems. We now describe four such
measures that are usually used in the mathematical
study of queueing systems. The reader should not
infer from our choices that these measures are
necessarily the most relevant or important in practice
(see Chap. 9 for further discussion). As a matter of
fact, for some real-world systems these measures
may not even be well defined; i.e., they may not
Simulation
Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
Slide 73 of 51
exist.
Let
Di = delay in queue of ith customer
Wi = Di + Si= waiting time in system of ith
customer
Q(t) = number of customers in queue at time t
L(t) = number of customers in system at time t
[Q(t) plus number of customers being served at
time t]the measures
Then
n
d
 lim
n

i 1
Di
w.p.1
n
n
W
i
w  lim
w.p.1
and
n
(if they exist) are called the steady-state average
delay and the steady-state average waiting time.
Similarly, the measures
i 1
n
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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Q  lim
T 
and

T
0
Q (t ) d t
w.p.1
T
L  lim
T 

T
0
L (t ) d t
T
w.p.1
(if they exist) are called the steady-state timeaverage number in queue and the steady-state
time-average number in system. Here and
throughout this book, the qualifier "w.p. 1" (with
probability 1) is given for mathematical
correctness and has little practical significance.
For example, suppose that (w.p. 1) for some
queueing system. This means that if one
performs a very large (an infinite) number of
experiments, then in virtually every experiment
converges to the finite quantity d. Note that p < 1
is a necessary condition for d, w, Q, and L to
exist for a GI/G/s queue.
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
Slide 75 of 51
Among the most general and useful results
for queueing systems are the conservation
equations
Q=λd
and L = λw
These equations hold for every queueing
system for which d and w exist [see Stidham
(1974)]. (Section 11.5 gives a simulation
application of these relationships.) Another
equation of considerable practical value is
given by
w = d + E(S)
(See Sec. 1.4.6 and also Sec. 11.5 Slide
for76 of 51
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
It should be mentioned that the
measures of performance discussed
above can be analytically computed for
M/M/s queues (s 1), M/G/1 queues for
any distribution G, and for certain other
queueing systems. In general, the
inter-arrival-time
distribution,
the
service-time distribution, or both must
be exponential (or a variant of
exponential, such as k-Erlang) for
analytic solutions to be possible [see
Gross and Harris (1998)].
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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For example, in the case of an M/M/1 queue,
it can be shown analytically that the steadystate average number in system is given by
L = p/(1 - p)
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
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which we plot as a function of p in Fig. 1.72.
Note that L is clearly not a linear function of p,
and for p > 0.8 the plot of L increases
exponentially. Although the formula for L is
specifically for the M/M/1 queue, the nonlinear
behavior seen in Fig. 1.72 is indicative of
queueing systems in general.
Another interesting (and instructive) example
of an analytical solution is the steady-state
average delay in queue for an M/G/1 queue,
given by
 Var S   E ( S )
d 
21  E ( S )
2
Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling

Slide 79 of 51
where Var(S) denotes the variance of the
service-time distribution [see, for example, Ross
(1997, p. 444) for a derivation of this formula].
Thus, we can see that if E(S) is large, then
congestion (here measured by d) will be larger;
this is certainly to be expected. The formula also
brings out the perhaps less obvious fact that congestion also increases if the variability of the
service-time distribution is large, even if the
mean service time stays the same. Intuitively,
this is because a highly variable service-time
random variable will have a greater chance of
taking on a large value (since it must be
positive), which means that the (single) server
will be tied up for a long time, causing the queue
to build
up.
Simulation
Modeling and
Analysis – Chapter 1 – Basic Simulation Modeling
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