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Chapter 1Basic 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
• Whats being simulated is the system
• To study system, often make assumptions/approximat
  ions, 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 (contd.)

• 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 (contd.)

• 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 (contd.)

• 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 (contd.)

• 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
• Theres 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 (contd.)

• 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 (contd.)

• Ways to study a system

• Simulation is method of last resort? Maybe
• But with simulation theres no need (or less
  need) to look where the light is

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1.2 Systems, Models, and Simulation (contd.)

• 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
  systems operation
• Can in principle be done by hand, but usually
  done on computer

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1.3 Discrete-Event Simulation (contd.)

• 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 (contd.)

• 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 doesnt exist for times between successive
  events periods of inactivity are ignored

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1.3.1 Time-Advance Mechanisms (contd.)

• 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 (contd.)
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1.3.2 Components and Organization of a
Discrete-Event Simulation Model (contd.)

• 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 theres 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 single-server 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 (contd.)

• 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 (contd)
Status shown is after all changes have been made
in each case
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1.4.2 Intuitive Explanation (contd)
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1.4.2 Intuitive Explanation (contd)
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1.4.2 Intuitive Explanation (contd)
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1.4.2 Intuitive Explanation (contd)
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1.4.2 Intuitive Explanation (contd)
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1.4.2 Intuitive Explanation (contd)
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1.4.2 Intuitive Explanation (contd)
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1.4.2 Intuitive Explanation (contd)
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1.4.2 Intuitive Explanation (contd)
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1.4.2 Intuitive Explanation (contd)
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1.4.2 Intuitive Explanation (contd)
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1.4.2 Intuitive Explanation (contd)
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1.4.2 Intuitive Explanation (contd)
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 (contd.)

• 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 Program1.4.6 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 SYSTEM1.5.1
Problem Statement

• Single-product inventory
• Decide how many items to have in inventory for
  the nextn 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, dont order (s is an input constant)
  no ordering cost
• If I lt 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 (contd.)

• Demand in excess of current (physical) inventory
  is backlogged so (accounting) inventory could
  be lt 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. Arrival of an order from the supplier
• 2. Demand for the product
• 3. End of the simulation after n 120 months
• 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

Why the ordering of event types 3 and 4?
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1.5.4 C Program1.5.5 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.

1. 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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Formulate problem and plan the study
1.
Collect data and define a model
2.
No
Assumptions document valid?
3.
Yes
Construct a computer programme and verify
4.
Make pilot runs
5.
No
Programmed Model valid?
6.
Yes
Design experiments
7.
Make production runs
8.
Analyze output data
9.
10.
Document, present, and use results
FIGURE 1.46 Steps in a simulation study
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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)
• Time frame for the study and the required
  resources

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c. Select the software for the model (see Chap.3)
2.Collect data and define a model.

1. Collect information on the system layout and
   operating procedures.

• No single person or document is sufficient.
• Some people may have inaccurate informationmake
  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).
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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
• Opinions of SMEs
• Time and money constraints

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f. There need not be a one-to-one correspondence
between each element of the model and the
corresponding element of the system.

• 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 Sec.
  5.4.2).

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3. Is the assumptions model valid?

1. 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

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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.
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b. Verify (debug) the simulation computer program
(see Sec. 5.3).
5. Make pilot runs.

1. Make pilot runs for validation purposes in Step 6.

6. Is the programmed model valid?

1. 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).
2. Regardless of whether there is an existing
   system, the simulation analysts and SMEs should
   review the model results for correctness.

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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.

1. 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 confidence
  intervals

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8. Make production runs.

1. Production runs are made for use in Step 9.

9. Analyze output data.

1. 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.
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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 credibility.

c. Results are used in decision-making process if
they are both valid and credible.
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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

63
1.8 Other Types of Simulation (contd.)

• 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 Ys this average
  will be an unbiased estimator of I

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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

65
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 discrete-event simulation
studies have involved the modeling of a
real-world queueing system, or at least some
component of the system being simulated was a
queueing system.
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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.
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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 Jobs
Manufacturing System Machines, workers Parts
Airport Runways, gates, security check-in-stations Airplanes, travelers
Communications network Nodes, links Messages, Packets
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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 ? 1 /E(A) the
arrival rate of customers.
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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.
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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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• 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
Ais and Sis are independent.
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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 Sis. If specific
distributions are given for the Ais and the Sis
(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 k-Erlang distribution (if X is a
k-Erlang random variable, 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.
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For any GI/G/s queue, we shall call the quantity
p ? / (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 exist.
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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
Then the measures
w.p.1
w.p.1
and
(if they exist) are called the steady-state
average delay and the steady-state average
waiting time. Similarly, the measures
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w.p.1
and
w.p.1
(if they exist) are called the steady-state
time-average 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
lt 1 is a necessary condition for d, w, Q, and L
to exist for a GI/G/s queue.
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Among the most general and useful results for
queueing systems are the conservation
equations Q?d and L ?w 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 for
further discussion).
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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).
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For example, in the case of an M/M/1 queue, it
can be shown analytically that the steady-state
average number in system is given by L p/(1 -
p)
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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 gt 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
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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.