Discrete Event Simulation

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Discrete Event Simulation

• Overview 4/5/09

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Outline

• What is discrete event simulation?
• Events
• Probability
• Probability distributions
• Sample application

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What is discrete event simulation?
Simulation is the process of designing a model
of a real system and conducting experiments with
this model for the purpose either of
understanding the behavior of the system or of
evaluating various strategies (within the limits
imposed by a criterion or set of criteria) for
the operation of a system. -Robert E Shannon
1975 Simulation is the process of designing a
dynamic model of an actual dynamic system for the
purpose either of understanding the behavior of
the system or of evaluating various strategies
(within the limits imposed by a criterion or set
of criteria) for the operation of a system.
-Ricki G Ingalls 2002 Definitions from
http//staff.unak.is/not/andy/Year20320 Simulati
on/Lectures/SIMLec2.pdf
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What is discrete event simulation?

• A simulation is a dynamic model that replicates
  the behavior a real system.
• Simulations may be deterministic or stochastic,
  static or dynamic, continuous or discrete.
• Discrete event simulation (DEVS) is stochastic,
  dynamic, and discrete.
• DEVS is not necessarily spatial it usually
  isnt, but the ideas are applicable to many
  spatial simulations

See many introductions online, such as
http//www.facsim.org/ Documentation/FAQ/Simulatio
nFAQ/S1.html
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What is discrete event simulation?

• DEVS has been around for decades, and is
  supported by a large set of supporting tools,
  programming languages, conventional practices,
  etc.
• Like other kinds of simulation, offers an
  alternative, often simple way of solving a
  problem simulate system and observe results,
  instead of coming up with analytical model.
• Many other benefits like repeatability, ability
  to use multiple parameter sets, experimental
  treatment of what may be unique system, cheap
  compared to physical experiment, etc.
• Usually involves posing a question. The
  simulation estimates an answer.

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What is discrete event simulation?

• Relationship between models and real system
• (Base model complete but partially known
  conceptual model lumped model modelers
  understanding of the system with assumptions and
  simplifications morphism structure-preserving
  transformation)

Diagram from Couclelis, H. A Theoretical
Framework for Alternative Models of Spatial
Decision and Behavior. Annals of the Association
of American Geographers, Vol. 76, No. 1, 1986.
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What is discrete event simulation?

• Stochastic probablistic
• Dynamic changes over time
• Discrete instantaneous events are separated by
  intervals of time
• Time may be modeled in a variety of ways within
  the simulation.

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Alternate treatments of time

• Time divided into equal increments
• Unequal increments
• Time as linked events, independent of intervals
  (most common in DEVS)
• Cyclical

(st moment in spacetime)
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Termination

• Simulation may terminate when a terminating
  condition is met.
• May also be periodic.
• Can also be conceptually endless, like weather,
  terminated at some arbitrary time.
• Requires spin-up time to remove initial
  variability due to small number of cases
• Usually converges or stabilizes on particular
  result.

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Events

• May change the state of the system.
• Have no duration.
• If time is event-based, events happen when time
  advances.
• Canonical example flipping a coin.
• Events usually have a value associated with them
  (e.g. coin true/false).
• The definition of an event depends on the subject
  of the model.

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Random Variables

• A random variable (X) is numerical value
  associated with a random event.
• X can have different values
• X1, X2, X3Xn
• In a stochastic DEVS, the value associated with
  event is probablistic, that is it occurs with a
  given probability.
• That probability is what allows us to use DEVS to
  estimate results when we dont know the precise
  value of input parameters to our model.

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Pseudorandom Number Generators

• A stochastic simulation depends on a pseudorandom
  number generator to generate usable random
  numbers.
• These generators can vary significantly in
  quality.
• This is a subject of a large body of research.
• For practical purposes, it may be important to
  check the random number generator you are using
  to make sure it behaves as expected.
• For example, if you are generating millions of
  numbers and expect them to be close to random, a
  generator that only can generate 32768 random
  numbers (as was the case with the old C rand()
  function) will be very un-random and could
  corrupt your results.

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(Very) Basic Probability

• The probability of X equaling some value is
  central to describing its behavior in a DEVS.
• All possible values for X is the sample space,
  e.g., heads/tails for coin flips 1,2,3,4,5,6 for
  dice.
• An event in that space could be a coin flip
  equaling true for example.
• S is the sample space of X. Probability is
  between zero and 1 the probability of X being in
  S is 1 the sum of the probabilities of all
  values of X is 1 if they are mutually exclusive
  the probability that an event does not occur is
  equal to 1 minus the probability that it does.

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(Very) Basic Probability
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(Very) Basic Probability

• The expected value (mean) will converge after a
  number of repetitions.
• We can demonstrate the process of flipping
  coinswith a simulation. Here are 4 runs of a
  simulated sequences of 1000 coin flips (see
  code), displaying the average result (sum of Xi /
  n)

Note that the variance is high initially and
diminishes with time.
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(Very) Basic Probability

• Joint probability is the probability of two
  events happening together, if they are
  independent
• Here for example is a display of a simulation
  where the event is two independent coin flips
  both being true in sequence (converges on .25
  see code)

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(Very) Basic Probability

• Conditional probability is the probability of two
  events happening in sequence. If they are
  independent, this is the same as the joint
  probability. If not, it is the probability of
  both happening divided by the probability of the
  second
• For example, if we our event is two consecutive
  tails, and the first flip is tails, the
  probability of the next one being tails is .25,
  so the result is .25 / .5, or .5.

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(Very) Basic Probability

• For mutually exclusive events
• P(B) P(BA1)P(A1) P(BAn)P(An).
• Suppose the chance of a fire per square km in a
  pine forest is .01, in a deciduous forest is
  .005, and in mixed forest is .0085 (these numbers
  are fictitious). 5 of all forest first reach
  the tree canopy. What is the probability that a
  sq. km of forest will have a canopy fire?
• P(A1) .01
• P(A2) .005
• P(A3) .0085
• P(BA1) .05
• P(BA2) .05
• P(BA3) .05
• P(B) P(BA)(PA1) P(BA3)(PA3) P(BA3)(PA3)
• .05 .01 .05 .005 .05 .0085
• .0005 .00025 .000425
• .001175

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(Very) Basic Probability

• Expected value average mean sum of X / n.
• Variance expected value of the squared
  difference between X and mean always positive
  and unit-less
• Covariance expected value of the difference
  between X and mean of X times the difference
  between Y and the mean of Y
• Graph summarizes change in variance over the
  course of a simulation

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Probability distributions

• In a DEVS, you need to decide what probability
  distribution functions best model the events.
• Pseudorandom number generators generate numbers
  in a uniform distribution
• One basic trick is to transform that uniform
  distribution into other distributions.
• There are many basic probability distributions
  are convenient to represent mathematically.
• They may or may not represent reality, but can be
  useful simplifications.

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Exponential distribution

• Many phenomena behave like an exponential
  distribution such as radioactive decay.
• Other examples?

Graphs from http//zoonek2.free.fr/UNIX/48_R/07.ht
ml
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Normal or gaussian distribution

• Ubiquitous in statistics
• Many phenomena follow this distribution
• When an experiment is repeated, the results tend
  to be normally distributed.

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Gamma distribution

• Phenomena with a sharp initial increase then long
  tail can be modeled with a gamma distribution.
• Useful for things like service times
• When an experiment is repeated, the results tend
  to be normally distributed.

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Poisson distribution

• Example of algorithm to sample from a
  distribution.
• X follows a Poisson distribution if
• An algorithm for sampling from a Poisson
  distribution
• 1. Generate a random number U
• 2. If i0, pe-lambda, Fp
• 3. If U lt F, return I
• 4. P lambda p / (i 1), F F p, i i 1
• 5. Go to 3
• There are similar tricks to sampling from other
• probability distributions. Some tools like
  Matlab
• have these built in.
• Algorithm from Ross, Sheldon M., Simulation,
  Fourth Edition, Elsevier 2006.

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Probability distributions

• Sampling values from an observational
  distribution with a given set of probabilities
  (discrete inverse transform method).
• Generate a random number U
• If U lt p0 return X1
• If U lt p0 p1 return X2
• If U lt p0 p1 p2 return X3
• Etc.
• This can be speeded up by sorting p so that the
  larger intervals are processed first, reducing
  the number of steps.

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Sample DEVS Application
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Congestion Pricing Plan

• Plan NYC modeling study claimed that only between
  .2 and 1.3 percent of commuting trips to NYC
  could be avoided by telecommuting incentives.
• Other sources claim 30-40
• Whats the maximum possible amount, without
  considering incentives?
• Many uncertainties, but amenable to estimation by
  simulation.

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Source Data

• Event is a commute
• Census residence-county to work-county data.
• Census work-industry data.
• Other estimates from hub-bound travel statistics.
• Assertionsespecially, estimates of per-industry
  telecommuting rates.

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Simplifying Assumptions

• County distances were used instead of actual or
  census tract distances.
• The maximum commutable distance was asserted to
  be 175 miles.
• Miles-to-CO2 conversion factors were applied
  uniformly without regard to specific efficiencies
  of different vehicles or of different routes.
• All forms of non-car transit were assigned a
  single CO2 conversion factor.
• Emissions were calculated per-passenger, not
  per-vehicle.
• Days-per-week telecommuting was folded into
  overall percentage.

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Data

Conversion factors 0.4 pounds of CO2 per person
mile for transit 1.2 pounds of CO2 per person
mile for car 1 passenger per car
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Commutes by County

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Commute by Distance from Sampled Distribution

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Sampled Counties Versus Data
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Sampled Industries Versus Data
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Sampled Versus Observed CDF
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95 Confidence Interval
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Mathematical Distribution

• Exponential, lambda 34.0418
• Formula arrived at by minimizing sum of
• absolute values of differences between
• exponential distribution and observed.

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Distance by County
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Distance From Exponential Distribution
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KS Comparison of Observed Versus Mathematical
Distributions
D 0.5501 P 0
Plot from www.physics.csbsju.edu/stats/KS-test.n.p
lot_form.html
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Results
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Results

• The mean emissions per trip were around 20
  pounds, plus or minus about .5 pounds at a 95
  confidence interval, as given in run A.
• The percentage eligible for telecommuting was
  approximately 20 percent, higher than PlanNYC but
  less than federal government high of 70.
• The mathematical distribution gave a slightly
  higher estimate, in run B.
• Increasing N to 1,000,000 narrowed the 95
  confidence interval to 0.06, with a result closer
  to that of the mathematical distribution, as
  given in run C.
• Doubling the number of riders per car in run D
  had a small effect.
• The original model assumed that only 10 of
  service jobs were telecommutable. Boosting that
  number to 50 in run E boosted the total
  estimated percent reduction up to 33.
• This suggests that researching the percentage of
  telecommutable jobs in the services sector would
  be especially useful in refining the estimate.
• Simplifying assumptions need to be tested.