Altiok / Melamed Simulation Modeling and Analysis with Arena

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SIMULATION MODELING AND ANALYSIS WITH ARENA T.
Altiok and B. Melamed Chapter 10 Correlation
Analysis
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What is Correlation Analysis?

• Correlation Analysis is a modeling and analysis
  approach that straddles both Input Analysis
  and Output Analysis
• Correlation Analysis consists of two activities
• modeling of correlated stochastic processes
• studying the impact of correlations on
  performance measures of interest via
  Sensitivity Analysis

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Correlation in Input Analysis

• Correlation Analysis as part of Input Analysis
  is simply an approach to modeling and data
  fitting that
• insists on high-quality models incorporating
  temporal dependence
• strives to fit correlation-related statistics in
  a systematic way
• To set the scene, consider a stationary time
  series , , that is, all statistics
  remain unchanged under the passage of time
• in particular, all share a common mean,
  and common variance,
• to fix the ideas, suppose that is to be
  used to model inter-arrival times at a queue
  (in which case the time series is non-negative)
• What statistical aspects of should
  be carefully modeled?

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Statistical Signatures

• Let a collection of statistics of a random
  variable or time series be referred to as a
  statistical signature (signature, for short)
• it is often possible to order signatures by
  strength, for example, signatures obviously
  become stronger under inclusion
• To clarify the signature strength notion,
  consider a time series of inter-arrival
  times, , and the following set of
  signatures in increasing strength
• the mean, , is a minimal signature,
  since its reciprocal, , is
  the arrival rate a key statistic in queueing
  models
• the mean, , and variance, , is a
  stronger signature
• adding moments of the inter-arrival
  distribution, such as the skewness and
  kurtosis, yields an even stronger signature
• the (marginal) distribution, , determines
  all its moments, and so is stronger than all
  of the above

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A Very Strong Signature

• Given a stationary empirical time series, our
  goal here is to to fit a particular very
  strong signature to a time series, ,
  which includes both of the following
  statistics
• the marginal distribution,
• the autocorrelation function,
• The marginal distribution,
• is a first-order statistic of , that
  is, it involves only a single random variable
  from (by stationarity)
• is estimated by an empirical histogram,
• The autocorrelation function,
• is a second-order statistic of , that
  is, it involves pairs of lagged random
  variables from , and
• serves as a statistical proxy for temporal
  dependence in , where each
  correlation coefficient,
  , measures of linear dependence
• is estimated by some estimator

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Correlation in Output Analysis

• Correlation Analysis as part of Output Analysis
  is the study of the sensitivity of output
  statistics to correlations in model components
• autocorrelation can have a major impact on
  performance measures
• consequently, they cannot always be ignored
  merely for the sake of simplified models
• however, modelers routinely ignore correlations
  to simplify model construction and its
  analysis
• A motivating example from the domain of queueing
  systems will illustrate the peril of ignoring
  correlations uncritically

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Example correlation Impact

• Consider a workstation operating as an M/M/1
  system with job arrival rate and
  processing (service) rate , such that
• since all job arrivals and processing times are
  mutually independent, all corresponding
  autocorrelations and cross-correlations are
  identically zero
• the system is stable with utilization
• it is known that the equilibrium mean flow time
  is and so the mean waiting time in the buffer
  is
• Next, modify the arrival process from a Poisson
  process to a (possibly autocorrelated) TES
  process, yielding a TES/M/1 system
• The merit of TES processes is that they
  simultaneously admit arbitrary marginal
  distributions and a variety of autocorrelation
  functions
• in particular, we can select TES inter-arrival
  processes with the same inter- arrival time
  distribution as in the Poisson process (i.e.,
  exponential with rate parameter ), but
  with autocorrelated inter-arrival times, yielding
  some
• TES/M/1 equilibrium mean waiting time in the
  buffer,

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Example correlation Impact (Cont.)

• We wish to gauge the impact of autocorrelations
  in the job arrival stream on mean waiting
  times via the relative deviation
•  
• the relative deviation is viewed as a function
  of the lag-1 autocorrelation,
  , in the TES arrival process
• The table below displays the relative deviations
  for two representative cases
• and (light traffic
  regime with utilization )
• and (heavy-traffic
  regime with utilization )

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Introduction to TES Modeling

• The definition of a TES process involves two
  related stochastic processes
• an auxiliary process, called the background
  process
• a target process, called the foreground process
  
• The two processes operate in lockstep in the
  sense that they are connected by a
  deterministic transformation
• more specifically, the state of the background
  process is mapped to a state of the foreground
  process
• this is done in such a way that the foreground
  process has a prescribed marginal distribution
  and a prescribed autocorrelation function

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Modulo-1 Arithmetic

• The definition of TES processes makes use of a
  simple mathematical operation, called
  modulo-1 arithmetic
• modulo-1 arithmetic is arithmetic restricted to
  the familiar fractions (values in the interval
  0,1), with the value 1 excluded)
• the notation
  is used to denote the
  fractional value of any number
• note that fractional values are defined for any
  real number (positive as well as negative)
• Examples
•  for zero, we simply have
• for positive numbers, we have the familiar
  fractional values, for example,
  
• for negative values, the fractional part is the
  complementary value relative to one, for
  example, .

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Outline of TES Processes Theory

• The Lemma of Iterated Uniformity is the
  foundation of the theory of TES processes
• let be a uniform random variable on 0,1)
  and let be any random variable
  independent of
• then is also uniform on 0,1),
  regardless of the distribution of !
• Define a stochastic process
• by the Lemma of Iterated Uniformity, each random
  variable above is uniform on 0,1)
• furthermore, each could be further transformed
  into a foreground process
• and by the Inverse Transform Method, each
  random variable above will have the prescribed
  distribution !

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Background TES Processes

• Define the following random variables
• let be a random variable with a uniform
  distribution on 0,1)
• let be an innovation sequence
  (that is, any iid sequence of random
  variables, independent of )
• TES background processes come in two flavors
• a background TES process, , is defined
  by the recursive scheme
• a background TES- process, , is
  defined by

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Visualizing Background TES Processes

• Background TES processes can be visualized as a
  random walk on the unit circle
• Consider a basic TES process, where the
  innovation variate is uniform on an
  interval , so its density is a single
  step of length not exceeding 1

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Basic TES Processes

• The following list summarizes qualitatively the
  effect of the parameters and on the
  autocorrelation of a basic background TES
  process
• the width, , of the innovation-density
  support (the region over which the density is
  positive) has a major effect on the magnitude of
  the autocorrelations the larger the support,
  the smaller the magnitude (in fact, when
  , then the autocorrelations vanish
  altogether)
• the location of the innovation-density support
  affects the shape of the autocorrelation
  function when the support is not symmetric
  about the origin, then the autocorrelation
  function assumes an oscillating form, and
  otherwise it is monotone decreasing

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Basic TES Processes (Cont.)
Autocorrelation function of a basic TES
process (symmetric innovation density and narrow
support)
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Basic TES Processes (Cont.)

Autocorrelation function of a basic TES
process (symmetric innovation density and wider
support)
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Basic TES Processes (Cont.)
Autocorrelation function of a basic TES
process (non-symmetric innovation density)
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Basic TES Processes (Cont.)
Autocorrelation function of a basic TES-
process (symmetric innovation density)
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Basic TES Processes (Cont.)
Autocorrelation function of a basic TES-
process (non-symmetric innovation density)
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Stitching Transformations

• A background TES process can produce marked
  visual discontinuities in its sample paths,
  which are noticeable when
• the innovation density has a narrow support
• successive background variates on the unit
  circle straddle the circles origin
• in the figure below we have a sudden drop from
  a relatively high value to a relatively small
  one as the process crosses the origin counter
  clock-wise

Sample path of a basic TES background process
with
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Stitching Transformations (Cont.)

• For modeling purposes, we would like sometimes
  to smooth (stitch together) such marked
  visual discontinuities
• To this end, define a family of so-called
  stitching transformations
• where is a so-called stitching parameter
  in the interval 0,1
• a stitching transformation preserves uniformity,
  that is, if , then
  for any
• therefore, any stitched background TES sequence
  is also a TES background sequence (and thus
  uniformly distributed), albeit a smoothed one

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Stitching Transformations (Cont.)

• The graph below displays typical stitching
  transformations for the following stitching
  parameters,
• for ,
• for , has a
  triangular shape
• for , is the identity

Stitching transformations for (dashed curve),
(dotted curve) and (solid curve)
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Stitching Transformations (Cont.)

• The graphs below illustrate the smoothing effect
  of stitching

Sample path of a basic TES background process
with and without stitching ( )
Sample path of a basic TES background process
with and with stitching ( )
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Foreground TES Processes

• A foreground TES process is obtained from a
  background TES
• process by a deterministic transformation,
  , called a distortion
• a foreground TES process, , is of the
  form
• a foreground TES- process, , is of the
  form
• In practice, one often applies a stitching
  transformation followed by an application of
  the Inverse Transform method via a distortion
  of the form
• where
• is a stitching transformation (often
  )
• is a cdf (typically, is an
  empirical histogram of data vector, )
• for example, for the exponential cdf,
  , the inverse is
  , and the
  stitching transformation might be

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Foreground TES Processes (Cont.)

• Example applying the exponential Inverse
  Transform formula above to basic background
  TES processes to obtain foreground TES
  processes
• two basic TES background processes are used
  with, respectively,
  and
• the Inverse Transform applied to these TES
  background processes uses the same parameter,
  
• The results are shown in the next few foils
• both foreground TES processes have the same
  exponential marginal distribution of rate 1,
  as attested by their histograms
• in contrast, the first foreground process
  exhibits significant autocorrelations, while
  the second has zero autocorrelations, as a
  consequence of its iid property (see the
  corresponding correlograms)

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Foreground TES Processes (Cont.)

Sample path (top), histogram (middle) and
correlogram (bottom) of an exponential basic
TES process with background parameters
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Foreground TES Processes (Cont.)

Sample path (top), histogram (middle) and
correlogram (bottom) of an exponential basic
TES process with background parameters
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Generation of TES Sequences

• TES processes are readily generated on a
  computer via algorithms that utilize random
  number generators (RNG)
• we assume that the availability of a function,
  called mod1(x), which implements modulo-1
  reduction of any real number, and returns the
  corresponding fraction
• For convenience, we separate the generation of
  TES processes from that of TES- processes
• the corresponding algorithms have considerable
  overlaps

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Generation of TES Sequences

• Inputs
• an innovation density // modeler choice
• a stitching parameter // modeler choice
• an inverse distribution // often inverse
  histogram (step) cdf,
• Outputs
• a background TES sequence,
• a foreground TES sequence,
• Algorithm
• 1. sample a value , uniform on 0,1),
  // initial background variate and set
  and // more initializations
• 2. go to Step 6. // go to generate initial
  foreground variate
• 3. set // bump up running index for next
  iteration
• 4. sample a value from // sample an
  innovation variate
• 5. set // compute a TES
  background variate
• 6. set // apply a stitching transformation
• 7. set // compute a TES foreground variate
• 8. go to Step 3. // loop to generate the next
  TES variate

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Generation of TES- Sequences

• Inputs
• an innovation density // modeler choice
• a stitching parameter // modeler choice
• an inverse distribution // often inverse
  histogram (step) cdf,
• Outputs
• a background TES- sequence,
• a foreground TES- sequence,
• Algorithm
• 1. sample a value , uniform on 0,1),
  // initial background variate and set
  and // more
  initializations
• 2. go to Step 7. // go to generate initial
  foreground variate
• 3. set // bump up running index for next
  iteration
• 4. sample a value from // sample an
  innovation variate
• 5. set // compute a TES
  background variate
• 6. if is even, then set ,
  // compute a TES- background variate if
  is odd, then set
• 7. set // apply a stitching transformation
• 8. set // compute a TES- foreground variate
• 9. go to Step 3. // loop to generate the next
  TES- variate

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Generation of TES Sequences in Arena

• Arena implementation of the algorithm to
  generate basic TES sequences with an
  exponential distribution (TES- is similar)
  assumes that the following parameters are given
  as inputs
• a pair of parameters some and such
  that , which determine a
  basic innovation density
• a stitching parameter (0.5 is
  typical)
• an inverse of an exponential distribution
  function,
• for some

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Arena Model for Basic TES
Arena model implementing the generation of basic
TES sequences with exponential marginal
distribution
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Arena Model for Basic TES (Cont.)
Arena Variable module for implementing the
generation of basic TES sequence with an
exponential marginal distribution
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Arena Model for Basic TES (Cont.)

• Arena variables in the model
• the variables L and R hold the parameters of the
  basic innovation density
• the variable xi holds the stitching parameter
• the variable lambda holds the rate parameter of
  the requisite exponential distribution
• the variable N holds the running index in the
  TES sequence (initially 0)
• the variable V_N holds an innovation variate
• the variable U_N holds an unstitched TES
  background variate
• the variable US_N holds a stitched TES variate
• the variable UP_N holds a stitched TES
  background variate
• the variable X_N holds a TES foreground
  variate
• The Arena Variable module
• lists all model parameters and variables and
  their initial values, if any
• those requiring initialization are identified by
  a 1 rows button label under the Initial Values
  heading), for example UP_N is initialized to a
  value between 0 and 1

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Correlation Analysis Example

• Consider a workstation subject to mutually
  independent failures
• in the case, the workstation can be modeled as
  an M/G/1 queueing system,
• where the processing time is, in fact, the
  process completion time, consisting of all the
  failures experienced by a job on the machine
• The mean job waiting time is given by the
  modified P-K formula
• where
• is the arrival rate
  
• is the average process
  completion time
• is the squared
  coefficient of variation of the process
  completion time, where is the second moment
  of repair times
• The probability that the machine is occupied
  (processing or down) is given by,
  and for stability we assume

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Correlation Analysis Example (Cont.)

• The table below displays the relative deviations
  
• as function of , where
• for
• for

Lag-1 Autocorrelation of Time-to-Failure 0.00 0
.14 0.36 0.48 0.60 0.68 0.74 0.99 0.66 15.8 11.4
51.3 90.1 260 543.7 3,232 4,115 0.81 36.3
11 151.3 229.8 564.7 766 3,429 7,800 Rela
tive deviations of mean waiting time in a
workstation with failures/repairs