Discrete/Stochastic Simulation

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Discrete/Stochastic Simulation

• Uisng PROMODEL

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Usages

• Business Process Re-engineering
• Manufacturing Process Design
• Service Process Design
• Operations
• Supply chains
• As a planning tool
• As an innovation and improvement tool

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More usages

• Queuing systems
• More general than analytic queuing theory
• Inventory systems
• PERT networks

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Applications of Discrete Stochastic Simulation

• Resource management systems
• Pollution management systems
• Urban and regional planning
• Transportation systems
• Health systems
• Criminal justice systems
• Industrial systems
• Education systems
• eCommerce systems

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What Discrete Stochastic Simulation isnt

• Stocks, states, rates, flows, information
• Continuously changing variables
• Causal loop diagramming
• stock-and-flow diagramming

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What Discrete Stochastic Simulation is

• Probabilistic occurrences
• Activity completions
• Processes
• Precedence relationships
• Probabilistic routing
• Events, Entities and Attributes

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An exampleSouthwest Airlines aircraft
turn--ACTIVITIES

• Disembark passengers
• Cabin cleanup
• Embark passengers
• Unload baggage
• Load Baggage
• Refuel
• Remove waste
• Refurbish snacks and drinks

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EVENTS for the aircraft gate turn

• Arrival at gate
• Beginning of unloading
• Completion of passenger unloading
• Beginning of cleanup
• Ending of cleanup
• Beginning of passenger loading
• Ending of passenger loading
• Beginning of baggage unloading
• Ending of baggage unloading

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ACTIVITIES EVENTS

• Activities always have time duration.
• That time duration is in general random
• Events are instants in time
• Activities are sometime engaged in by entities
• Activities, events and entities have attributes

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Events, Entities and Attributes

• Entities may be permanent or temporary
• Customers, students, piece parts, messages,
  boxes, items,--TEMPORARY
• Universities, cities, companies, facilities,
  servers, professors, service areas --PERMANENT
• Both entities and events possess ATTRIBUTES
• Attributes of a server entitymean service time,
  std. dev. of service time, distribution type
• attributes of an event--event type, no of
  entities assoc. with it

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A typical service system scenario

• In the early morning hours between 7 and 9 a.m.,
  arrivals to convenience stores are larger than
  normal. If there are more than 6 people waiting
  in line, new arrivals will balk and go somewhere
  else. People arrive at the rate of 1 every
  second, but the time is exponentially
  distributed. Patrons shop for a time period that
  is uniformly distributed between 3 and 5 minutes.
  

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Convenience Store Scenario, continued

• It takes the checkout clerk an average of 43 sec
  to collect money from a customer and provide them
  with a receipt, but this time is normally
  distributed with a std dev. of 30 sec. . People
  will automatically en-queue themselves in front
  of the checkout stand.
• The manager can hire a second clerk, who is less
  well paid but also much slower

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Convenience Store Scenario, continued

• The store manager is interested in
• the average waiting time of his patrons in the
  queue
• the average number of customers that balked

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With one store clerk

• average waiting time is 128 seconds
• number of balked customers is 76 out of 1000
  customers
• Check out clerk is busy 86 of the time checking
  out customers

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With two store clerks

• average waiting time is 42 secs for the first
  server
• 69 secs for the second
• There are no balked customers
• Servers are busy 70 and 40 of the time,
  respectively

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How does randomness come into play?

• Probabilistic activity durations
• Probabilistic routing decisions
• Probabilistic arrivals

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How is randomness created within a digital
computer?

• Monte Carlo--the computer generation of random
  numbers
• Sample the clock?--no--not replicable
• maintain a huge file of random numbers--no
• Takes up too much space in primary memory
• On secondary storage, its too slow
• (when deciding to fetch from disk as opposed to
  primary memory, the time required. is 500,000
  times longer
• Use an ALGORITHM --YES, YES

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Why use an algorithm?

• The sequence it generates will be deterministic
• Doesnt take up much space in primary storage
• Takes up no space on secondary storage

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An Algorithm for Generating Random Numbers

• Must be fast (short and sweet)
• Must be capable of generating numbers that have
  all of the characteristics of randomness, but in
  fact are deterministic
• Multiplicative Congruence is one method

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About Random Numbers

• Uniform on the interval zero to one
• They are completely independent and therefore
  un-correlated
• We represent them this way U(0,1)

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Multiplicative Congruence Algorithm

• CI1 KCi
• function random(float u, int I)
• I I 1220703125
• if Ilt0 then
• I I 2147483647 1
• else
• U I 0.4656613E-9
• return and end

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Notes

• Generates a uniform sequence on the entire
  interval of 32-bit integers--0 to 2147483647
• Maps these onto the real interval of 0 to 1
• If the first multiplication causes integer
  overflow, the resultant number I will be
  negative--it is made positive by adding the
  largest 32-bit integer representable 1
• The last multiplication is like dividing the
  number by the largest integer possible
• 1/2147483647 .4656613x10 to the minus 9

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You can easily generate random numbers in an
EXCEL spreadsheet using the function RAND()
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What about non-uniform random numbers?

• Exponential
• Normal
• Gamma
• Poisson
• Lognormal
• Rectangular
• Triangular

• Beta
• Poisson
• Binomial
• Hyper-geometric

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ONE ANSWER Use the inverse transformation method

• Every non-uniform random variate has an
  associated cumulative distribution function F(x)
  whose values are contained within the interval 0
  to 1 and whose values are uniformly distributed
  over this interval
• If x is a non uniform random variate, y F(x) is
  uniformly distributed over the interval 0 to 1.

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Strategy

• If the inverse of the cumulative distribution
  function F(x) exists so that x F-1(y) can be
  determined, then
• 1) simply generate a random number uniformly
  distributed on the interval 0 to 1
• 2) call this number y and apply the inverse
  transformation F-1(y) to obtain a random number x
  with the appropriate distribution.

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Exponential Random variates

• 1) generate a random number U using the program
  given above
• 2)then apply EXPRND -XMEAN ALOG(U)
• (This is simply using the inverse distribution
  method)

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When Analytic inverses of the cumulative
distribution function are unavailable

• You can use a table function
• You can use specialized algorithms that have been
  developed by academics over thirty-five years of
  cumulative research

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Outputs

• The animation reveals bottlenecks, idleness
• We are also interested in
• productivity, cycle time (time in the system),
  wait time,blocked time,
• number of trips made in a given period of time,
• system throughput within a given period of time
• We can get this from the statistical reports
  provided after the simulation is finished

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Discrete stochastic simulation time advance

• Time is advanced from event to event.
• The only time instants looked at are the event
  times
• The events are stored chronologically in time in
  an event file known as an event calendar
• Corresponding to each event type is an event
  subroutine
• When an event occurs, its subroutine is called

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Discrete-stochastic simulation as

• A statistical experiment
• Running times must be long enough to ensure
  sufficient samples are collected
• Several runs are often averaged together
• The starting random number seeds are changed and
  the model is rerun
• The basic idea is to get the variance to converge
  to the actual real-world variance

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Another scenario

• A mufflers-shocks-brakes shop is turning away
  business. It is considering hiring another
  mechanic or adding another bay. It currently has
  four bays.

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Another scenario

• A shipping company has just picked up additional
  customers and needs to add capacity. At its
  loading warehouse, it has four loading docks. It
  also has 10 trucks. Trucks currently wait upon
  return for four hours before they can go out on
  another trip. Should the company add docks,
  remove trucks, or both.

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Another scenario

• A ski rental shop fits customers for boots and
  then skiis. It currently has four people working
  in the boot area and four people working in the
  ski area. Lines are very long and waiting times
  unacceptable. Should the shop hire more help or
  just shift some of its existing help from skis to
  boots or vice versa.

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GO BACK TO 7-11 STORE example

• Consider a 7-11 store in which the 7-9 a.m.
  period is of interest. Management is considering
  hiring a second clerk. Patrons arrive at the
  rate of one every 45 secs. Arrivals are
  Exponentially distributed. Patrons shop for a
  period that is uniformly distributed between 3
  and 5 minutes. The check out time for each
  customer is normal with a mean of 43 secs and a
  std. dev. of 30 secs. Customers who encounter
  a queue of six customers or more upon arrival
  will walk away without shopping.

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What are the activities?

• Arrivals
• Shopping
• Checkout
• What about waiting in Queue?
• This is not an activity
• This is handled automatically by the simulation

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What units on time?

• Secs or mins?
• Lets go with SECONDS
• We must be consistent!!!!

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Must also identify

• Locationspoints assoc with the starting and
  stopping events of an activity
• Path networkthe network the entity travels
• Resourcespermanent entities that act on ordinary
  temporary entities
• Processesthe activities

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PROMODEL

• SELECT BACKGROUND--optional
• BUILD--gtlocations
• BUILD--gtentities
• BUILD--gtPATH NETWORK
• BUILD--gtresources
• BUILD--gtprocesses and routing
• BUILD--gtarrivals
• RUN IT

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Locations

• Places where an event of importance to the model
  occurs
• Like an arrival
• A beginning of customer checkout
• An ending of customer checkout

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Entities

• These are the temporary items that pass through
  the model of the system
• Widgets, chits
• Mail pieces
• Piece parts
• Students
• Cars
• People

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Path network

• The network that will be followed by the entities
  and/or the resources

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Resources

• Mobile permanent entities that can move over a
  network

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Processes

• A process is required everywhere the entity
  undergoes an operation
• An exit process is always required

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Routing

• You must specify how the entities move through
  the model
• Usually you inform PROMODEL what path network to
  use

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Arrivals

• The statistics of the arrival process for each
  entity type must be communicated to PROMODEL

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Now lets look at PROmodel
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• Exercise 1. (15 points) A local convenience
  store has a self-service island from which it
  dispenses gasoline. Two lines of cars may form
  on either side of the island. The island will
  accommodate no more than two cars being filled
  with gas on a single side. There is space for no
  more than three cars in each of the two queues of
  cars waiting for each of the two service areas.

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• Cars arrive at the rate of one every minute with
  a distribution that is exponential. Service
  times are normal with a mean of seven minutes and
  a standard deviation of two minutes. Cars will
  drive away if more than six cars total are either
  waiting or in service (regardless of the line
  they are in). Once cars have entered the stores
  gasoline facility, they will en-queue themselves
  into the shortest queue. Formulate a model in
  BLOCKS to determine how many cars are turned away
  in a day. For BRANCH/ TRANSFERS, be sure to
  indicate the type, such as UNCONDITIONALLY to
  block 12. Assuming the store is open 24 hours,
  setup the model to determine how many cars are
  turned away in one 24-hour day.