Lec 17A--Discrete Stochastic Simulation (2)

7/30/2019 Lec 17A--Discrete Stochastic Simulation (2)

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Discrete/Stochastic Simulation

Using 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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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

airline 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 airline 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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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 9a.m., arrivals to convenience stores arelarger than normal. If there are more than 6people waiting in line, new arrivals will

balk and go somewhere else. People arriveat the rate of 1 every second, but the time isexponentially distributed. Patrons shop fora time period that is uniformly distributedbetween 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 isnormally distributed with a std dev. of 30

sec. . People will automatically enqueue

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 primarystorage

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

CI+1 = K*Ci

function random(float u, int I)

I = I * 1220703125;

if I


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Notes

Generates a sequence on the entire interval of32-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 madepositive by adding the largest 32-bit integerrepresentable +1

The last multiplication is like dividing the numberby the largest integer possible

1/2147483647 = .4656613x10 to the minus 9

ou can eas y genera e ran om


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ou can eas y genera e ran omnumbers in an EXCEL

spreadsheet using the functionRAND()


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What about non-uniform random

numbers?Exponential

Normal

Gamma

Poisson

Lognormal

Rectangular

Triangular


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ONE ANSWER: Use the inverse

transformation methodEvery non-uniform random variate has an

associated cumulative distribution

function F(x) whose values are containedwithin 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) isuniformly 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, then1) simply generate a random number

uniformly distributed on the interval 0 to 1

2) call this number y and apply the inversetransformation 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 unavailableYou 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

We are also interested in

idleness, 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 oftime

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. Itcurrently 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 hasfour loading docks. It also has 10 trucks.

Trucks currently wait upon return for four

hours before they can go out on anothertrip. 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 skis. It currently has four people

working in the boot area and four peopleworking in the ski area. Lines are very long

and waiting times unacceptable. Should

the shop hire more help or just shift someof 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 normalwith 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


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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 entitytravels

Resourcespermanent entities that act on

ordinary temporary entities Processesthe activities


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PROMODEL

SELECT BACKGROUND--optional

BUILD-->locations

BUILD-->entities

BUILD-->PATH NETWORK

BUILD-->resources

BUILD-->processes and routing

BUILD-->arrivals

RUN IT


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Locations

Places where an event of importance to the

model occurs

Like an arrivalA 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

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 pathnetwork to use


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Now lets look at PROmodel


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Exercise 1. (15 points) A local conveniencestore has a self-service island from which

it dispenses gasoline. Two lines of carsmay form on either side of the island. Theisland will accommodate no more than twocars being filled with gas on a single side.

There is space for no more than three carsin each of the two queues of cars waitingfor 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 astandard deviation of two minutes. Cars will driveaway if more than six cars total are either waitingor in service (regardless of the line they are in).

Once cars have entered the stores gasolinefacility, they will en-queue themselves into theshortest queue. Formulate a model in BLOCKS todetermine how many cars are turned away in a

day. For BRANCH/ TRANSFERS, be sure to indicatethe type, such as UNCONDITIONALLY to block 12.Assuming the store is open 24 hours, setup themodel to determine how many cars are turnedaway in one 24-hour day.


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Promodel

What do we need to know??


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What the following are

Locations

Entities

Path networks

Resources

Processes

Arrivals


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What has to be specified before

resources can be specified Path networks


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In order to get more than one

arrival, the freq must be set to INFINITE

I d d bl h i f


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In order to double the capacity of

the number of turning machines

and machining centers you

would

Go to locations and increase the capacity

from 1 to two for both of these locations


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Failed Arrivals means

Arrivals that could not even get their foot in

the door, in this case because the pallet

was full


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The important measures for this

model were Throughput for the product

Utilization of the resource

Utilization of the locations

Failed arrivals

Amount of blocked time there is

On the final you will be given a


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On the final you will be given a

scenario like the ones above and

asked to determine

Locations

Entities

Path networksmay be asked to draw

these

Resources

Processes

Arrivals


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You will also need to know

what events and activities are

How random numbers are generated

How random variates (non-uniform) aregenerated

What is meant by MONTE CARLO

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Lec 17A--Discrete Stochastic Simulation (2)

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