Discrete/Stochastic Simulation

Using PROMODEL

Usages

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

tool

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

What Discrete Stochastic Simulation isn’t• Stocks, states, rates, flows,

information• Continuously changing variables• Causal loop diagramming• stock-and-flow diagramming

What Discrete Stochastic Simulation is• Probabilistic occurrences• Activity completions• Processes• Precedence relationships• Probabilistic routing• Events, Entities and Attributes

An example—Southwest Airlines airline turn--ACTIVITIES• Disembark passengers• Cabin cleanup• Embark passengers• Unload baggage• Load Baggage• Refuel• Remove waste • Refurbish snacks and drinks

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

ACTIVITIES & EVENTS

• Activities always have time duration. – That time duration is in general random

• Events are instants in time• Activities are sometimes engaged in

by entities• Activities, events and entities have

attributes

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 entity—mean service

time, std. dev. of service time, distribution type – attributes of an event--event type, no of entities

assoc. with it

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.

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 enqueue themselves in front of the checkout stand.

• The manager can hire a second clerk, who is less well paid but also much slower

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

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

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

How does randomness come into play?• Probabilistic activity durations• Probabilistic routing “decisions”• Probabilistic arrivals

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

Why use an algorithm?

• The sequence it generates will be deterministic

• Doesn’t take up much space in primary storage

• Takes up no space on secondary storage

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

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)

Multiplicative Congruence Algorithm

• CI+1 = K*Ci

function random(float u, int I)I = I * 1220703125;if I<0 then

I = I + 2147483647 + 1;elseU = I * 0.4656613E-9;

return “and” end;

Notes• Generates a 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

You can easily generate random numbers in an EXCEL spreadsheet using the function RAND()

What about non-uniform random numbers?ExponentialNormalGammaPoissonLognormalRectangularTriangular

ONE ANSWER: Use the inverse transformation methodEvery 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.

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.

Exponential Random variates1) generate a random number U

using the program given above2)then apply EXPRND = -XMEAN *

ALOG(U)

(This is simply using the inverse distribution method)

When Analytic inverses of the cumulative distribution function are unavailableYou can use a table functionYou can use specialized algorithms

that have been developed by academics over thirty-five years of cumulative research

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 of time

• We can get this from the statistical reports provided after the simulation is finished

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

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

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.

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.

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 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.

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.

What are the activities?

• Arrivals• Shopping• Checkout• What about waiting in Queue?

– This is not an activity– This is handled automatically by the

simulation

What units on time?

• Secs or mins?• Let’s go with SECONDS• We must be consistent!!!!

Must also identify

• Locations—points assoc with the starting and stopping events of an activity

• Path network—the network the entity travels

• Resources—permanent entities that act on ordinary temporary entities

• Processes—the activities

PROMODEL

• SELECT BACKGROUND--optional• BUILD-->locations• BUILD-->entities• BUILD-->PATH NETWORK• BUILD-->resources• BUILD-->processes and routing• BUILD-->arrivals• RUN IT

Locations

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

Entities

• These are the temporary items that pass through the model of the system

• Chits• Mail pieces• Piece parts• Students• Cars• People

Path network

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

Resources

• Mobile permanent entities that can move over a network

Processes

• A process is required everywhere the entity undergoes an operation

• An exit process is always required

Routing

• You must specify how the entities move through the model

• Usually you inform PROMODEL what path network to use

Arrivals

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

Now let’s look at PROmodel

• 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.

• 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 store’s 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.

Promodel

What do we need to know??

What the following are

• Locations• Entities• Path networks• Resources• Processes• Arrivals

What has to be specified before resources can be specified• Path networks

In order to get more than one arrival, the freq must be set to• INFINITE

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

Failed Arrivals means

• Arrivals that could not even get their foot in the door, in this case because the pallet was full

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 scenario like the ones above and asked to determine

• Locations• Entities• Path networks—may be asked to

draw these• Resources• Processes• Arrivals

You will also need to know

• what events and activities are• How random numbers are

generated• How random variates (non-

uniform) are generated• What is meant by MONTE CARLO