(Monté Carlo) Simulation

MGS 3100 – Chapter 4

What is Simulation?

A definition of simulation from Dictionary.com– Imitation or representation, as of a potential

situation or in experimental testing.– The basic idea is to build an experimental device, or

simulator, that will “act like” the system of interest … in a quick, cost-effective manner.

– Representation of the operation or features of one process or system through the use of another: computer simulation of an in-flight emergency.

Simulation in Business Analysis

• Uses mathematical models

• Probabilistic (as opposed to deterministic)

• Uses the entire range of possible values of a variable in the model

• Imitates a system or situation (like a coin-flip, or how long a person might have to wait in a line at a restaurant)

Why Simulate?

• Safety – flight simulator• Cost – easier to simulate adding a new

runway and find out effects than to implement in reality and then find out

• Time – Boeing uses simulated manufacturing before the real thing, with tremendous savings in time and money – can discover parts that do not fit and fix them before actual production

How does it work?

• Simulation requires you to know – What variable is to be simulated– Is the variable discrete or continuous? – The distribution of the variable – values it can take on

and the probabilities of those values occurring.

• Step 1: Generate a variable containing uniformly distributed random variables between 0 and 1 (the rand() function in Excel).

• Step 2: Create a rule to map the random numbers to values of the variable desired in the right proportion, and apply the rule.

Types of Variables in Simulation Models

• Discrete– Used for simulating specific values or specific

points.– Example: Number of people in a waiting line

(queue). • Continuous

– Used for simulating any value (between specific points)

– Example: The amount of time a person spends in a queue.

Discrete Example 1– coin toss

• Variable to be simulated is “Outcome of a coin toss”. It takes on values “Heads” and “Tails”, each with 0.5 probability.

• Generate 100 random numbers (100 tosses of coin).

• Make a rule like – if random number > 0.5, then “Heads”, else “Tails”. This will create the right distribution of outcomes.

Discrete Example 2: Machine Failures

• Simulate machine failures based on this historical data

Number of Failures per month

Frequency

(# of months this occurred)

0

1

2

3

36

20

3

1

Total 60

Discrete Example 2: Machine Failures

Number of Failures per month

Frequency

(# of months this occurred)

Probability Cumulative

Probability

0

1

2

3

36

20

3

1

0.600

0.333

0.050

0.016

0.600

0.933

0.983

1.000

Total 60 1.00

Create the following cumulative probability table.

Discrete Example 2: Machine Failures

• Now map the random numbers between 0 and 1 using the cumulative prob. Column as the cutoffs.

• Random numbers between 0 and 0.6 represent 0 failures, between 0.6 and 0.933 represent 1 failure, and so on.

0.60 0.93 0.980

0 failures 1 failure 2 3 failures

Discrete Example 2 Solution –

Random Number Mapping

Random #

Number of

Failures

0.345

0.008

0.985

0.878

0

0

3

1

The random numbers are now mapped to number of failures as follows.

Continuous Example – arrival time

• Variable to be simulated is arrival time at a restaurant which can literally take on infinite individual values

• For example someone could arrive at:

• 12:09:37

• 12:09:37:52

• 12:09:37:52:14, etc.

Continuous Example – arrival time con’t

To simulate this situation, we must specify intervalsAt the restaurant the intervals could be all people arriving

between 11am and 12pm, 12pm and 1pm, or 1pm and 2pm.

As with the coin toss, generate random numbers in Excel ( =RAND() )

• Make a rule – if random number:• <=.333, then =11am-12pm• >.333 up to =.666, then 12pm-1pm• >.666 up to 1, then 1pm to 2pm

each category is equally likely

Continuous Example – arrival time con’t

If the random number is.47, then this would fall in the 12pm to 1pm category,

If the random number is .88, then this would fall in the 1pm to 2pm category, etc.

Because each category is equally likely, if we run enough trials, each category will contain about the same number of random numbers, which will tell the restaurant owner that it is equally likely that a person will arrive at any of the three times.

Continuous Example – arrival time con’t

• The owner looks at historical information and says that on an average day, 225 people eat lunch at his restaurant , and that typically

• 47 people arrive between 11am and 12pm

• 112 people arrive between 12pm and 1pm

• 66 people arrive between 1pm and 2pm

How do we map these numbers?

Continuous Example – arrival time con’t

count percent11am to 12pm 47 0.2112pm to 1pm 112 0.501pm to 2pm 66 0.29total 225 1.00

To complete the mapping, we need to make a cumulative distribution function (CDF)

Continuous Example – arrival time con’t

count percent CDF11am to 12pm 47 0.21 0.2112pm to 1pm 112 0.50 0.711pm to 2pm 66 0.29 1.00total 225 1.00

Make a new rule – if random number:<=.21, then =11am-12pm>.21 up to =.71, then 12pm-1pm>.71 up to 1, then 1pm to 2pm

Note on Random Numbers in Excel Spreadsheets

• Once entered in a spreadsheet, a random number function remains “live.” A new random number is created whenever the spreadsheet is re-calculated. To re-calculate the spreadsheet, use the F9 key. Note, almost any change in the spreadsheet causes the spreadsheet to be recalculated!

• If you do not want the random number to change, you can freeze it by selecting: tools, options, calculations, and checking “manual.”

Evaluating Results

• Simulation measures the quality of a solution because it gives the probability of a certain event occurring

• Simulation also shows the variability

• Simulation does not necessarily give the best possible answer. It gives the most likely answer.

Appendix

• Some useful information on very popular probability distribution function

Probability DistributionsA probability distribution defines the behavior of a variable by defining its limits, central tendency and nature– Mean– Standard Deviation– Upper and Lower Limits– Continuous or Discrete

Examples are:– Normal Distribution (continuous)– Binomial (discrete)– Poisson (discrete)– Uniform (continuous or discrete)– Custom (created to suit a specific purpose)

Uniform Distribution

• All values between minimum and maximum occur with equal likelihood

• Conditions– Minimum Value is Fixed– Maximum Value is Fixed– All values occur with equal likelihood

• Excel function: RAND() – returns a uniformly distributed random number in the range (0,1)

Normal Distribution

• Conditions:– Uncertain variable is symmetric about the mean– Uncertain variable is more likely to be in vicinity of the

mean than far away

• Use when:– Distribution of x is normal (for any sample size)– Distribution of x is not normal, but the distribution of

sample means (x-bar) will be normally distributed with samples of size 30 or more (Central Limit Theorem)

• Excel function: NORMSDIST() – returns a random number from the cumulative standard normal distribution with a mean of zero and a standard deviation of one [e.g., NORMSDIST(1) = .84]

Simulation

Continuous Variables

Distributions

• Variables to be simulated may be normal (e.g. height) or exponential (e.g. service time) or various other distributions.

• Task is to convert uniform distribution to the required distribution.

0 10 infinity

Freq

Freq

Application - Queuing Systems

• A queuing system is any system where entities (people, trucks, jobs, etc.) wait in line for service (processing of some sort) – – retail checkout lines, jobs on a network

server, phone switchboard, airport runways, etc.

Queuing System Inputs

• Queuing (waiting line) systems are characterized by:– Number of servers / number of queues

• SSSQ – Single Server Single Queue• SSMQ – Single Server Multiple Queue• MSSQ – Multiple Server Single Queue• MSMQ - Multiple Server Multiple Queue

– Arrival Rate (Arrival Intervals)– Service Rate (Service Times)

Performance Variables (outcome)

• Performance of a queuing system is measured by– Average time waiting in queue/system– Average number of entities in queue/system

Arrival time Service Begins Service Ends

Time in Queue Service Time

Time in System

Distributions in Queuing

• Arrival Intervals (time between two consecutive arrivals) and Service Time (time to serve one customer) are exponentially distributed.

• Confirm it yourself by watching cars on a street!

Sample Problem

• A loading dock (SSSQ) has trucks arriving every 36 minutes (0.6 hrs) on average, and the average service (loading / unloading) time is 30 minutes (0.5 hrs). A new conveyer belt system can reduce that to 15 minutes (0.25 hours). Simulate the arrival of 200 trucks to see how performance would be affected by the new system.

Simulating Exponential Distributions

• To convert the uniform distribution of the random numbers to an exponential distribution, take the negative natural log of the random numbers.– This creates an exponential distribution with an

average of 1.00.– To get an average of 0.6 (to represent average arrival

interval in hours), simply multiply result by 0.6.– Thus, the conversion formula is:

–ln(rand())*µwhere µ is the mean of the exponential distribution desired.

Sample Conversion

0.500645 1.040982

Random -ln(rand())

0.449796 0.798962

0.858464 0.15261

0.828061 0.188668

0.938751 0.063206

0.84637 0.166798

0.428408 0.847678

0.357574 1.028412

0.63932 0.447351

Average:

….. …..

0 1 0 infinity