Monte Carlo Methods

• A Monte Carlo simulation creates samples from a known distribution

• For example, if you know that a coin is weighted so that heads will occur 90% of the time, then you might assign the following values:

X 0 1fX(x) 0.10 0.90

Monte Carlo Methods

• If you tossed the coin, the expected value would be 0.9

• However, a sample simulation might yield the results 1, 1, 1, 0, 1, 1, 0, 1, 0, 1

• The average of the sample is 0.7 (close, but not the same as the expected average)

Monte Carlo Methods

• Another type of simulation can be run using the RAND function

• RAND chooses a random number between 0 and 1

• Entered as RAND( )

• Used for continuous random variable simulations

Monte Carlo Methods

• The outputs will include as many decimal places as Excel can keep

• This is used to model situations where you have a continuous random variable

• There would be an infinite number of possible outcomes

Monte Carlo Methods

• The IF function in Excel determines a value based upon a logical TRUE/FALSE scenario

• If math formula is true, then one outcome happens

If math formula is false, then another outcome happens

Monte Carlo Methods

• Ex. The situation where heads occurs 90% of the time can be simulated by using RAND and IF functions.

=IF(RAND()<=0.90,1,0)

• We can use COUNTIF to count the number of times an outcome occurs

Monte Carlo Methods

• If we have a variable with a known distribution, we may construct the c.d.f. function

• Once we have this, a simulation can be run from the inverse of the c.d.f.

Monte Carlo Methods

• For example, if we have an exponential function with a known value

• The inverse function is • Here x would be replaced by RAND( )

)1ln(

)1ln(/

1

1/

/

yx

yx

ye

eyx

x

)1ln(1 xxF

Monte Carlo Methods

• Focus on the Project:

• Enter mean time between arrivals for variable A in cell B31 of the sheet 1 ATM for the Excel file Queue Focus.xls.

Monte Carlo Methods

• Focus on the Project:

• The formula in cell G35 of the sheet 1 ATM for the Excel file Queue Focus.xls needs to be changed

• Original:

=IF(ISNUMBER(F35),VLOOKUP(RANDBETWEEN(1,7634),

Data!$G$45:Data!$H$7678,2),"")

Monte Carlo Methods

• Focus on the Project:

• Change the numbers indicated to match your data

• Copy your new formula into cells G36:G194

Monte Carlo Methods

• Focus on the Project:

• Note that my simulation (from my posted SampleData.xls) must accommodate 170 customers

• Drag the information in cells B195:C195 down until the last value in column B is one more than the number of customers (for me, 171)

Monte Carlo Methods

• Focus on the Project:

• Drag the information in cells E195:F195 down until the last values are at the same row as the values in columns B and C.

• Drag the information in cells G194:L195 down until the last values are one row above the values in columns E and F.

Monte Carlo Methods

• Focus on the Project:• The finished columns E through L should look like:

• Note: columns

E and F have one

extra cell

Monte Carlo Methods• Focus on the Project:• The formulas in column L need a special modification• The formulas in cell L193 is:

=IF(ISNUMBER(F193),DCOUNT($I$34:I192,,Y349:Y350),"")

• The formula in cell L194 is: =IF(ISNUMBER(F194),DCOUNT($I$34:I193,,Y351:Y352),"")

• Notice as we go down 1 row, Y349:Y350 becomes Y351:Y352

Monte Carlo Methods• Focus on the Project:• You must modify the formulas according to this pattern• So for cell L195, the formulas would be:

=IF(ISNUMBER(F195),DCOUNT($I$34:I194,,Y353:Y354),"")

• Continue this pattern for the extra rows you added . . .• In my example, I added 10 rows in column L, so my last

modification appears in cell L204:

=IF(ISNUMBER(F204),DCOUNT($I$34:I203,,Y371:Y372),"")

Monte Carlo Methods

• Focus on the Project:

• Cells Y351 and Y352 should be copied and pasted several times

• My simulation must accommodate 170 customers (compared to 160 from the original class file)

• This means I must copy and paste Y351 and Y352 ten times

Monte Carlo Methods

• Focus on the Project:

• Cell Y351 is blank, so new cells Y353, Y355, Y357, etc. will also be blank

• Cell Y352 contained the formula

=($F$194<=I35)

Monte Carlo Methods

• Focus on the Project:

• Cell Y352 contained the formula =($F$194<=I35)

• Cell Y354 should have the formula =($F$195<=I35)• Cell Y356 should have the formula =($F$196<=I35)• Cell Y358 should have the formula =($F$197<=I35)• And so on … (Be careful, you must carefully change all of the

new formulas)

Monte Carlo Methods

• Focus on the Project:

• Finally, we need to modify the formulas in cells N35:S35

• N35 contains (# of customers plus 1)

=IF(MAX(E35:E195)=161,"Overflow",MAX(E35:E195))

(new ending cell in column E)

Monte Carlo Methods

• Focus on the Project:

• O35 contains

=SUM(J35:J194)

(new ending cell in column J)

• P35 contains

=MAX(J35:J194)

(new ending cell in column J)

Monte Carlo Methods

• Focus on the Project:

• Q35 contains

=COUNTIF(K35:K194,”yes”)

(new ending cell in column K)

• R35 contains

=SUM(L35:L194)

(new ending cell in column L)

Monte Carlo Methods

• Focus on the Project:

• S35 contains

=SUM(L35:L194) (new ending cell in column L)

• Finally, run the Macro One_ATM

• Save the results in a folder (do not change the name of the Excel file Queue Focus.xls)