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 1 f
X
(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
 y

1
 e
 x /
 e
 x /
 
1
 y
 x /
  x
 ln( 1

  y )
 ln( 1
 y )
• The inverse function is
F

1     ln( 1

• Here x would be replaced by RAND( ) x )

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$194<=I35)
•
Cell Y354 should have the formula
•
Cell Y356 should have the formula
=($F$19 5 <=I35)
=($F$19 6 <=I35)
•
Cell Y358 should have the formula =($F$19 7 <=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:E
195
)=
161
,"Overflow",MAX(E35:E
195
))
(new ending cell in column E)

Monte Carlo Methods
• Focus on the Project:
• O35 contains
=SUM(J35:J
194
)
(new ending cell in column J)
• P35 contains
=MAX(J35:J
194
)
(new ending cell in column J)

Monte Carlo Methods
• Focus on the Project:
• Q35 contains
=COUNTIF(K35:K
194 ,”yes”)
(new ending cell in column K)
• R35 contains
=SUM(L35:L
194
)
(new ending cell in column L)

Monte Carlo Methods
• Focus on the Project:
• S35 contains
=SUM(L35:L
194
)
(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
)