1. 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: X01 fX(x)fX(x)0.100.90

2. 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)

3. 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

4. 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

5. 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

6. 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

7. 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.

8. 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( )

9. 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.

10. 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),"")

11. Monte Carlo Methods Focus on the Project: Change the numbers indicated to match your data Copy your new formula into cells G36:G194

12. 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)

13. 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.

14. 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

15. 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

16. 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),"")

17. 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

18. 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)

19. 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)

20. 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)

21. 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)

22. 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)

23. 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)