Erlang C Calculation

grogmi1

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Sep 17, 2010
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49
I'm hoping someone on here can help me. I have read through forums, watched endless videos and downloaded umpteen Excel spreadsheets to try and achieve a calculation that will work out how many staff I need in my call centre. Is there anyone on here that can help me, Please? I am simply looking for an Excel formula or even an explanation on how to work this figure out. I'd prefer to stay away from writing VBA if I can (it's not my strongest skill) but if it will get the job done then I'm ready and willing.
Thanks
 

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Using something similar to the approach outlined here:
...do you have some estimates/statistics that would allow you to use the inputs described in that description, namely something like this?
  • Number of calls – 100
  • In a period of minutes – 30
  • Average Handling Time (seconds) – 180 (3 minutes)
  • Required Service Level 80%
  • Target Answer Time (Seconds) 20
  • Maximum Occupancy 85%
  • Shrinkage 30%
If so, then I think a formula-based spreadsheet could work. What is the approximate size of the call center (i.e., number of agents)?
 
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Here is spreadsheet version based on the method outlined in the reference above. You should be able to click on the clipboard icon in the upper left at the intersection of rows/columns to copy this and then paste it into cell A1 in your workbook. The terms of art are described at the URL above. Your inputs are shaded blue.
ErlangC.xlsx
ABCDEFGHIJKLMNO
1Tool for Estimating Staffing for Call Center using Erlang C Method
2
31. Inputs:Helper area
4Number of calls100N = 111213141516
5In a period of minutes (min)30iΣ(Ai/i!)Σ(Ai/i!)Σ(Ai/i!)Σ(Ai/i!)Σ(Ai/i!)Σ(Ai/i!)
6Average Handling Time (AHT) (s)1800111111
7Required Service Level80%1111111111111
8Target Answer Time (s)202616161616161
9Maximum Occupancy85%3227.6667227.6667227.6667227.6667227.6667227.6667
10Shrinkage30%4644.3333644.3333644.3333644.3333644.3333644.3333
1151477.6671477.6671477.6671477.6671477.6671477.667
122. Determine number of calls per hour62866.5562866.5562866.5562866.5562866.5562866.556
13n_avg (# calls/h)20074850.6834850.6834850.6834850.6834850.6834850.683
1487330.8417330.8417330.8417330.8417330.8417330.841
153. Determine the Traffic Intensity (A)(hour basis)910086.5710086.5710086.5710086.5710086.5710086.57
16# of calls in hour basis2001012842.3112842.3112842.3112842.3112842.3112842.31
17Avg time per call (min)311 15347.5215347.5215347.5215347.5215347.52
18Call-minutes per hour60012  17435.1917435.1917435.1917435.19
19Traffic intensity (A) , call-hours per hour (Erlang unitless)1013   19041.119041.119041.1
2014    20188.1720188.17
214. Estimate the raw number of agents required (N)15     20952.89
22One agent max capability is 1 call-hour per hour actual time16      
23Initial guess for N agents required11121314151617      
24X term27557.3212526.056958.9194014.7612294.1491274.52718      
25Y term12842.3115347.5217435.1919041.120188.1720952.8919      
26Probability a call has to wait, Pw68.21%44.94%28.53%17.41%10.20%5.73%20      
2721      
285. Calculate the Service Level (list item #13 in ref)22      
29Service Level38.96%64.02%79.56%88.84%94.15%97.06%23      
30Target Service Level80%80%80%80%80%80%24      
31Target Service Level Met?FALSEFALSEFALSETRUETRUETRUE25      
3226      
336. Average Speed of Answer (ASA) (list item #15 in ref)27      
34Avg Speed of Answer (ASA) (seconds)122.781340.4449417.116237.8359373.6735251.7202128      
3529      
367. Percentage of Calls Answered Immediately (list item #16 in ref)
37Immediate Answer31.79%55.06%71.47%82.59%89.80%94.27%
38
398. Check Maximum Occupancy (list item #17 in ref)
40Occupancy90.91%83.33%76.92%71.43%66.67%62.50%
41Max Occupancy85%85%85%85%85%85%
42Max Occupancy constraint met?FALSETRUETRUETRUETRUETRUE
43
449. Factor in Shrinkage (list item #18 in ref)
45Number agents needed considering shrinkage of 30%161819202223
46
4710. Summary
48Number agents needed20
49Service Level88.8%
50Probability a call has to wait17.4%
51Average Speed of Answer (seconds)7.8
52% of calls Answered Immediately82.6%
ErlangC
Cell Formulas
RangeFormula
J4:O4J4=B23
I6:I35I6=ROWS(I$6:I6)-1
J7:O35J7=IF($I7<=J$4-1,J6+($B$19^$I7)/FACT($I7),"")
B13B13=B4/(B5/60)
B16B16=B13
B17B17=B6/60
B18B18=B16*B17
B19B19=B18/60
B23B23=B19+1
B24:G24B24=B$23/FACT(B$23)/(B$23-$B$19) * $B$19^B$23
B25:G25B25=MAX(INDEX($J$6:$S$35,,MATCH(B23,$J$4:$S$4,0)))
B26:G26B26=B24/(B24+B25)
B29:G29B29=1-B26*EXP(-(B23-$B$19)*($B$8/$B$6))
B30:G30B30=$B$7
B31:G31B31=B29>=B30
B34:G34B34=B26*$B$6/(B23-$B$19)
B37:G37B37=1-B26
B40:G40B40=$B$19/B23
B41:G41B41=$B$9
B42:G42B42=B40<B41
A45A45="Number agents needed considering shrinkage of "& TEXT($B$10,"0%")
B45:G45B45=ROUNDUP(B23/(1-$B$10),0)
B48B48=AGGREGATE(15,6,(B45:G45)/((B31:G31)*(B42:G42)),1)
B49B49=INDEX($B$29:$G$29,MATCH(B$48,$B$45:$G$45,0))
B50B50=INDEX($B$26:$G$26,MATCH(B$48,$B$45:$G$45,0))
B51B51=INDEX($B$34:$G$34,MATCH(B$48,$B$45:$G$45,0))
B52B52=INDEX($B$37:$G$37,MATCH(B$48,$B$45:$G$45,0))
Cells with Conditional Formatting
CellConditionCell FormatStop If True
B42:G42Cell Value=TRUEtextNO
B31:G31Cell Value=TRUEtextNO
 
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I have read through forums, watched endless videos and downloaded umpteen Excel spreadsheets to try and achieve a calculation that will work out how many staff I need in my call centre. Is there anyone on here that can help me, Please?

Presumably the "umpteen spreadsheets" work. So I don't see any explanation of the problem that you are having.

Are you simply running into computational limits, for example (using the notation in the article that KRice cites): A^N exceeds about 1.8E+308; or N exceeds 171, exceeding the limit of FACT(170)?

Either might cause #NUM errors.

The root cause might be that you are simply trying to use the Excel implementation for a "raw number of agents required" (N) that exceeds the limits of computation.

If KRice's spreadsheet (#umpteen+1 :rolleyes:) does not help you, please explain your problem.



Here is spreadsheet version based on the method outlined in the reference above.

For the calculation of the "Y term" in row 25, you can eliminate the helper cells in columns I:O and replace the formulas in B25:G25 with the following formula in B25, then copy across:

=1 + SUMPRODUCT($B$19^ROW($A$1:INDEX($A:$A, B$23-1)) / FACT(ROW($A$1:INDEX($A:$A, B$23-1))))

This follows from the mathematical Sigma expression directly. The value 1 replaces the term A^0/FACT(0), which is 1/1 = 1.

Note: B23 ("N") must be 2 or more. The formula could be tweaked to allow for B23=1, if necessary, to wit:

=IF(B$23<2, 1, 1 + SUMPRODUCT($B$19^ROW($A$1:INDEX($A:$A, B$23-1)) / FACT(ROW($A$1:INDEX($A:$A, B$23-1)))))
 
Last edited:
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Too late to edit....
If KRice's spreadsheet (#umpteen+1 :rolleyes:) does not help you, please explain your problem.

Include data similar to the input parameters described in the article that KRice cites.
 
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@joeu2004 , thanks for the improvement to eliminate the helper cell area. I too was wondering whether computation limits were the issue, which is why I asked about the approximate size (number of agents). Some large number approximations may need to be incorporated, but I didn't want to go to that trouble unless it is necessary.

To recap the use of the spreadsheet, @grogmi1, read through the link I referenced and then adjust the blue cells in B4:B10. You may want to leave B9:B10 as is...according to the guidance, those are fairly typical numbers. But you'll need to make some estimates about the other cells based on your particular business and the nature of the support. Once those upper blue cells have been reviewed, jump down to the lower blue cells (B23:G23). The first is fixed based on an initial optimistic guess from the upper blue inputs. You'll want to change the numbers in C23:G23 until you see that other targets/constraints are met (conditional formatting turns those cells green in rows 31 and 42). Once you have green highlighting for a common number of agents (in this example, that occurs at N=14), the smallest value (in green in both rows) is further burdened by the shrinkage factor to account for staffing outages due to any number of reasons...and that is why N=14 (on row 23) becomes N=20 as the final estimate.

If you are dealing with significantly larger numbers, some some other adjustments will be necessary...so more insight about the issues you've encountered will be necessary to offer further guidance.
 
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The easiest way is to click the clipboard icon in the upper left of the spreadsheet snippet (at the intersection of row and column headings) and then paste that into a fresh worksheet in the same cell as the upper left corner (in this case, A1). That should give complete functionality. Let me know if you run into any issues.
 
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For convenience, here is the file with the improvement in the factorial formula offered by @joeu2004, so the helper columns aren't needed.
 
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Hello, seeking someone to help me on this.

I'm trying to understand the number of agents required per day with the volume of 4620 (in 12 hours) SL 80% within 20 secs, AHT at 308 secs, max occupancy set at 85%, and shrinkage at 18.9%.

Based on the online Erlang calculator from the contact center helper i'm getting 78.4 FTE per day (i've attached the image below for reference),

I've also used the Excel calculator from contact center helper, where i'm getting 49 agents.
Erlang - CCHelper.xlsm

i've also use the excel file that was provided by Kirk, where i'm getting 61 agents.
Erlang - Kirk.xlsx


I'm actually trying to understand the number of agent required per day with the above assumptions, and i've have to propose multiple simulation. Seeking assistant to all excel + wfm expert over here.


Thanks a lot in advance.
 

Attachments

  • ErlangCC - 1.png
    ErlangCC - 1.png
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  • ErlangCC - 2.png
    ErlangCC - 2.png
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