Erlang C Calculation

grogmi1

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Joined
Sep 17, 2010
Messages
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
 
I think the underlying assumptions may not quite fit your application. It sounds as if you may have more than one person working on a given item during the 3-day answer period. I also don't think my one question was clear:
What is the best way to describe this operation's working period? It appears that 1 month consists of 20 working days. Does one work day (the time during which work is actually accomplished) consist of 8 h or 24 h?
  • The operation can take 3 days to complete after initially being assigned which historically on average has been 2 days.
I could see that you are considering a "month" to be 20 working days. My question was trying to clarify what is meant by a "working day". Do you have staff working around the clock, 24 hours in a day, or is your operation best described as an 8-hour workday, and then no one is working for the remaining 16 hours of that day? Your answer to this question appears to focus on the number of workdays necessary, on average, to address an inquiry, but that was the focus of my 2nd question...rather than the 1st one.
 
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I think the underlying assumptions may not quite fit your application. It sounds as if you may have more than one person working on a given item during the 3-day answer period. I also don't think my one question was clear:

I could see that you are considering a "month" to be 20 working days. My question was trying to clarify what is meant by a "working day". Do you have staff working around the clock, 24 hours in a day, or is your operation best described as an 8-hour workday, and then no one is working for the remaining 16 hours of that day? Your answer to this question appears to focus on the number of workdays necessary, on average, to address an inquiry, but that was the focus of my 2nd question...rather than the 1st one.
1 work day is composed of an 8hr period and 1 month consists of 20 days. The remaining 16hrs has no one monitoring our queues which at the moment does get factored into the total cycle time.
 
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Thanks for the clarification...that is what I was assuming (1 work day = 8 h, 1 mo = 20 work days). From my understanding, the equations developed for Erlang C are assuming a fairly homogeneous situation where call (inquiry) demand and the "machinery" to handle the demand are consistent. For that reason, in your analysis, I would consider a "day" to be an 8-hour block (out of a 24-h calendar day) during which tangible work is performed. I'm not sure how your system works regarding inputs, but I'll assume that inquiries might be made anytime during a calendar day, and if so, they are queued, with no action taken until the next workday. Then the inquiry rate (in cell B13 on my worksheet) is a somewhat artificial number that distributes inquiries over the 8-hour workday.

In post #18, you mentioned:
...it looks like I would need to further expand the number of FTEs that would normally be applicable...
I'm not sure what you mean, but wanted to point out that you can modify cells C23 (and those to the right) to consider any number of staff. B23 represents the baseline minimum number, but manually change C23 (and those to the right) to anything that might be of interest. I typically change the formula to increment by something other than 1 to identify approximately where other metrics (Target Service Level, Occupancy, Shrinkage) all appear to be met, and then explore the range in that neighborhood at a smaller resolution to identify the estimated optimal number of staff.

If we set up a sheet using this example:
Using May's numbers as a baseline, a count of 22 FTEs on average each closed 28 requests with a cycle time of 11 days. The impacting factor was the wait time for routed requests, where from a hands to keys perspective they would still be individually taking 3 days to complete the work after assigning within <=2 days. The ability to close the 28 requests is what I would like to somehow factor in to the WFM table to give me a true number of FTE's to confirm whether or not staffing should be 22 or another number.
...using these redefined "days" values, converted into units expected by the worksheet
ErlangC_20240220_20240801.xlsx
HIJ
511d, redefine "day" as 8 h then convert to min5280
63days, redefine day as 8 h, convert to s86400
7
82days, redefine day as 8 h, convert to s57600
ErlangC_not_a_swim_fan2 (3)
Cell Formulas
RangeFormula
J5J5=H5*8*60
J6,J8J6=H6*8*60*60

...the model suggests that 16 staff would be needed to handle the 28 requests while satisfying the three metrics (target service level, max occupancy, shrinkage). Does this make sense to you?...that you might be able to close out 28 requests over 11 days with fewer staff, and without those staff feeling too overwhelmed, while also having sufficient time to deal with other things?
ErlangC_20240220_20240801.xlsx
ABCDEFG
31. Inputs:
4Number of inquiries28
5In a period of minutes (min)5280
6Average Handling Time (AHT) (s)86400
7Required Service Level95%
8Target Answer Time (s)57600
9Maximum Occupancy85%
10Shrinkage30.0%
11
122. Determine number of calls per hour
13n_avg (# calls/h)0.318
14
153. Determine the Traffic Intensity (A)(hour basis)
16# of calls in hour basis0.318
17Avg time per call (min)1440
18Call-minutes per hour458.18
19Traffic intensity (A) , call-hours per hour (Erlang unitless)7.636
20
214. Estimate the raw number of agents required (N)
22One agent max capability is 1 call-hour per hour actual time
23Initial guess for N agents required8910111213
24X term6309.51606.1786.2421.9225.8116.9
25Y term1046.01332.81576.11761.91890.91973.0
26Probability a call has to wait, Pw85.78%54.65%33.28%19.32%10.67%5.59%
27
285. Calculate the Service Level (list item #13 in ref)
29Service Level32.69%77.98%93.12%97.95%99.42%99.84%
30Target Service Level95%95%95%95%95%95%
31Target Service Level Met?FALSEFALSEFALSETRUETRUETRUE
32
336. Average Speed of Answer (ASA) (list item #15 in ref)
34Avg Speed of Answer (ASA) (seconds)203813346261216549622112901
35
367. Percentage of Calls Answered Immediately (list item #16 in ref)
37Immediate Answer14.22%45.35%66.72%80.68%89.33%94.41%
38
398. Check Maximum Occupancy (list item #17 in ref)
40Occupancy95.45%84.85%76.36%69.42%63.64%58.74%
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%121315161819
46
4710. Summary
48Number agents needed (minimum of #9 where #5 and #8 are satisfied)16
49Service Level97.9%
50Probability a call has to wait19.3%
51Average Speed of Answer (seconds)4962.2
52% of calls Answered Immediately80.7%
ErlangC_not_a_swim_fan2 (3)
Cell Formulas
RangeFormula
B13B13=B4/(B5/60)
B16B16=B13
B17B17=B6/60
B18B18=B16*B17
B19B19=B18/60
B23B23=ROUNDUP(B19,0)
C23:G23C23=B23+1
B24:G24B24=B$23/FACT(B$23)/(B$23-$B$19) * $B$19^B$23
B25:G25B25=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$23-1)))))
B26:G26B26=B24/(B24+B25)
B29:G29B29=1-B26*EXP(-(B23-$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))
Press CTRL+SHIFT+ENTER to enter array formulas.
Cells with Conditional Formatting
CellConditionCell FormatStop If True
B31:G31Cell Value=TRUEtextNO
B42:G42Cell Value=TRUEtextNO
B45:G45Expression=AND(B31,B42)textNO

I have another draft version of this worksheet that might be useful for cases where the staff size is so large that the factorial function (FACT) in Excel fails because the value exceeds Excel's limit. That version revisited Erlang's original development to rewrite expressions in terms of other functions available in Excel to avoid the limitations with FACT. I never quite finalized that effort, but could if errors are generated in the row 24/25 computations.
 
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