# Erlang Calculator

## How To Use This Erlang Calculator

If you have 200 calls per hour, then enter the number of incoming contacts as 200 and the period is 60 minutes.
The Average Handling Time is the amount of time that a person (an agent) takes to handle a phone contact. This includes the talk time as well any paperwork time (wrap-up time) before they are able to answer the next call. This should be in seconds.
Put in your Service Level target and time. So if you wanted to handle 90% of calls in 15 seconds, put in 90 and 15. If you are uncertain of this the industry “average” is 80% of calls answered in 20 seconds.
This contact centre staffing Erlang calculator is a hybrid model based on both the Erlang C formula the Erlang A formula. The Erlang C formula was invented by the Danish Mathematician A.K. Erlang and is used to calculate the number of advisors and the service level. Call Abandons are calculated using the Erlang A formula which was devised by Swedish statistician Conny Palm in 1946. This assumes an Average Patience – also know as Average Time to Abandon (ATA).
This calculator works on probabilities, so may appear to overstate the number of agents needed at low levels. So for example, if you enter O calls per hour it will say that you need 1 agent. This is quite correct, as there may be a possibility that one call may come in. In practicality, you may decide to not schedule any staff.
The maximum number of agents that the calculator can calculate before shrinkage is applied is 10,000 Agents.
Shrinkage is a factor designed to take into account holidays, sickness etc.
The maximum occupancy is designed to improve accuracy. If you take Occupancy over 85%-90% for long periods you will find that it gets hidden in a longer AHT figure, and agent burn out happens.
Call Abandons are calculated using the Erlang A formula, which assumes an Average Patience -also know as Average Time to Abandon (ATA).
The Calculator can deal with up to 10,000 agents, thanks to some help with the maths from Philip Wright CEng – (Former Technical Director & CTO Europe at Aspect Telecommunications/Communications 1988-2001).

An Erlang calculator is one of the most useful Workforce Management (WFM) tools in the call centre toolkit.
An Erlang Calculator is a mathematical calculation that allows you to calculate the number of staff that you need for a given number of calls, to meet a given service level.
It is based on the Erlang C formula (a derivative of the Poisson distribution) that was designed by the Danish Mathematician A.K. Erlang around 100 years ago. The formula is quite involved, but is relatively easy to follow if you studied maths to a reasonable level at school.
You simply enter in the number of phone calls that you receive in a period of time (say per half hour), along with the average duration of the calls and also the service level that you are looking for.
Typical Inputs
Number of phone calls
Time period (e.g. per half hour)
Average Call Duration (Average Handling Time)
Service Level (Percentage of calls answered within a period of time, e.g. 80% of calls in 20 seconds)
Some Erlang calculators also include a shrinkage input.
Typical Outputs
Number of agents (advisors) needed to meet the service level target

### Inputs

Number of calls 100
In a period of minutes 30
Average Handling Time (seconds) 180
Required Service Level 80%
Maximum Occupancy 85%
Shrinkage 30%

### Lets start by working out N!

``````So let's break down the formula into parts
N! is mathematical notation for N factorial
A factorial is the  product of an integer (whole number) and all the integers (whole numbers) below it
For example 3! = 3 Factorial = 3 x 2 X 1 = 6
11! = 11 Factorial = 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 =  39916800

Luckily Excel has a formula that can do this for you
So for 11 Factorial  it is =FACT(11)
N! = 11 Factorial = FACT (11)   39916800                ``````

### Be careful with large factorials

Note that factorials get very large as you increase the number of agents and very soon can cause Excel to overload.
10 Factorial 3628800
100 Factorial 9.3326E+157
170 Factorial 7.2574E+306
171 Factorial #NUM! This will cause errors in your calculations

``````So it won't easy work in Excel above 170 Agents
Luckily we have devised a Fast Erlang formula  for high number so agents``````

### Work out the Powers – AN

So AN means A multiplied by A, N times
e.g. 2 to the Power of 2 = 22 = 2 x 2 = 4

``````So if A = 10 and N = 11 then AN = 1011 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10
Luckily Excel has a formula for this = POWER (10,11)
AN = 1011 = POWER (10,11)   1E+11               ``````

### Let’s simplify the Erlang Formula

``````We can use a substitution
X = and
Y = So Pw = X / ( Y + X)``````

### Work out the top row of the Erlang C Formula (X)

`````` Top Row = AN / N! * (N / (N-A))
Top Row = 1011 / 11! * (11 / (11-10))  27557.32``````

### Work out the Sum of a series (Y)

``````Now this looks scary but is simply adding together a sum of numbers looping  through a number of times
In this case you start with i=0 and keep going through until i= N-1
so if N = 11 , N - 1 = 10 and we loop through from i= 0 until i = 10
and Σ is the mathematical symbol for the sum of a series so you add together all of the results of the individual loops

i   i!  Ai  Ai/i!   Σ Ai/i!
0   1   1   1   1
1   1   10  10  11
2   2   100 50  61
3   6   1000    166.7   227.7
4   24  10000   416.7   644.3
5   120 100000  833.3   1477.7
6   720 1000000 1388.9  2866.6
7   5040    10000000    1984.1  4850.7
8   40320   100000000   2480.2  7330.8
9   362880  1000000000  2755.7  10086.6
10  3628800 10000000000 2755.7  12842.3

Which gives the sum of the series Y =   12842.30511

Now note that the maths becomes simplified                  ``````

### Put X and Y into the Erlang C Formula (The probability a call has to wait)

`````` So Pw = X / ( Y + X)

Pw = 27557 / (12842 + 27557)    0.6821
Or in percentage terms  68.2%               ``````

### Calculate Service Level

``````Service Level Follows the Formula

SL= 1 - (Pw *e -(N - A) * (TargetTime / AHT))

Where e is the mathematical constant (Euler's number) = 2.71828

Luckily Excel has an Equation for this =EXP()
Lets work out -(N - A) * (TargetTime / AHT)
Lets work out -(11 - 10) * (20 /180)    -0.111

Then work out Service Level
SL =  1- (0.6821 * EXP(-0.11111))   0.390
Service Level as a Percentage   39.0%
This is below the target of 80% so we need to increase the number of agents                 ``````

### Increase the Number of Agents by 1 to see If Service Level is achieved

#### Number of Agents N 12

Traffic Intensity A 10
Work out X 12526.1

``````i   i!  Ai  Ai/i!   Σ Ai/i!
0   1   1   1   1
1   1   10  10  11
2   2   100 50  61
3   6   1000    166.7   227.7
4   24  10000   416.7   644.3
5   120 100000  833.3   1477.7
6   720 1000000 1388.9  2866.6
7   5040    10000000    1984.1  4850.7
8   40320   100000000   2480.2  7330.8
9   362880  1000000000  2755.7  10086.6
10  3628800 10000000000 2755.7  12842.3
11  39916800    100000000000    2505.2  15347.5

Work out Y  15347.5
Probability Call Waits  0.4494
Service Level    0.640
Service Level as a Percentage   64.0%
This is below the target of 80% so we need to increase the number of agents                 ``````

#### Increase the Number of Agents by 1 to see If Service Level is achieved

Number of Agents N 13
Traffic Intensity A 10
Work out X 6958.9

``````i   i!  Ai  Ai/i!   Σ Ai/i!
0   1   1   1   1
1   1   10  10  11
2   2   100 50  61
3   6   1000    166.7   227.7
4   24  10000   416.7   644.3
5   120 100000  833.3   1477.7
6   720 1000000 1388.9  2866.6
7   5040    10000000    1984.1  4850.7
8   40320   100000000   2480.2  7330.8
9   362880  1000000000  2755.7  10086.6
10  3628800 10000000000 2755.7  12842.3
11  39916800    100000000000    2505.2  15347.5
12  479001600   1000000000000   2087.7  17435.2

So value of Y   17435.2
Probability Call Waits  0.2853
Service Level    0.796
Service Level as a Percentage   79.6%
This is below the target of 80% so we need to increase the number of agents                 ``````

#### Increase the Number of Agents by 1 to see If Service Level is achieved

Number of Agents N 14
Traffic Intensity A 10
Work out X 4014.8

``````i   i!  Ai  Ai/i!   Σ Ai/i!
0   1   1   1   1
1   1   10  10  11
2   2   100 50  61
3   6   1000    166.7   227.7
4   24  10000   416.7   644.3
5   120 100000  833.3   1477.7
6   720 1000000 1388.9  2866.6
7   5040    10000000    1984.1  4850.7
8   40320   100000000   2480.2  7330.8
9   362880  1000000000  2755.7  10086.6
10  3628800 10000000000 2755.7  12842.3
11  39916800    100000000000    2505.2  15347.5
12  479001600   1000000000000   2087.7  17435.2
13  6227020800  10000000000000  1605.9  19041.1

So value of Y   19041.1
Probability Call Waits  0.1741
Service Level    0.888
Service Level as a Percentage   88.8%
This is above the target of 80%

So the Raw Number of Agents      14                 ``````

ASA = Pw x AHT / (N-A)
ASA = 0.1741 x 180 / (14 – 10) 7.8 seconds

### Percentage of Calls Answered Immediately

Immediate Answer = (1- 0.1741) 82.6%

### Check Maximum Occupancy

Raw Agents 14
Traffic Intensity A 10
Occupancy = Traffic Intensity (Erlangs) / Raw Agents 71.4%
This is less than 85% so we can keep it at this.
If this is more than 85% the number of Raw Agents is Traffic Intensity/ (Occupancy %/100)
If you have an occupancy of more than 85% your agents will likely get burnt out and you will find that Average Handling Time may rise to cover it.

### Factor in shrinkage

Shrinkage is a factor that is widely used in the industry to include holidays, sickness, training and meetings etc. The industry average is around 30 – 35%
Shrinkage 30%
Number of Agents Required = Raw Agents/ (1- (Shrinkage% /100))
Number of Agents Required = 14/ 0.7 20

### Summary

So the total number of Agents Required 20
Service Level 88.8%
Probability a call has to wait 17.4%
Average Speed of Answer 7.8 Seconds
% of calls Answered Immediately 82.6%