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Erlang Library for Excel Function Reference |
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Erlang Library for Excel Function Reference |
Returns the probability that a customer arriving at the queueing facility will experience a delay before beginning service.
ErlcFractionDelayed( nsrv, trafficInErlangs)
nsrv is the number of servers (can be any non-negative number).
trafficInErlangs is the offered traffic in erlangs (can be any non-negative number).
| Formula | Description | Return Value |
|---|---|---|
| =ErlcFractionDelayed(11, 10.1) | Returns the probability that a customer will experience a wait, when there are 11 servers and 10.1 erlangs of offered traffic. | 0.71095477 |
| =ErlcFractionDelayed(133.78, 130.1) | Returns the probability that a customer will experience a wait, when there are 133.78 servers and 130.1 erlangs of traffic. | 0.65932309 |
Public Function ErlcFractionDelayed(nsrv As Double, trafficInErlangs
As Double) As Double This is the classical "Erlang C" function. Its definition in the Erlang Add-In for Excel uses the Erlang B function ErlbBlockage, and is as follows::
ErlcFractionDelayed(n, x) = n * ErlbBlockage(n,x) / (n - x*(1 - ErlbBlockage(n, x))) if n > x,
ErlcFractionDelayed(n, x) = 1 otherwise.
Here, n is the number of servers, and x is the offered traffic in erlangs. The function is undefined if n<0 or x<0. Also, you normally should call this function only with n>x. Mathematically it makes no sense to call the function with n ≤ x. However it is convenient for the function to return 1 in that case, rather than triggering an error.
The queueing model that applies here is the M/M/n/∞ queue, also known as the Erlang C queueing model. That is: Poisson arrivals, exponential service time, n servers, and infinitely slots for customers in the system.
If this worksheet function is not available, and returns the #NAME? error, then you must install and load the Erlang Library for Excel from Abstract Micro Systems.
How?
Analysts often use Erlang C to estimate the probability of delay for incoming calls in a call center. Suppose that we have a call center with these parameters during a certain period of time:
| Maximum number of calls that can be waiting: | Very large |
| Period length: | 30 minutes |
| Number of agents: | 103 |
| Incoming calls offered during the period: | 1000 |
| Average handle time: | 3 minutes |
Assume further that the calls arrive in a Poisson process, that the handle times are exponentially distributed, and that the queue of waiting calls is processed in a FIFO manner. Finally, assume that if a caller finds that all agents are busy, then the caller will wait until an agent picks up the call.
We are interested in estimating the probability of delay p, defined as the fraction of callers that are not served immediately. We use the Erlang C function as follows. First, we compute the offered load x in erlangs:
x = (calls per period)*(call length) / (period length) = 1000 * (3 minutes) / (30 minutes) = 3000/30 = 100 erlangs
Then the probability of delay is
p = ErlcFractionDelayed( nsrv, trafficInErlangs) = ErlcFractionDelayed( 103, 100) = 0.6808
In other words, we find that the probability of delay is approximately 68 percent. In other words, a little over 2/3 of the callers will experience a wait before being served.