# How to Use the Extended Erlang B Formula

## Extended Erlang B Calculator

In telecommunication science, traffic is measured in dimensionless units called Erlangs. The number of Erlangs of traffic is equal to the average number of calls received per unit of time, multiplied by the average duration of a call. If you know the traffic level in Erlangs, the number of trunks (e.g. phone lines or servers), and the retry rate, then you can use the Extended Erlang B formula to calculate the probability that a call is dropped. This probability is known as the Grade of Sevice (GoS) for the call center, and it is always a number between 0 and 1.

The Extended Erlang B formula differs from the regular Erlang B formula in that some callers retry after their call is denied. By factoring in the rate of repeat callers, one can more accurately calculate the GoS. The retry or recall rate is a number between 0 and 1, for example, if 30% of the blocked callers hang up and call again, the retry rate is 0.3. If you are unsure how many callers try again, use a retry value of 0.5.

When the GoS is low, i.e., less than .05, then the call center is efficient at handling traffic. GoS values higher than 0.1 indicate that a call center does not have enough trunks to adequately handle its traffic.

The Extended Erlang B formula calculates GoS with a recursive application of the regular Erlang B formula. Though it is possible to perform these recursions by hand, it is faster to use an Extended Erlang B calculator, such as the one at left.

## The Extended Erlang B Formula

If the Erlangs of traffic is E and the number of trunk lines is M, then recall the Erlang B formula for the probability that a call is dropped:GoS = (E

^{M}/M!)/(∑

^{M}

_{n=0}E

^{n}/n!).

To calculate GoS that is adjusted to account for a recall factor of R, follow this procedure:

**(1)**First, calculate P

_{0}= GoS using regular Erlang B formula and the known values of E and M. Call the initial Erlang value E

_{0}.

**(2)**Next, calculate successive values of the Erlang with the recursive relation E

_{n+1}= E

_{0}+ R*E

_{n}*P

_{n}, where P

_{n}is the GoS for a traffic level of E

_{n}Erlangs.

**(3)**Repeat the recursion until the values of E

_{n}and P

_{n}stabilize. The stable value of P

_{n}is the true GoS for the system with a retry rate of R.

**Example:**Suppose a call center receives 480 calls per day, and the average duration of a call is 15 minutes. The center has 10 phone lines and the retry rate for dropped calls is 40%. Since 15 minutes = 1/96 days, the number of Erlangs in this system is (480)(1/96) = 5. The recall rate R equals 0.4. Thus, we start out with E

_{0}= 5 and

P

_{0}= (5

^{10}/10!)/(∑

^{10}

_{n=0}5

^{n}/n!) = 0.0183846

Now E

_{1}= 5 + (0.4)(5)(0.0183846) = 5.0367692. If we reapply the Erlang B formula to this value of E, we get

P

_{1}= (5.0367692

^{10}/10!)/(∑

^{10}

_{n=0}5.0367692

^{n}/n!) = 0.019081

Thus, E

_{2}= 5 + (0.4)(5.0367692)(0.019081) = 5.0384426 and so on. If we carry out this iteration several more times, the value of E stabilizes at 5.0385245, and the value of P stabilizes at 0.019115. Thus, the GoS for the system is about 1.91%.

When the initial GoS of the system is already quite low, retries do not have much of an impact on the adjusted GoS. When the initial GoS is high, retries make the system have higher rate of blocked calls. For systems in which blocked calls do not make multiple attempts, use the Erlang B Formula instead.

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