# How to Use the Engset Blocking Function

## Engset Blocking Calculator

In computing telecommunications traffic stats, there are the Erlang functions which assume infinite sources of traffic, and the Engset function which assumes a finite source of traffic. If you know the level of traffic measured in Erlangs, the number of available lines, and the size of the finite population, then you can use the Engset formula to calculate the probability that a call is blocked. The probability is also known as the Grade of Service, GoS.

The Engset function was developed by Tore Olaus Engset, who like Agner Krarup Erlang also studied the mathematics of traffic engineering. For extremely large sources of traffic, the Engset formula and Erlang B formula yield almost identical values for the GoS.

The Engset Blocking function can also be used to determine how many trunks are needed to achieve a desired Grade of Service, given fixed values for the population size and Erlangs.

The first calculator tells you how many phone lines are needed when input the traffic in Erlangs, desired probability of blocked calls, and size of the traffic source. The second calculator gives the GoS when you input the traffic in Elangs, number of lines, and size of the traffic source.

## The Engset Blocking Equation

If the Erlangs of traffic*per capita*is A, the number of trunk lines is M, and the source size S, then the Engset Blocking formula gives the probability that a call is dropped, aka the GoS:

GoS = (A

^{M})(S ch M)/[∑

^{M}

_{n=0}(A

^{n})(S ch n)],

where (X ch Y) is the combination formula--the number of ways of forming subsets of size Y from a larger set of size X. Notice that the Engset function does not use the same traffic input as the Erlang B, Erlang C, and Extended Erlang B formulas. For the Engset calculation, we must divide the total Erlangs by S to compute A.

**Example 1:**Suppose a call center only receives traffic from the residents of a small town with a population size of 500. Also suppose that the call center has 4 phone lines and receives 48 calls per day with an average duration of 30 minutes. Since 30 minutes = 1/48 days, the total number of Erlangs is (48)(1/48) = 1. The number of Erlangs per resident is 1/500 = 0.002. Thus, the probability that a call is blocked is

GoS = (0.002

^{4})(500 ch 4)/[∑

^{4}

_{n=0}(0.002

^{n})(500 ch n)] = 0.01538.

This means about 1.54% of the calls get dropped. As S becomes larger, the Engset function yields the same GoS as the Erlang B function.

**Example 2:**Recursive iterations can be used to determine M from a given values of A and S, and a desired value of GoS.

Suppose that a call center receives 4 calls per hour, the average call duration is 24 minutes, the total population of callers is 250, and the center would like a GoS value of 0.01.

First, we calculate the number of Erlangs per person as (4)(24/60)/250 = 0.0064. Using a recursive procedure, we find that the call center must have a minimum of 6 phone lines to accommodate the callers with a GoS of 0.01.

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The analogous formula for a system whose source is effectively infinite is the Erlang B formula.

If the system places calls in a queue rather than drops them, you must use the Erlang C Formula instead. If a significant portion of dropped calls retry, then use the Extended Erlang B Formula, which factors in a recall rate.

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