-- Erlang B --

June 1998

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In the 1920's, a Danish mathematician and telephone traffic engineer named A. K. Erlang studied delays in telephone traffic and came up with several models which are still in use today. These models deal with the following terms:

**Arrival Rate**- lambda- The mean number of arrivals per unit time. If five hundred calls arrive in an hour, on average, then the
*Arrival Rate*is 500 calls/hour. The reciprocal (1/lambda) is then the average amount of time that seperates incoming calls. **Service Rate**- mu- The mean number of calls serviced per unit time. If it takes 20 minutes to service one call, then 3 calls may be serviced in one hour and the
*Service Rate*is 3 calls/hour. The reciprocal (1/mu) is the average time taken to service one call, given in units of time per call (ie. 0.333 hours/call or 20 minutes/call in the current example) **Number of Lines**- c- The number of independent phone lines. This determines the number of calls that can be serviced concurrently.
**Capacity**- The size of the call queue. This represents the total number of calls that can be serviced at a given time as well as the total number of calls that can be held until a line becomes free. Notice that in a system with no holding queue, the capacity is equal to the number of lines.
**Busy Hour**- The busiest, one-hour period of the day. This is the period when incoming calls are most likely to be blocked or turned away, so this is the time for which statistics are calculated.
**Busy Hour Traffic (BHT)**- rho- The amount of traffic in the system during the
*Busy Hour*. This can be measured in several ways, including call minutes and call hours. In our example system, where 500 calls come in per hour and each call takes 20 minutes (0.333 hours) to service, then the system has to handle 500 * 0.333 = 166.5 call hours during the busiest hour of the day. **Erlang Rho>**- Another common unit is the
**Erlang**, representing the ratio of*Arrival Rate*to*Service Rate*(lambda/mu). For example if we expect 500 calls/hour, the*Arrival Rate*lambda = 500. If the average duration of a call is 20 minutes, or 0.333 hours, then the*Service Rate*mu = 3. BHT is then calculated as lambda/mu = 500/3 = 166.67 **Grade of Service**- P(c)- The percentage of incoming calls turned away during the
*Busy Hour*because all lines are busy at the time of the call. Notice that this statistic is a function of the number of lines, with more lines resulting in a lower and therfore better*Grade of Service*.

The Erlang B model deals with a traffic system in which:

*Arrival Rate*is Poisson distribution with parameter lambda (l);*Service Rate*is exponential with parameter mu (u);*Number of Lines*is c;*Capacity*is also c.

Other common usages include calculating various traffic load statistics such as the maximum service time and the minimum number of phone lines necessary to maintain a given Grade of Service.

Traditional, straightforward evaluation of the given equation results in overflow errors unless rho and c are relatively small (between 100 and 200). To circumvent these problems, we can reformulate the Erlang B formula as:

Implementing this rewritten formula efficiently vastly reduces the magnitude of computation, thus allowing larger values for the system. A more detailed discussion of this approach can be found in the companion paper, Qiao and Qiao (1998).

This calculator will handle systems at least as large as rho=1e8,c=1e7, but become prohibitively slow as the systems's magnitude increases. The calculation time becomes noticable when (rho*c) exceeds 10^7. If the calculator's behaviour becomes confusing, clear the fields and try again.

**Calculate**button will calculate the specified value for the corresponding inputs.**Clear All Values**button will clear all values in every fields.**Use Exist Values**button will allow you to use already inputed or calculated values for other input fields.

- Qiao, L. and Qiao, S. 1998,
*A Robust and Efficient Algorithm for Evaluating Erlang's Formula*, Manuscript. - Intel Support for ProShare Video Traffic

Last updated on Mar. 13 2000