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Chapter: Cellular Telephone Networks
Section: Performance Analysis, Telephone Traffic Analysis

Erlang B

Call Blocking

In an Erlang-B telephone system, N channels are available. New calls are assigned a channel until all channels are full. Whenever all channels are occupied, a new call is blocked. That is, it is denied a channel. the assumption is that that the calls is lost and that the calling subscriber will not retry again. This in contrast to an Erlang-C system, in which new calls are queued, until they can be served.

Model

The number of active calls is a Markov process, i.e., at any instant t the number of active calls statistically only depends on the number of active calls at t - Dt, with very small D t. New calls arrive according to Poisson process with rate l calls per unit of time. Calls have a (memoryless) exponential duration with mean 1/m. Hence a call terminates with "rate" m and for i calls the rate of termination of one call is i timesm.

  Figure: Markov model. The state represents the number of occupied channels in a network with a total of N channels.  
The probability of being in state i, thus with i channels occupied, can be found from solving the "balance equations" (at equilibrium, the rate of moving into a state should equal the rate of moving out of a state). One finds 

          Ai           1
Pi   =   ---   -----------------
          i!         N      Ai
                     S    ----
                    i=0     i!
where A is the offered traffic expressed in erlang (l/m)
Number of Users

Calls blocked thus far:

%

The random process running here illustrates the number of simultaneous telephone calls in an Erlang B system. There are N = 8 channels available.


Percentage of time spent thus far in state 0 .. 8. Probability of being in state 8 should asymptotically equal the percentage of new calls that is blocked.

Traffic: erlang. Call duration: on average.

Note how it can take a fairly long time before the measured blocking rate resembles the expected value. Click here to stop this animation and remove the animation display.

PASTA

The "Poisson arrivals see time-averages" (PASTA)-rule says that the probability that a newly arriving call sees a full system equals the time-average probability that the system is in state N.

Thus the blocking probability equals PN.

Computer Software

Example and Exercise

A GSM base station uses one carrier of 8 TDMA users. Find the traffic load (in erlang) for which the blocking rate is 1%. Answer


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