# Erlang C

## Call Queueing

In an Erlang-C 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 queued until it can be served. This in contrast to an Erlang-B system, in which new calls are blocked.

## Model

New calls arrive according to Poisson process with rate calls per unit of time. Calls have a (memoryless) exponential duration with mean 1/mu. The number of active calls is a Markov process.

Figure: Markov model for number of occupied channels in a network with N channels.

### Exercise

Explain the transition rates the above Markov chain.
Why is the rate of going from state i to i-1 equal to
• i*mu for i =< N
• N*mu for i >= N

### Exercise

Compute the state probabilities and the call queuing probability.

Show that the probability of delaying a call is

```         A^N        1
P    =   ---   ------------
N!     1 - A/N

```
where A is the offered traffic expressed in erlang (/mu)