# Teletraffic engineering/Blocking

Module by: Nomfundo N. Dlamini

## Summary

The telephone system is not dimensioned such that all subscribers can be connected at the same time [2]. Providing sufficient resources to carry all traffic that could be offered in a telecommunications system would be very uneconomical [1]. Subscribers have to share resources because equipment at the exchanges is expensive. As a result of this, telecommunications systems are likely to experience problems at times, for example calls fail to get through. This phenomenon is termed blocking. Indiscernible amounts of blocking are acceptable in the telecommunication systems.

## What is Blocking?

Blocking in telecommunication systems is when a circuit group is fully occupied and unable to accept further calls [1]. It also referred to as congestion. Due to blocking in telecommunications systems, calls are either queueued (but not lost) or are lost (all calls made over congested group of circuits fail). Such systems are called queueing systems (delay systems) and lost-call systems respectively.
* An example of a queueing system: a message-switched exchange
* An example of a lost-call system: a circuit-switched exchange
The proportion of calls that are lost or delayed during blocking portray the measure of the grade of service which is basically the measure of the service provided. A large grade of service indicates a poor service offered to the customer. The grade of service is always specified at the busy hour.
The grade of service (B) in a lost-call system is defined as [1]:
B = Number of lost calls / Number of offered calls
B may also be defined as [1]:
* B = Lost traffic / Offered traffic
* B = Proportion of time in which congestion exists
* B = Probability that a call will be lost through congestion
There are acceptable grade of service standards for different telecommunication systems [1]. Values which are lower than the stipulated values imply the systems offer poor service.
* 0.001 for cheap tie line circuits
* 0.002 for within building inter-exchange connections
* 0.01 for expensive international circuit groups
* 0.02 for cellular circuit groups
The grade of service is the blocking probability. A higher grade of service implies high probability of loss during the busy hour. Blocking probability is the chance that a customer will be denied service due to lack of resources. A blocking probability of 0.01 means 1% of customers will be denied service. It should be as low as possible and can be decreased by [3]:
* Increasing resources in the system
* Offering incentives and discounts during off-peak hours to encourage usage of resources outside the busy hour.
Two formulae are used for calculating the blocking probability: the Erlang-B and Erlang-C. The choise of formula is dependent upon the method of handling of customers when all resources are busy.
* Erlang-B: used for lost-call systems whereby calls are lost should all resources be busy.
* Erlang-C: used for queueing systems whereby calls are queued should all resources be busy.
The Erlang-B formula is:
$P_{B}={\frac {\frac {A^{N}}{N!}}{\sum _{i=0}^{N}{\frac {A^{i}}{i!}}}}$ .............(1)
where
The Erlang-C formula is:
    $P_{c}={{{\frac {A^{N}}{N!}}{\frac {N}{N-A}}} \over \sum _{i=0}^{N-1}{\frac {A^{i}}{i!}}+{\frac {A^{N}}{N!}}{\frac {N}{N-A}}}\,$.................(2)
where:

#### Example

Let us assume that a teletraffic engineer wants to design a telecommunications system that will be a lost-call system with a blocking probability of 0.02 (2%). The maximum amount of busy-hour traffic (BHT) that the engineer wants the system to support during the busy hour is 78 Erlangs. Using the Erlang B calculator, we can find the number of circuits or circuits that the engineer is supposed to budget for as being 90.
Click here for Erlang-C calculator

#### Exercises

* Exercise adopted from this link
Consider the resource dimensioning of DTMF (Dual Tone Multi Frequency) receivers in the Xenon switching system. DTMF receivers are used to receive tones from the phone keypad and recognize the dialled digits. Thus a DTMF receiver should be allocated before dial-tone is fed to the subscriber. The DTMF receiver can be freed after digit dialling has been completed. The average duration of the digit dialling phase is 30 seconds. The total number of circuits in the system is 180. A XEN processor shall handle at least 20,000 originations in the busy hour. Calculate the blocking probability of the system stated using: The Erlang-B model .
    Solution