# Teletraffic engineering/Trunking

Author: Michel Le Vieux
Module 22 of the Teletraffic Hyperlinked Textbook

## What is Trunking?
### Summary

In telecommunications systems, trunking is the aggregation of multiple user circuits into a single channel. The aggregation is achieved using some form of multiplexing. Trunking theory was developed by Agner Krarup Erlang, Erlang based his studies of the statistical nature of the arrival and the length of calls. The Erlang B formula allows for the calculation of the number of circuits required in a trunk based on the Grade of Service and the amount of traffic in Erlangs the trunk needs cater for.

### Definition

In order to provide connectivity between all users on the network one solution is to build a full mesh network between all endpoints. A full mesh solution is however impractical, a far better approach is to provide a pool of resources that end points can make use of in order to connect to foreign exchanges. The diagram below illustrates the where in a telecommunication network trunks are used.

#### Erlang and Trunking Theory

The Danish mathematician Agner Krarup Erlang is the founder of teletraffic engineering. Erlang developed the fundamentals of trunking theory while investigating how a large population can be serviced by a limited number of servers. Trunking theory leverages off the statistical behaviour of users accessing the network, these characteristics discussed in the assumptions of the Erlang B equation.
Grade of Service is a measure of the probability that a user may not be able to access an available circuit because of congestion. The busy hour is the time when the network is the most busy and is dependent on the users. The highest traffic may not occur at the same time every day so the concept of time consistant busy hour is defined, TCBH, as those 60 minutes (within 15 minute accuracy) that has the highest traffic. Business users may have their busy hour between 8:30am and 9:30am while residential users may have their busy hour between 6:00pm and 7:00pm.
Erlang B formula
    $GoS={\frac {\frac {A^{C}}{C!}}{\sum _{i=0}^{C}{\frac {A^{i}}{i!}}}}$ [5]
where:
* GoS Grade of Service is the probability of blocking during the busy hour
* C is the number of resources such as servers or circuits in a group
* A = λh is the total amount of traffic offered in Erlangs
and based on the following assumptions, taken from Kennedy 2007.
* The assumption of pure-chance traffic means that call arrivals and terminations are independent, identically distributed random events. The number of call arrivals in a given time also has a Poisson distribution.
* Statistical equilibrium assumes that the probabilities do not change with time.
* Full availability means an arriving call can be connected to any free outgoing circuit. If switches make the connection from incoming calls to outgoing, each switch must have sufficient outlets to provide connection to every outgoing circuit.
* Any attempted call that encounters congestion is lost because the derivation assumed lost-calls. If this congestion did occur, the customer is likely to make another attempt in a short while, thus increasing the traffic offered when there is congestion.

#### Multiplexing

In order to have multiple communication channels use the same medium some form of multiplexing needs to be used. The two main types of multiplexing are;
The two main types of multiplexing are: -
* Time Division Multiplexing
    Multiple channels are combined onto a single medium for transmission. The channels are separated in the medium by their time slot.
* Frequency Division Multiplexing
    Multiple channels are combined onto a single medium for transmission. The channels are separated in the medium by their frequency.

#### Example

Assume a fictitious residential telephone network with 10 users connected to Local Exchange A and 10 users connected to Local Exchange B. If we would like 10 users on LE A to connect to 10 users on LE B a proposed architecture might use a connection between the two exchanges with 10 circuits. If the circuits were to be investigated it would be seen that there utilisations would in fact be very low. The low usage of the circuits in this scenario leads us to look at Trunking theory.
Looking at the assumptions made by the Erlang B equations we have:
* Pure-chance traffic - A user may make a call at any time of the day
* Statistical equilibrium - A user may or may not make a call directly after a previous call
* Full availability - If there is a circuit on the trunk availiable an incoming call may make use of it
* Calls which encounter congestion are lost - If there were no available circuits on the trunk the call will be lost and the user will recieve a busy tone
Investigations have also indicated that a residential user generates 0.02 erlangs of traffic and if a Grade or Service of 0.001 (1 in a thousand calls will be lost) is selected.
Applying this information into the the Erlang B equation above
A = 0.2 (10 x 0.02 Erlangs)
GoS = 0.001
We can calculate that the actual number of circuits required between LE A and LE B is 4. Trunking theory has been a major driver in making communication networks economically viable and affordable to service providers and users.

#### Exercises

The management of a fixed line voice operator would like to try a decrease costs, they have suggested reducing the Grade of Service on their E1, E3, T1, T3 and STM-1 trunk circuits.
(a) Calculate and graph the efficiency per circuit of each of the carrier’s trunks with the following GoS 0.001, 0.002 and 0.005. You may make use of the following Erlang-B calculator
(b) Does reducing the grade of service significantly increase circuit efficiency? Could you suggest a better way of reducing costs?
Solution :
(a) The table below summaries the results and the graph below illustrates the increase in the amount of traffic a circuit may carry for different Grades of Service.
T1
E1
E3
T3
STM-1
Circuits
24
32
512
672
2430
Erlang
At 0.01% GoS
10.2
15.6
442.8
593.3
2287.4
At 0.02% GoS
10.7
16.3
447.3
598.6
2298.6
At 0.05% GoS
11.5
17.3
453.9
606.4
2315.5
Link Effiency (circuits per Erlang)
At 0.01% GoS
0.43
0.49
0.86
0.88
0.94
At 0.02% GoS
0.45
0.51
0.87
0.89
0.95
At 0.05% GoS
0.48
0.54
0.89
0.90
0.95
Table 1: - TABLE OF THE LINK EFFICIENCY PER CIRCUIT WITH DECREASING GRADES OF SERVICE
Graph 1: - GRAPH OF THE LINK EFFICIENCY PER CIRCUIT WITH DECREASING GRADES OF SERVICE
(b) Dropping the Grade of Service on a trunk does not seem to significantly increase the efficiency per circuit. If you do not change the Grade of Service but compare the link efficiencies of larger trunks. A better way to reduce cost would to be aggregate smaller circuits into larger circuits as close to the edge of the network as possible. Simply aggregating multiple E1's into an E3 significantly increase link efficiency so by reducing the amount of circuits required and ultimately reducing costs, this illustates the importance of design in order to make networks efficient.
Graph 2: - GRAPH OF THE LINK EFFICIENCY PER CIRCUIT WITH INCREASING CIRCUIT SIZES :