# Teletraffic engineering/How is Telephony Traffic Modelled?

Author: Jorenjeye Arubayi
Module 20 of the Teletraffic Textbook

## How is Telephony Traffic Modelled?
### Summary

Telephony networks like all other networks have finite resources (circuits). This therefore makes it possible that an incoming call may not be served at a particular time, that is, the call may be blocked and cleared due to a limitation on the availability of resources. Call blocking is possibly the most noticeable phenomenon in telephony networks. The aim of modelling telephony traffic is to compute the best estimate of network blocking in order to obtain a reasonable balance between dimensioning of resources and acceptable grade of service. The most widely used traffic models are the Erlang models. Some other commonly adopted traffic models are the Engset and Poisson models.

### The Poisson Traffic Model

The arrival and dropping of a telephone call occurs randomly. It is often assumed that the arrival of a call occurs according to a Poisson process, with a rate of $\lambda$ calls per unit of time [2]. However, this assumption may not be realistic if blocked calls result in immediate new attempts by impatient subscribers. Calls are assumed to have a memoryless property, with a mean of 1/$\mu$ [2]. Under such assumptions, the number of active calls is described as a Markov process [2].
The number of calls arriving within a given period of time, T, may be represented as [3]:
P(a) = ($\mu$ a/ a! ) $e^{-\mu }$.................(1)
where a is the number of calls arriving within a period of time T and $\mu$ is the mean number of call arrivals in time T.
The number of calls departing within a given period of time, T, also has a Poisson distribution given by [3]:
P(d) = ($\lambda$ d/ d! ) $e^{-\lambda }$ .................(2)
where d is the number of call departures in time T and $\lambda$ is the mean number of call departures in time T.
It can also be shown that calls have a negative exponential distribution given by [3]:
P(T >= t) = $e^{-t/h}$.................(3)
where h is the call mean holding time (MHT).
"The Poisson model is commonly used for over engineering standalone trunk groups" [4]. The following formula is used to derive the Poisson traffic model [4]:
    $P(i,a)=1-e^{-a}{\sum _{k=0}^{i-1}{\frac {a^{k}}{k!}}}$ ........(4)
where:

#### Example 1[4]

You are required to create a new trunk group to be used by an office and you need to determine how many lines are needed. The office is expected to make and receive about 200 calls per day with an average holding time (AHT) of about 3 minutes (180 seconds). A blocking probability not more than 1 percent is required. Let’s assume that approximately 20 percent of the calls happen during the busy hour.
Solution
Number of calls during the busy hour = 200 calls * 20% = 40
Busy hour traffic = (40 calls * 180 AHT)/3600 = 2 erlangs
From the Poisson table we find that at 1% blocking probability, 7 trunks would be required carry slightly over 2 erlangs of traffic. This answer can be double checked by substituting the values into equation 4 as follows:
    $P(7,2)=1-e^{-2}{\sum _{k=0}^{7-1}{\frac {2^{k}}{k!}}}$

### Erlang Traffic Models

A telephone system has a limited number of channels available for carrying user traffic. Arriving calls are assigned a channel until all the channels are occupied, after which any new calls will either be blocked or queued [2].
Erlang traffic models are formulae that can be used to estimate the number of channels required in a network, or between two PSTN exchanges, for a given grade of service and user traffic. There is also an Erlang formula for modelling queuing situations, which is useful in estimating the agent staffing requirements of call centres [1].
The main Erlang traffic models are:
* Erlang B
* Extended Erlang B
* Erlang C

#### The Erlang B Model

This is the most common Erlang model and is used to determine the number of circuits required to carry user traffic during the busy hour for a given grade of service and traffic load [1]. The Erlang model assumes that all blocked calls will be immediately cleared [1].
In an Erlang-B telephone system, when all the available channels are occupied a new call trying to access a channel is blocked. The basic assumption is that that the call is lost and that the calling subscriber will not try again [2].
The number of active calls is described as a Markov process [2] and new calls arrive according to a Poisson process at a rate of $\lambda$ calls per unit of time and terminate a rate of $\mu$ [2]. "For i calls the rate of termination of one call is i times $\mu$ [2]."
At equilibrium the rate of moving into a state is the same as that of moving out of a state [2]. The probability of being in state i, and therefore having i channels occupied, can be found to be:
    $P_{i}={\frac {\frac {A^{i}}{i!}}{\sum _{k=0}^{N}{\frac {A^{k}}{k!}}}}$ ............(5)
where:

##### Example 2

There is a need to redesign trunk groups that are currently experiencing blocking during the busy hour. The current trunk group is offered 80 erlangs of traffic during the busy hour. The design requirement is for low blocking, so at a blocking probability of 0.001, the number of required circuits can be found using an Erlang B calculator based on equation 5 to be 106.

#### The Extended Erlang B Model

The Extended Erlang B model takes retried calls into account . This traffic model is based on the following assumptions [4]:
* Calls arrive randomly
* Blocked callers make multiple attempts to complete their calls
* No traffic overflow is allowed
The Extended Erlang B model is usually used for standalone trunk groups that have retry probability (for example, a modem pool) [4].

##### Example 3 [4]

You are required to determine how many circuits you need for your dial access server. You are informed that the expected traffic during the busy hour is 35 erlangs and you require not more than 2% blocking probability during that period. You also expect that 60 percent of the users will retry immediately.
Solution
Using an Extended Erlang B calculator you find that you would require 45 circuits.

#### The Erlang C Model

In systems modelled according to the Erlang-C formula, when all the available channels are occupied a new call is queued until it can be served [2]. "The Erlang C model is designed around queuing theory" [4]. The Erlang C traffic model is based on the following assumptions [4]:
* There is an infinite number of sources
* Traffic arrives randomly
* Blocked calls are delayed
* The holding time has an exponential distribution
In the Erlang C model, you need to know the number of calls , the average call length, and the expected amount of delay in seconds [4]. This model is usually used for conservative automatic call distributor (ACD) design to determine the number of call agents needed [4]. The following formula is used to derive the Erlang C traffic model:
    $P_{c}={{{\frac {A^{N}}{N!}}{\frac {N}{N-A}}} \over \sum _{i=0}^{N-1}{\frac {A^{i}}{i!}}+{\frac {A^{N}}{N!}}{\frac {N}{N-A}}}\,$.................(6)
where:

##### Example 4 [4]

A call centre is expected to have about 350 calls lasting approximately 2 minutes each, and each agent has an after-call work time of 15 seconds. You would like the average time in the queue to be approximately 10 seconds. Calculate the amount of expected traffic and the delay factor.
Solution
You know that you have about 350 calls of 2 minutes duration, 15 seconds must be added to the call duration because each agent is not answering a call for approximately 15 seconds. The additional 15 seconds is part of the amount of time it takes to service a call.
Expected traffic = (350 calls * (2 * 60) + 15 seconds AHT)/3600 = 13.125 erlangs
The delay factor is computed as follows:
Delay factor = (10 sec delay)/(135 seconds) = 0.074

### The Engset Model

The Engset traffic model is based on the Engset calculation named after T. O. Engset. This model is similar to the Erlang B traffic model already discussed, but contains one major difference. While the Erlang B model assumes an infinite number of callers, the Engset model specifies a finite number of callers [5]. The Engset formula requires that the expected peak traffic, the number of sources (callers) and the number of circuits in the network be known [5]. The Engset formula is given by [5]:
    $P_{b}={\frac {\left[{\frac {\left(S-1\right)!}{N!\cdot \left(S-1-N\right)!}}\right]\cdot M^{N}}{\sum _{X=1}^{N}\left[{\frac {\left(S-1\right)!}{X!\cdot \left(S-1-X\right)!}}\right]\cdot M^{X}}}$.......................(7)
    $M={\frac {A}{S-A\cdot \left(1-P(b)\right)}}$.................(8)
where:
The Engset model also assumes that the call arrivals can be modelled by a Poisson process, and that the call holding times are described by a negative exponential distribution [6]. However, because there are a finite number of sources, the arrival rate of new calls, $\lambda$, decreases as more sources become busy and therefore new calls cannot be accommodated [6].

### Exercises

1. Design a trunk group for an office with the following specifications:
* Approximately 500 calls will be received per day with an average holding time (AHT) of about 2 minutes.
* A blocking probability not more than 1 percent is required.
* Approximately 20 percent of the calls happen during the busy hour.
2. Determine the required number of circuits for a trunk group that is offered 150 erlangs of traffic during the busy hour with a blocking probability of 0.05.
3. A call centre is expected to have about 1000 calls lasting approximately 3 minutes each, and each agent has an after-call work time of 35 seconds. You would like the average time in the queue to be approximately 10 seconds. Calculate the amount of expected traffic and the delay factor.