# Traffic Theory

Traffic theory describes, through models and mathematical equations, the flow of traffic through a switching device. Many established models describe the various switching conditions of central offices and systems. These models use various assumptions about the characteristics of the switch (blocking, non-blocking), the availability of channels (trunks, lines, receivers), and the characteristics of use (call-holding time, lost call handling).

## Traffic Analysis

The application of traffic theory to practical situations requires some form of measurement facility within a system. This measurement facility monitors and keeps statistics on equipment usage. The analysis of its output forms the basis of Traffic Analysis. Traffic Analysis provides a method of interpreting traffic measurements in a way that allows the optimization of a switching system to provide adequate service to users while maintaining the greatest possible economy.

Blocking and Non-Blocking Switches

Central to traffic theory is the distinction between blocking and non-blocking switches. In providing switching services to a group of users, two situations are possible:

1. Any user may issue a request for service to the system and obtain service, regardless of how many users are currently using the switch. A switch providing this service must have at least as many service channels as there are potential users. This switch is known as a non-blocking switch.

2. A user may at any time initiate a function at the system and may, under most circumstances, receive service. This is provided by equipping the system with only enough switching equipment (trunks, receivers, etc.) to guarantee that a user will be able to complete a given percentage of all service requests. Such a switch is known as a blocking switch (some services are blocked or delayed because of a lack of facilities to process the request).

Grade of Service

A switching system, in practical situations, blocks calls to a certain extent. The amount of blocking in a system is expressed as a Grade-of-Service figure which indicates the ratio of unsuccessful call attempts with respect to the total amount of calls handled by the system within a certain span of time.

Performance Specification

The ability of a system to guarantee a certain percentage of completion of service attempts is a measure of its performance and results from a compromise between equipment cost and the need to insure a high completion rate for service requests. The ability of a switching system to provide a certain performance level is expressed through various parameters such as: Dial Tone Delay, Post-Dialing Delay, Dial Tone Removal on First Dialed Digit, and Connecting Delay on Answer. Many more parameters can be used in characterizing the performance of a switching system, but the four mentioned here are specifically relevant to the MITEL system and are defined in the following table.

Performance Specifications
Term | Definition
Dial Tone Delay | The time taken to receive the dial tone after a station goes off-hook.
Post-Dialing Delay | The time taken following the end of dialing before reception of ringback.
Dial Tone Removal on First Dialed Digit | The time taken for the removal of dial tone following the first dialed digit.
Connecting Delay on Answer | The time taken to establish a voice path following an answer from the called party.

Traffic Measurement Parameters

Traffic is a measure of the total flow of calls through a switching system. The two most important characteristics of calls for the purpose of traffic measurement are the number of calls in a given period of time and their duration. Three terms which are used frequently in traffic measurement are defined as follows:

1. Peg counts: This refers to the total number of calls of a given type (i.e., trunk calls, calls to attendant, etc.) during a traffic measurement period (the span of time during which traffic data is collected).

2. Call Holding Time: This refers to the length of time for which a call holds a service channel busy. This figure shows the various phases of a call, and how these affect Call Holding Time.

3. Usage: Peg counts and call holding times, when looked at individually, do not yield a complete picture on how heavily a service channel, or group of channels is used. To this end, a usage value is used extensively in traffic measurement. Usage (in Erlangs) is defined as the Total Call Holding Time (sec./peg X peg count/hour) divided by 3600 (sec./hour). In North America, the Erlang unit is replaced by the Centi Call Second (CCS), which evaluates traffic over 100 second periods, as opposed to 1 hour periods. Usage (in CCS) is defined as the Total Call Holding Time (sec./peg X peg count/CCS period) divided by 100 (sec./CCS period). Note: Usage (Erlang) = Usage (CCS) / 36 or 1 Erlang = 36 CCS.

Summary of Traffic Theory

Traffic theory is based on the statistical analysis of calls in a switching system. The details of traffic theory are complex and are only briefly covered in this section. Two important results of Traffic Theory are the Erlang B and C formulas which are used extensively in Traffic Analysis. The Erlang B formula assumes that the system offers no queuing facilities to blocked calls, while the Erlang C formula assumes that queuing is provided.

Equipment Provisioning

The Erlang formulas are both used, under different circumstances, to determine the quantity of equipment required to meet given traffic levels. The Erlang B formula is used in the original provisioning of the switch to determine if eight or 16 circuit switch links will be connected to each peripheral pair. The Erlang C formula is used both in initial provisioning and subsequent traffic evaluations to calculate the quantity of DTMF receivers required. Because of the complexity of the formulas, their results have been tabulated and published for use in Traffic Analysis. These tables should be consulted to extrapolate equipment quantities for the system.

## Traffic Flow Graphs

In studying traffic flow through a switching system, two assumptions are made: first that service attempts occur at random, and second that service attempts are requested uniformly by the users. Traffic Variation With Time of Day and Traffic Distribution Curve show graphs of traffic flow plotted against two different parameters. A simple graph of how traffic varies with time of day is given. This graph shows several peaks and valleys where service requests reach maximum and minimum values, respectively.

The graph is generated by plotting a series of discrete traffic flow values taken at regular intervals against the time axis. Taking the information from this first graph, it is possible to obtain the graph shown in Traffic Distribution Curve if the time over which traffic values are taken is sufficiently large. Traffic flow organized not with respect to time of day, but rather with respect to probabilities of occurrence. Probabilities of occurrence are obtained simply by counting how many times a given traffic flow value occurs and dividing by the total number of traffic measurements. The graph has a distinct bell shape and indicates that there is one traffic value which is most likely to occur at any time (the value corresponding to the peak in the curve). This curve is indicative of the loading of a switch and the mathematical description of this graph is central to traffic theory.

Depending on what assumptions are made on the exact properties of the bell-shaped curve (also known as a traffic distribution), several mathematical descriptions are possible. The most widely-used mathematical formulation is the Poisson Distribution function, if the number of service channels is large, as is the case when investigating traffic at the level of the peripheral switch of thesystem. If the number of service channels is small, as is the case when traffic for an ONS card is measured, the Bernouilli distribution provides a more accurate description. Both distribution functions yield approximately the same bell-shaped curve, but make different assumptions concerning the variation in the probability of further channels being used as traffic increases.

## Truncated Distributions

Because the Truncated Traffic Distribution Curve shows the probability of occurrence of various traffic intensities, in Erlangs or CCS, it has a direct relationship to the quantity of service channels. Traffic flow is dependent on equipment quantities because the ability to cater to a given traffic intensity is directly related to the quantity of equipment provisioned in the switching system. This argument points out one deficiency in the Poisson distribution, that of allowing arbitrarily large traffic flow, implying an arbitrarily large number of service channels. If the finite quantity of channels in a switching system is taken into account, graphs are obtained. Traffic Distribution Curve shows how the traffic variation in Traffic Variation With Time of Day is modified by the imposition of a fixed number of channels, and hence a maximum traffic flow value. In Truncated Traffic Variation Curve, the distribution curve ends abruptly at the maximum traffic flow value permitted by the switch. This Truncated Distribution is described mathematically by the Erlang B Formula (if the blocked calls indicated in Traffic Distribution Curve are dropped) or the Erlang C Formula (if the blocked calls are queued for later processing when the overload condition disappears).