The use of mathematical modeling to predict line, equipment, and staff capacities for telephone systems is an accepted technique for fine-tuning existing systems, as well as designing new ones. Through sensitivity analysis, such predictions can also provide a comprehensive overview of a particular design. A knowledge of secondary school mathematics is sufficient to do the calculations in this paper. Traffic engineering techniques are used most often to determine:

• Line and trunk quantities required for a PBX or computer

• Number of DTMF (Dual Tone Multi-frequency) registers, conference trunks, RAN (Recorded Announcement Route) trunks, etc. required

• Traffic capacity of a PBX, given the number of speech paths (simultaneous conversations) available

• Quantities, service levels, and usage of such special service trunks as foreign exchange (FX), discounted toll trunks, and tie trunks (leased lines between PBXs

• Operator staffing levels and performance predictions as well as the impact of system change on staff quantities

• Automatic call distributor (ACD) staffing and service levels

Arrival Rate: - The arrival rate is the number of calls that will arrive at a facility during a finite time period. The Greek letter lambda (λ) is generally used to represent arrival rate. The distribution of calls to a server group will vary with the source. People calling to a line group often do so at random, with each call independent of the others. This is called a Poisson arrival process and is the most common assumption used in traffic engineering for the distribution of call arrivals. (Calls to a power utility during a power failure would not be considered a Poisson distribution.)

Blocking:- Blocking occurs whenever the number of calls, in or out, exceeds the number of facilities (lines, trunks, agents, operators) available to support them. A blocked call is given a busy signal, which requires the caller to disconnect, and try again. Blocking probability is expressed as a percentage of denial, e.g. for 1 call in 100 blocked, it is expressed as P.01 (1% of the offered calls will expect to be blocked).

Centum Call Seconds (CCS):- This is a measure of telephone traffic in 100 second increments. It originated in the early days of electromechanical switching, and was developed to make the traffic volume quantities more manageable, For example:

10 minutes of traffic = 600 seconds (60x10)

600 seconds

100 = 6 CCS

Erlang:

An erlang is defined as a dimensionless unit of traffic intensity. The key to this definition is that, dimensionless means no specific time period. A CCS is exactly 100 seconds, whereas an Erlang is dependent on observation time. The maximum that a facility can be in use is 100% of the time. If the observation time is 10 minutes, and the facility is in use for the full time, then that is 1 Erlang

Holding time:

Holding time is the call length, call overhead time, plus queuing time, if any. Overhead includes the activities necessary on the transmit/receive sides of the call.

Outgoing calls incur different activities than do incoming calls.

Queuing:-

Queuing is waiting in a holding facility until a server becomes available. When, for example, an ACD (Automatic Call Distributor has more lines than agents (e.g., 50 lines serving 42 agents) and all agents are busy, the extra lines become the holding facility.

Measure of ability of a user to access a trunked system during the busiest hour. Measure of the congestion which is specified as a probability.

The probability of a call being blocked

•Blocked calls cleared or Lost Call Cleared (LCC).

Erlang B systems–

The probability of a call being delayed beyond a certain amount of time before being granted access

•Blocked call delayed or Lost Call Delayed (LCD) or

Erlang C systems

Blocked Call Cleared Systems

•Queuing is not provided for call requests

•When a user requests service, there is a minimal call setup time and the user is given immediate access to a channel if one is available

•If channels are already in use and no new channels are available, call is blocked without access to the system

•The user does not receive service, but is free to try again later

•All blocked calls are instantly returned to the user pool

•Mathematical modelling of such systems is done by Erlang B formula

The Erlang B model is based on following assumptions:

•Calls are assumed to arrive with a Poisson distribution

•There are nearly an infinite number of users

•Call requests are memory less, implying that all users, including blocked users, may request a channel at any time

•All free channels are fully available for servicing calls until all channels are occupied

•The probability of a user occupying a channel (called service time) is exponentially distributed. Longer calls are less likely to happen.

•There are a finite number of channels available in the trunking pool.

Blocked Call Cleared Systems

•The assumptions on the previous slide lead to the Erlang B formula which determines the probability that a call is blocked and is a measure of the GOS for a trunked system which provides no queuing for blocked calls.

Blocked Call Delayed Systems

•Queues are used to hold call requests that are initially blocked.

•When a user attempts a call and a channel is not immediately available, the call request may be delayed until a channel becomes available.

•Mathematical modelling of such systems is done by Erlang C formula.

Modelling of Blocked Call Delayed Systems

The Erlang C model is based on following assumptions:

•Similar to those of Erlang B.

•Additionally, if offered call cannot be assigned a channel, it is placed in a queue of infinite length.

•Each call is then serviced in the order of its arrival.