54-10-10 Equipment Sizing and Traffic Engineering
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Gilbert Held
Payoff
One of the most common problems associated with the acquisition of such
communications networking devices as multiplexers, concentrators, and data PBXs is
determining the configuration or size of the device. If equipment with too few ports is
acquired, users will either be queued or be prevented from accessing the organization's
communications facilities. If equipment with too many ports is acquired, those ports that
are unused represent money spent unnecessarily. Thus, it is important for the network
manager to understand the equipment-sizing process, which is based on a defined
communications discipline known as traffic engineering.
Introduction
This article focuses on the application of telephone traffic formulas to the sizing of data
communications equipment and line facilities. Although most telephone traffic formulas
were developed during the 1920s, many are applicable to such common problems as
determining the number of dial-in business and Wide-Area Telecommunications Service
lines required to service terminal users as well as the number of ports or channels that
should be installed in communications equipment connected to dial-in lines. More formally
referred to as traffic dimensioning formulas, several common formulas—including the
erlang B formula—for sizing data communications equipment and line facilities are
discussed.
Telephone Terminology
Most of the mathematics used for sizing data communications equipment evolved from
work originally performed to solve the sizing problems connected with telephone
networks. Exhibit 1 illustrates the interconnection of two telephone company Central
Office, a network configuration that was the basis for the development of traffic
engineering. The subscriber lines represent the local loops (connections from each
customer location to a shared electronic switch) that service the telephones installed at the
customer's premises.
Telephone Traffic Sizing Problem
The electronic switch installed in the telephone company's central offices is designed to
service a large number of telephone company subscribers located within a defined
geographical area. Most telephone traffic is between subscribers connected to the same
switch; thus, the number of trunks—a term meaning the lines connecting central offices to
each other—should be less than the number of subscriber lines serviced by each central
offices. The determination of the number of trunks needed is called dimensioning and is
critical for the efficient operation of the telephone network.
If insufficient trunks are available, customers will encounter an unacceptable
number of busy signals. They will become angry at the telephone company, and their
inability to place long distance calls will result in a loss of revenue to the company. With
too many trunks, all or most calls will go through, but the unoccupied trunks will represent
a waste of the telephone company's construction funds.
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Traffic Measurements
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Telephone activity can be described by the calling rate (the number of times a particular
route or path is used per unit time period) and the holding time (the duration of the call).
Three other terms that warrant attention are the offered traffic, the carried traffic, and the
busy hour.
The offered traffic is the volume of traffic routed to a particular telephone exchange
during a predetermined time period; the carried traffic is the volume of traffic actually
transmitted through the telephone Central Office during a predefined period of time. The
busy hour is the busiest one-hour period of the day. It is the busy-hour traffic level that is
employed in dimensioning telephone central offices and transmission routes to size the
office or route with respect to its busiest period.
Telephone traffic can be defined as the product of the calling rate per hour and the
average holding time per call. This measurement can be expressed mathematically as T =
C * D, where T is the traffic, C is the calling rate per hour, and D is the average duration
per call. With this formula, traffic can be expressed in Call-Minutes or Call-Hours, where a
call-hour is the quantity represented by one or more calls having an aggregate duration of
one hour.
Erlangs and Call-Minutes
The preferred unit of measurement in telephone traffic analysis is the erlang, named after
A.K. Erlang, a Danish mathematician. The erlang is a dimensionless unit that represents
the occupancy of a circuit, with one erlang of traffic intensity on one circuit representing a
continuous occupancy of a circuit. For a one-hour unit interval, the relationship between an
erlang and Call-Minutes and Call-Hours becomes 1 erlang = Call-Minutes= 1 call-hour.
If a group of 20 trunks is measured, a call intensity of 10 erlangs means that one-half of all
trunks were busy during the measuring period. If a call intensity of 5 erlangs is measured,
one-quarter of all trunks were busy during the measuring period.
Grade of Service
One of the most important concepts in the dimensioning process is what is known as the
grade of service. If a subscriber attempts to place a long distance call when all trunks are in
use, that call is said to be blocked. The probability that a call will be blocked can be
computed from the traffic intensity and the number of trunks available to serve calls.
Call blocking is most likely to occur during the busy hour—the period when the
greatest amount of activity occurs. Thus, telephone-company trunk capacity is engineered
to service a portion of the busy-hour traffic, the exact amount of service being dependent
on economics as well as the level of service the company is obliged to provide to
customers. Overdimensioning the number of trunks by providing a trunk for every
subscriber would ensure that a lost call would never occur. This would be equivalent in the
data communications world to providing a multiplexer port for every terminal . Because a
1:1 subscriber-to-trunk ratio is expensive and results in most trunks being idle a large
portion of the day, a smaller number of trunks is used. As the number of trunks decreases
and the subscriber-to-trunk ratio correspondingly increases, some call blockage will result.
The number of blocked calls allowable during the busy hour can be specified. This
specification is known as the grade of service and represents the probability (P) of having a
call blocked. If a grade of service of 0.05 between Central Office is specified, a sufficient
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number of trunks is required so that only 1 call in 20, or 5 calls in every 100, will be
blocked during the busy hour.
Route-Dimensioning Parameters
Several formulas can be used to determine the number of trunks required to service a
particular route. Each formula's use depends on the call arrival and holding time
distribution, the number of traffic sources, and the handling of lost or blocked calls.
Regardless of the formula employed, as the number of trunks and the traffic intensity
increase, each formula yields results that converge. Each formula provides the probability
of call blockage or grade of service, which, because it is based on a given number of trunks
and level of traffic intensity, presents a technique for sizing equipment.
The number of traffic sources can be considered either infinite or finite. If the
subscriber population is large and subscribers tend to redial when their call is blocked, the
calling population can be considered infinite. An infinite traffic source means that the
probability of a call arriving is constant and does not depend on the state of traffic in the
system. In fact, the two most commonly used traffic-dimensioning equations, erlang and
Poisson, are both based on an infinite calling population.
Blocked (also known as lost) calls can be considered cleared, delayed, or held.
When calls are considered held, it is assumed that the telephone subscriber, upon
encountering a busy signal, immediately redials the desired party. The lost call-delayed
concept assumes that each subscriber is placed in a waiting mechanism for service and
forms the basis for queueing analysis. For the purpose of this article, a service or
nonservice condition is assumed, so the lost call-delayed concept is disregarded. (It would
be relevant if access to a network resource occurs through a data Private Branch eXchange
or port selector that has a queuing capability.)
The Erlang Traffic Formula
The most commonly used telephone traffic dimensioning equation is the erlang B formula.
This formula assumes that a subscriber encountering a busy signal will hang up the
telephone and wait a certain amount of time prior to redialing, a condition known as lost
call-cleared. This assumption is equivalent to stating that traffic offered to but not carried by
one or more trunks vanishes. This is the key difference between the erlang B and Poisson
formulas. (Poisson assumes that lost calls are held.)
If E is used to denote the traffic intensity in erlangs and T represents the number of
trunks, channels, or ports designed to support the traffic, the probability P(T,E) represents
the probability that T trunks are busy when a traffic intensity of E erlangs is offered to
those trunks. The probability is equivalent to specifying a grade of service and can be
expressed by the erlang traffic formula as follows:
Assume, for example, that a traffic intensity of 3 erlangs is offered to a three-position
rotary. The grade of service is the probability that all three ports will be busy when a traffic
intensity of 3 erlangs is offered to those trunks.
On the average during the busy hour, 34.6 out of every 100 calls will encounter a
busy signal. For most organizations this is an unacceptable grade of service.
If the dial-in rotary is expanded by adding additional lines that are connected to
modems, which in turn are connected to ports on a multiplexer or another type of
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communications device, the probability that a call will encounter a busy signal should
decrease.
An expansion of the number of dial-in lines from three to five reduces the
probability that users will encounter a busy signal from 34.6% to 11.1%. By considering
the cost associated with adding additional lines and of the equipment connected to those
lines and comparing that cost to the value of lost calls, an optimal level of service can be
determined. This optimal level of service is illustrated in Exhibit 2.
Finding the Optimal Level of Service
Optimal Level of Service
According to Exhibit 2, the revenue the organization loses if customers get busy signals
when they call or the cost in lost productivity (of employees who are idle as they wait for
service) increases as the number of lines and ports decrease. This is to be expected, because
a lower level of service causes more persons to encounter busy signals. Similarly, a higher
level of service in the form of additional lines and ports enables more persons to receive
service without delay, reducing lost revenue or lost productivity.
Cost of service increases as the number of lines and ports increases. By combining
the cost of service and the cost of lost revenue and productivity, the total cost curve is
calculated. Where that curve has its minimum value is the optimum level of service with
respect to the equipment's number of lines and ports.
Because the optimum level of service depends on lost revenue or productivity, it
varies between organizations. Therefore, the remainder of this article focuses on the grade
of service, a figure critical for determining the optimum level of service and common to all
organizations. Direct computation of the erlang B formula can be both tedious and timeconsuming as the traffic intensity and the number of ports increase, so tables are used
instead. Many books have extensive tables giving computed grades of service based on
various levels of traffic intensity and numbers of trunks or ports.
Table Extract
Exhibit 3 is an extract from a table of erlang B grade of service values. An example of the
use of Exhibit 3 is the following situation: customers require a grade of service of 0.1
when the specific traffic density is 7.5 erlangs. From Exhibit 3, 10 channels or trunks are
required (the use of the table requires some interpolation and rounding to the highest port
or channel).
Erlang B Grade of Service (probability all ports are busy when call is
attempted)
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Port
---1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
6.00
---0.85714
0.72000
0.59016
0.46957
0.36040
0.26492
0.18505
0.12188
0.07514
0.04314
0.02299
0.01136
0.00522
0.00223
0.00089
Traffic in Erlangs
-----------------6.50
---0.86667
0.73799
0.61523
0.49994
0.39391
0.29910
0.21737
0.15010
0.09780
0.05977
0.03412
0.01814
0.00899
0.00416
0.00180
7.00
---0.87500
0.75385
0.63755
0.52734
0.42472
0.33133
0.24887
0.17882
0.12210
0.07874
0.04772
0.02708
0.01437
0.00713
0.00332
7.50
---0.88235
0.76792
0.95751
0.55214
0.45302
0.36154
0.27921
0.20746
0.14740
0.09954
0.06356
0.03821
0.02157
0.01142
0.00568
For a 0.01 grade of service when the traffic intensity is 7 erlangs, between 13 and 14
channels are required. It is not possible to install a fraction of a trunk or channel, so 14
channels is the correct number of trunks.
Equipment Sizing
As an example of the application of the erlang B formula to the sizing of communications
equipment, the number of ports on a multiplexer, which will determine the number of
telephone lines and modems required to provide access to those ports, is determined.
A survey of terminal users in a geographic area indicates that during the busy hour
an average of six terminals are active. This represents a traffic intensity of 6 erlangs. If the
goal is to size the multiplexer so that at most only 1 out of every 100 calls to the device
encounters a busy signal, the desired grade of service is 0.01. The 6-erlang column in
Exhibit 3 indicates that to obtain a 0.01136 grade of service requires 12 channels; a
0.00522 grade of service results if the multiplexer has 13 channels. The multiplexer would
be configured for 13 channels. If a higher grade of service is acceptable, such as 1 in 25
calls receiving a busy signal (a grade of service of 0.04) and traffic intensity is 6 erlangs, 10
ports provides a grade of service of 0.04314 and 11 ports provides a grade of service of
0.02299. The multiplexer can then be sized for 11 ports instead of 13, thereby reducing the
number of lines and modems connected to the multiplexer from 13 to 11.
Recommended Course of Action
The erlang B formula is a means of computing the grade of service for a level of traffic
offered to a group of trunks or lines. In addition, if the potential loss of revenues or worker
productivity when busy signals are encountered can be estimated, the optimum number of
ports based upon the cost of equipment, facilities, loss of productivity, and lost revenue can
be calculated. Although erlang B formula computations can be tedious and timeconsuming, there are many books that contain precalculated tables. Network managers
should use the information presented in this article to systematically approach the
equipment-sizing process.
Author Biographies
Gilbert Held
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Gilbert Held is chief of data communications for the US Office of Personnel Management.
He is an internationally recognized lecturer and has written 11 books and more than 50
technical articles. Held received the Karp award for best conference paper at Interface '84
and at Interface '87. He has a BSEE degree from Pennsylvania Military College, an MSEE
degree in computer science from New York University, and MBA and MSTM degrees
from American University of Washington DC.