Part 2: Traffic Assumptions Of
The Public Voice Network
In Part 1 of this series of articles,
we have examined the Public Switched Telephone Network, and understand
how it works. In this article we will attempt to characterize the telephone
traffic and how to engineer the network to serve that kind of traffic.
An understanding of the nature of
the telephone traffic and its distribution with respect to time and destination
is essential in determining the amount of telephone facilities required
to serve the subscriber's needs. Telephone traffic varies greatly from
one period to another, not in any uniform manner but according to the needs
of the telephone subscribers. Traffic volume varies from season to season,
from month to month, from day to day, and from hour to hour. There are
also variations from minute to minute in the same hour, as well as variations
from subscriber to subscriber.
Busy-season traffic varies differently
from cities to cities, both in amplitude and in duration. There are many
reasons for these variations such as: Where is the city located (resort
area, or retirement area), or what industry does the city have (agricultural,
or high-tech).
The amount of equipment and facilities
that is needed to serve each community is calculated based on the normal
busy season. Occasionally, however, the traffic will exceed that level.
Some unusual peaks can be expected during special days such as Mother's
Day and Christmas. Some other peaks may be due to special events such as
The Republican Convention or a natural disaster (flood, hurricane). I am
told that the peak day for "collect calls" is Father's Day. To
engineer the network facilities for all those eventualities would be very
costly and, furthermore, the magnitude of such events is usually unpredictable.
As a result there will be peak periods when the traffic exceeds the volume
that has been assumed; while most of the time the facilities will handle
all calls with capacity to spare. Usually, the seasonal variations in the
same community will have the same pattern from year to year.
Within the busy season, each community
also experiences weekly and daily traffic variations. Due to condition
peculiar to the community, some weeks may have more traffic than others.
The traffic volume within the week in each community usually forms a fairly
consistent pattern, such as high on Mondays and Fridays, low on Wednesdays,
etc. However, holiday occurring within the week, or other variable influences
such as weather, can greatly affect the traffic distribution pattern.
There are also the traffic variations
between the hours of the day. The degree of hourly variations is greater
than that of any other period. Volume ratios between the busiest hour to
the least busy hour can be as high as 100:1. In business area, peak busy
hour can be expected in the late morning hours; while in the residential
area, the busy time usually falls in the evening hours.
Although the traffic volumes in
all hours are significant, only the busiest hours are of primary interest
to the traffic engineer. The variations discussed so far in this article
are fairly systematic. Their occurrence can be predicted with some reasonable
assurance. A combination of historical records, experience, and good judgment
can produce data on which sound engineering decision can be made. There
is another traffic variation that is not systematic and cannot be predicted.
It is the variation that occurred within an hour. This variation is very
different from hour to hour, or from the same hour in different days. This
random distribution is because the subscribers can originate calls independently
of each other. One other important source of fluctuation in the traffic
pattern is that some subscribers use their telephones more than others.
In general business subscribers make more calls than residential subscribers.
The lengths of conversations also vary considerably among subscribers.
Historical data show that the conversation times (or call holding time)
between one and three minutes are relatively frequent, while long holding
time of 10 minutes or more occur much less often.
How is the traffic measured?
Telephone traffic is defined as
the aggregate of telephone calls over a group of circuits or trunks with
regard to their durations of the calls as well as their numbers. Traffic
flow through a switch or a trunk group is defined as the product of the
number of calls during a period of time and their average holding times.
In traffic theory, the unit of time is one hour. Let C be the number of
calls originated in one hour, and T be the average holding time then the
traffic flow intensity A = C x T. For example, if there are 200 calls of
average length of 3 minutes between Atlanta and Los Angeles in one hour,
then the traffic intensity is: 200 x 3 = 600 minute-calls. Expressed in
hour, A = 600/60 = 10. This value is dimensionless but a name was given
to it. The international unit of telephone traffic is called "erlang,"
named after the Danish mathematician A. K. Erlang, founder of the theory
of telephone traffic.
From the example above A = 10 erlangs.
This number represents:
- The average number of calls in
progress simultaneously during the period of one hour, or
- The average number of calls originated
during a period of time equal to the average call holding time, or
- The total time, expressed in hours,
to carry all calls.
In the US, the term Unit Call (UC)
or its synonym "Centium (Hundred) Call-Second," abbreviated CCS
is generally used. Remember that with over 100 years of history, the telephone
network did not have the modern sophisticated computer equipment to measure
traffic accurately until very recently. To estimate traffic intensity,
mechanical devices were invented to sample or observe the number of busy
circuits. These devices can sample each trunk group once every 100 seconds
(or 36 times per hour). If the measuring device found that in one hour,
all 36 samples show that a particular trunk is being used, we conclude
that the trunk is being used the whole hour, thus by definition this trunk
carries 1 erlang or 36 CCS (i.e., 1 erlang = 36 CCS).
In the above example, if the average
holding time is 5 minutes instead of 3 minutes, then the traffic intensity
is then: A = (200 x 5)/60 = 16.67 erlangs. According to definition 1) above,
the average number of busy trunks between Atlanta and LA has just increased
to 16.67 from 10 because the average subscriber holds a conversation 2
minutes longer
How many trunks are needed?
Before calculating the number of
trunks between 2 cities (e.g., Atlanta and LA) we have to determine what
kind of service we would like to provide to our subscribers. As discussed
in the previous article, it is possible to provide some service between
Atlanta and LA with one trunk.
The subscriber would not be happy
because most of the time the trunk will be busy. The Grade of Service (GOS)
is the proportion of the number of calls finding a busy trunk vs. the total
number of calls. A GOS of 0.5 means that 50% of the call will not get through
right away because of unavailable facilities. In the US, a GOS of 1% is
generally adopted by all telephone companies.
The next question is: what is the
calling pattern of our subscribers? Obviously if all subscribers call at
the same time then we have no choice but to provide one trunk between the
2 cities for each subscriber in the originating city. After observing the
calling pattern for a long time, we have come to realize that since the
number of calls arriving at each trunk group is relatively large at the
busy hour, and the average holding time of each call is small (2 to 3 minutes),
the traffic distribution can be modeled fairly accurately as a Poisson
distribution.
What the Poisson formula, named
after the French Mathematician S. D. Poisson (1781-1840), tells us is that:
a) calls originate individually and collectively at random, i.e., knowing
when one or a set of calls arrives doesn't tell us anything about any other
calls, or b) a call is as likely to originate at any moment as any other
call.
The third question is what to do
with the calls that find all trunks busy. This is the same problem that
we have when we arrive at the bank and see that all tellers are busy. We
have two choices: 1) Leave and come back some other time, or 2) Wait in
line to be served when one teller becomes available. In the telephone networks,
option 1) is usually assumed for several reasons. First, the callers are
unlikely to wait; they tend to hang up and try again. Second, it is much
simpler and more economical to discard a call when all circuits are busy
than to hold it and implement some forms of queuing disciplines. So now
we have a GOS, a calling pattern that looks like a Poisson distribution,
and any call that finds all trunks busy will be discarded.
Based on those assumptions, the
Danish mathematician A. K. Erlang in 1917 developed the following (Erlang
B) formula that expressed the relationship between the number of trunks,
the traffic intensity, and the probability of all trunks busy:
(a^c)/c!
Blocking probability B = -------------------------------------
1 + a + (a^2)/2! + ...... + (a^c)/c!
with a = traffic intensity expressed
in erlang, and c = number of trunks (^ denotes exponential and ! denotes
factorial)
To return to the original question:
how many trunks do we need between Atlanta and LA given that the traffic
intensity is 10 erlangs (It is much more than that in current real network),
and a GOS of 1%? This is just a matter of plugging in the numbers, with
a = 10, B = 0.01. We would need to keep trying various values for c until
the results is less than or equal to our desired GOS. For those of you
who are trying to calculate this number using a spreadsheet, the result
should be 18 trunks. For large traffic intensity a, this is a time consuming
task, and various approximation schemes have been developed to calculate
this number faster. Also various tables have been developed to find this
value based on a number of pre-defined GOS.
Now we know what it takes to compute
the number of trunks required between 2 cities. For a large network with
many cities, and the possibility to route traffic from A to B via C, D
and E, for example, the problem becomes much more complicated. All this
and we have not even touched on how we would handle the day-to-day variation
of the traffic, the non-randomness of some traffic (e.g., retrial immediately
after a busy signal), etc.
How does the Internet traffic fit
into this network? What characteristic does the Internet traffic have that
is different from what we described above? How does that affect the public
voice network? We will talk about that in the next article in this series.
Again, I have omitted many technical
details out of the discussion to make the article readable for everyone.
I welcome any technical question or comment on this article and any other
articles in our regular columns. Please send them to us at [email protected]
and we will discuss them on [email protected].
Luc T. Nguyen, Ph.D.
[email protected]
For discussion on
this column, join [email protected]
Copyright ©
1996 by VACETS and Luc T. Nguyen
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