Goeller on Telecom
Traffic
'Introduction' to
Ted Frankel, Tables for Traffic
Management
and Design
(Webmaster’s note:
abcTeletraining, the original publisher of Ted Frankel’s book,
which includes all the tables referred to in Lee Goeller’s
Introduction, was acquired some years ago by
AVO Technical Resource Center. It still offers some abc titles,
but it appears that Ted Frankel's book is out of print.)
Unlike Ted Frankel, I'm not much of a
mathematician. However, this lack may well be of value to the
reader; after 20 years of struggle to understand telecommunications
traffic theory and practice, I now know exactly where mathematical
details, beloved of pedagogues, blinded me to what was actually
happening. Thus the first part of this treatise deals with certain
items that are so basic that few teachers or textbooks even mention
them. The rest shows how to find the right mathematical model for a
given problem. Had I had this information all along, it would have
been much easier to make sense of the mathematics.
Things my teachers never taught me
Traffic theory is concerned with predicting
future behavior on the basis of the past, so it consists of two
parts. The first deals with obtaining numerical information that
gives a good description of the past, and the second deals with the
construction of mathematical models that relate information about
the past to the needs of the future. Both aspects of traffic theory
require a number of assumptions; after almost a hundred years, these
assumptions have been found to match pretty well with reality.
It is not always obvious that obtaining raw
data is ex-pensive. Further, data collection is a continuing
endeavor to allow past predictions to be checked and, as changes
occur, new predictions to be made. Thus, effort is required to hold
to a minimum the cost of obtaining data; as a result, raw traffic
data is usually (but not always) obtained in the form of averages.
The purpose of mathematical models is to provide a simple means of
relating such less costly average traffic values to maximum values.
The number of "servers" required, whether they are trunks in a
telephone system or checkout counters in a super-market, are related
to the peak values of traffic to be served rather than to the more
easily obtained averages.
To obtain a meaningful measurement, certain
characteristics of the measuring plan must be understood. First, if
something is to be measured, the period of time over which it is
measured must be correctly selected. Thus the concept of the "busy
hour" has been introduced in telephony.
A busy hour must be appreciably longer than
the length of each telephone call, and yet it must be short enough
so that traffic during the interval is relatively constant. It
happens that, in many instances, telephone calls average about five
minutes or so in length, and traffic from a large enough number of
individuals is fairly constant for several hours at a time in the
morning and again in the afternoon during the business day, and in
the early evening when residential users take advantage of reduced
rates. Thus an hour is a convenient duration for averaging use.
Of course, not just any hour will do. The
hours from 7 a.m. to 10 a.m., for instance, would be ill-chosen for
studying PBX traffic because such traffic would build up almost
linearly during the entire interval, and the average for any given
hour would not represent a "stationary random proc ess."
On the other hand, traffic between about 9:30
a.m. and 11:30 a.m. might well be fairly constant, unaffected by the
morning build-up of business or the variation produced by the lunch
hour.
Units in which traffic is measured are of
some importance. There are three basic units: hours per busy hour,
minutes per busy hour, and hundred-seconds per busy hour. Hours per
busy hour are called Erlangs after the great Danish traffic engineer
and mathematician. The erlang is used largely by design engineers
and mathematicians; it is particularly convenient in that it shows
directly the average occupancy during an interval.
Minutes per busy hour are used by business
communication managers, usually in connection with WATS and toll
studies. In the telephone industry in the United States, the most
commonly used traffic unit is the CCS or hundred call seconds (C for
hundred as in Roman numerals). CCS per busy hour is implied, but is
seldom stated.
An erlang is 60 minutes or 36 CCS of use per
busy hour, and a CCS is 1 and 2/3 minutes. Conversions from one unit
to another are given in a chart inside the back cover of this book;
in the tables themselves, all traffic values are given in erlangs.
Total traffic offered a group of circuits,
whether measured in erlangs, minutes or CCS, is all that is needed
to do most kinds of traffic engineering. Thus traffic tables,
including the ones in this book, are all based on offered traffic.
Unfortunately, monitoring equipment measures traffic carried; so
allowances must be made when field data is used.
Holding time per call is another very
important concept. It includes the time required for dialing and
ringing (establishing the call), conversation time, and the time
required to take the connection down. Traffic monitors measure the
whole thing — all the time the circuit is in use. Equipment for
automatic message accounting (AMA), however, measures only the
conversation time. Thus average holding time may be quite different
if obtained from AMA equipment rather than traffic recorders.
Holding time is particularly important in
systems where calls must queue up to use expensive facilities. It
turns out that delay in the queue must be measured in units of
average holding time if any generalized mathematical theory is to
work. This is also true at toll booths, check-out counters, and
other systems where users wait in line for service. To appreciate
the impact, consider a situation where the number of calls is cut in
half but the holding time is doubled, thus keeping the offered
traffic constant. The delay in a queue will NOT remain constant — it
will be doubled to reflect the longer holding time per call.
Once raw data is available from measurements,
mathematical models must be used to relate the offered (average)
traffic to the number of circuits required. "Required" involves
another assumption regarding "grade of service." Grade of service is
the probability that a given call attempt will be blocked or delayed
during the busy hour.
Since an unused server of any sort serves no
useful purpose and is, as a result, uneconomical, it is obviously
undesirable to provide servers in such quantities that blocking or
delay is never encountered in the busy hour. As a practical matter,
grades of service on the order of one to three calls per hundred
finding facilities unavailable are about right. If a higher
proportion meets busies or delays, excessive retries will result. If
a lower proportion is encountered, circuits will go to waste and the
system will be uneconomical. Selection of a grade of service is
often beyond the capability of traffic theory; once grade of service
is available, however, traffic theory can provide the information
necessary to engineer systems as required.
Selecting the right traffic model
Traffic theory, as normally expounded in
textbooks and journal articles, represents a formidable challenge to
the reader. But, even when the theory is mastered, one is still
seldom in a position to DO anything. Most of the mathematics, even
when understood, is totally intractable without an electronic
computer, and even then, some sophistication is needed in handling
numbers like 50! (50 factorial, or 50x49x48x ... x3x2xl) which are
frequently en-countered. Thus, to move from theory to practice,
tables which summarize the results of theoretical equations are
needed. However, outside of the Bell System's Traffic Engineering
Practices (which are proprietary), only a few scattered tables have
been available, and these only to people who know where to look. The
present volume is, thus, unique.
It contains 14 different tables which permit
the user to obtain immediate results even without a thorough
knowledge of traffic theory. But choosing the right table is
some-times a problem. For a solution, the reader is referred to the
Master Flow Chart on page 9.
I constructed an early version of the Flow
Chart some years ago when Ted Frankel and I were teaching a course
on Telephone System Design and Traffic Theory. Over the years, the
Flow Chart has been modified to reflect teaching experience and
suggestions from Ted and others; during this interval, too, Ted has
computed more and more tables. With the publication of this book,
both the tables and the Flow Chart have reached a stage of
completion, and the Flow Chart has become a handy one-page guide to
the tables. I'd like to thank Ted for remembering my Flow Chart and
giving it a chance to appear with his tables where it all started in
the first place.
Using the flow chart
Starting at the top of the
Flow Chart, there are two
choices to be made initially: Are the users offering traffic to the
system finite in number, or "infinite?" That is, are there less than
ten times as many users as servers, or are there perhaps 100 times
more users than servers? In between finite and infinite, of course,
judgment must be used as to which leg of the chart to follow.
After the first choice is made, three more
choices are required depending on how we dispose of calls that find
all servers busy. These are usually referred to as the lost calls
cleared, held and delayed assumptions. If we assume lost calls are
cleared, a call finding all servers busy simply vanishes. It goes
away and does not come back. This is nice from a mathematical
standpoint, since we only have to calculate the probability that all
N servers in the group are busy; if less than N are busy, the group
isn't blocked, and if more than N are needed, the calls generating
that need vanish.
This is a bad assumption in that it does not
reflect user behavior; upon being blocked, users try again and
again. However, when blocking probability is low (good grade of
service), the effect is small.
There is one place where the assumption is
good, however: in alternate routing theory. If a call finds all
circuits in the initial route busy, it can vanish as far as that
route is concerned when it is applied immediately to an alternate
route.
The "lost calls held" assumption is slightly
different. It assumes a call is in the system for an average holding
time, whether it finds all trunks busy or not; if all servers are
busy initially but one comes free in less than one holding time, the
user will seize it and use it for the remainder of his holding time.
The lost calls held assumption is generally
used by the Bell System, while lost calls cleared is generally used
throughout the rest of the world. For a good grade of service, the
difference is small.
The "lost calls delayed" assumption says that
once a call is placed, it will remain in a queue until a server is
ready to handle it; then it will use the server for the full holding
time of the call. By queuing, higher occupancy of the facility can
be assured. However, no server can handle more than 60 minutes of
traffic during an hour. If more than 1 erlang (36 CCS, 60 minutes)
of traffic is offered per circuit per hour, the queue will grow
without limit.
The names of the formulas used for each leg
of the flow-chart are given next. These formulas are mathematical
models that fit the assumptions. With infinite sources and lost
calls cleared and held we use the Erlang B and Poisson formulas,
respectively. Delayed calls from infinite sources are treated in the
bottom of the Chart. For finite sources, the formulas are Engset,
Binomial and an un-named formula called "Delay" in this book.
Following the formula names, the appropriate
tables are identified. Since a number of parameters are involved in
each formula, different expressions of the same information are
possible. For instance, under the Engset formula, if we know the
number of servers, N, the number of sources, L, and the average
traffic per source, a, we can find the probability of blocking, P,
in Table 11. On the other hand, with N and L and the acceptable
blocking probability, P, given, we can use Table 12 to find A, the
total traffic which can be offered from the L sources.
Tables 13 and 14 work with the Binomial and
Delay formulas, respectively, and relate N, L, and a to the
probability of blocking or delay, P.
The difference between an infinite number of
sources delivering A erlangs of traffic to N servers, and L sources
each delivering a erlangs of traffic at the same grade of service
can be quite striking. The phenomenon is called "limited source
gain," and is illustrated on page 12.
With infinite sources, the lost calls cleared
and held assumptions require the use of the Erlang B and Poisson
models. When L goes to infinity and L times a becomes A, a finite
number of erlangs of offered traffic, the number of parameters is
reduced.
Tables 1 and 4 are similar in that they
relate a given N to the desired traffic that can be handled, A, at a
given grade of service, P; Table 1 is Erlang B, while Table 4 is
Poisson, lost calls cleared and held respectively. Tables 2 and 5
are again for Erlang B and Poisson, but assume that the offered
traffic, A, is given, and the number of trunks required, N, is to be
found.
Table 3 is a little different. It shows
traffic offered to each circuit in a group, assuming traffic is
always offered to the same trunk first and to the rest in the same
order when busy trunks are found. For each trunk, the offered
traffic is shown, the traffic carried on that trunk, and the traffic
that overflows. Obviously, the traffic that overflows trunk K is the
traffic offered to trunk K+1. These Tables are particularly useful
in designing groups of WATS lines or alternate routing systems.
When we come to lost calls delayed, infinite
sources, we now add a new dimension to the information, and continue
to the lower section of the Flow Chart where more decisions must be
made.
The first concerns holding times. The study
of statistics tells us that the mean is the best single description
we can get of a group of numbers. However, until we know how the
individual numbers from which the mean is computed are distributed
around the mean, its use is dangerous. In statistics, many different
probability distributions are studied; for holding times, only two
appear to be of general use: the exponential and the constant.
The exponential distribution applies rather
well, as many measurements have shown, to the length of local
telephone calls. Thus we can use the mean with some confidence,
knowing that there will be many shorter calls, and longer calls will
become less frequently encountered at an exponential rate. This is
fortunate since the exponential distribution is very easy for
mathematicians to use.
The constant distribution, on the other hand,
applies to various kinds of telephone equipment such as markers,
decoders, senders, etc., and to CAMA operators, intercept operators,
and other personnel who perform very specialized and limited
functions that require a fixed amount of time.
Once we have selected exponential or constant
holding time, a three-way choice is required as to how calls in a
queue are to be treated. The three choices that have been
extensively studied are first-in first-out (FIFO), random and
Last-in first-out (LIFO). FIFO is the most logical and fair, and
tables are provided here for this case only. If the other orders are
required, data is available in the references cited.
Tables 6 and 7 are based on the Erlang C
formula which applies to exponential holding times and FIFO queue
order. P(0) is the probability that a call will be delayed; Dl is
the number of holding times an average call will be delayed; and D2
is the number of holding times a delayed call will have to wait. Dl,
unlike D2, includes those calls which happen to find a server
available. D1/D2 = P(0), or, the relationship between the delay on
all calls and the delay on all delayed calls is the probability that
a call will actually be delayed. Because of rounding in the tables,
this ratio is not always exact. The right half of each of these
tables shows P(t), the probability that a call will be delayed for
longer than t holding times.
Tables 8 and 9 are similar to the left-hand
side of Tables 6 and 7, and provide P(0), Dl and D2 for constant
holding times under the Crommelin-Pollaczek model. Table 8 also
includes the average occupancy per circuit at the offered traffic.
Tables 6 and 8 are used where offered traffic
is given and the number of trunks is to be found. Tables 7 and 9
start with the number of trunks as given, and show the traffic that
can be carried. Table 10 is actually a continuation of Tables 8 and
9. It corresponds to the right hand side of Table 7, and shows the
probability that calls are delayed for t holding times or longer.
This brief introduction to the traffic tables
is certainly not intended to be a complete course. It is merely a
quick survey of some points that I have found to be important, and a
guide to the proper use of the more popular traffic models. With
luck, the reader can now solve many of the common traffic problems
that confront him, even if he can't derive the formulas or pronounce
Pollaczek. In addition to the tables, some interesting solved
problems have been included as typical examples, along with
additional background information and a bibliography arranged in
lists to compare with sections of the Master Flow Chart. We hope
that the reader will find this book useful and, perhaps even
entertaining.
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