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- Networks and Telecommunications: Design and Operation, Second Edition.
Martin P. Clark
Copyright © 1991, 1997 John Wiley & Sons Ltd
ISBNs: 0-471-97346-7 (Hardback); 0-470-84158-3 (Electronic)
31
Trafic Monitoring and
Forecasting
Having explained the underlying principles of electrical communication and the statistical ‘laws’
oftelecommunicationstraffic, we cannowconsiderthepracticaldesignandoperationof
networks. A prime concern is to ensure that there are adequate resources to meet the traffic
demand, or to prioritize the use of resources when shortfalls are unavoidable. Two things have to
be done to keep abreast of demand. These activities are the monitoring and future forecasting of
network use. In this chapter we shall review the parameters to be applied in measuring traffic
activity and go on provide an overview of some forecasting models for the prediction of future
to
demand. Used as an input to the teletraffic engineering formulae of Chapter 30, the forecast
values of future traffic can be used to predict the future network equipment requirements on
which the overall planning process will be based.
31.1 MEASURING NETWORK USAGE
Chapter 30.
We discussed different methodsof defining the volume of network usage in
These methods in various ways served to measure, over a predetermined period time,
of
any of the following parameters
0 the total number of calls or messages
0 thetotalconversationtime
0 thetotal holding time (holding includes conversation and
time both time call
set-up time)
0 the total number of data characters (e.g. packets, frames or cells conveyed)
We have covered the concept of trafic intensity, which is a measure of the average call
demand. Wehaveshownhow traffic intensitymeasuredin Erlangs determinesthe
number of circuits needed on a route in a circuit-switched network to maintain given
a
grade o service. For this reason it is usual practice to monitor the magnitude of the
f
555
- 556 TRAFFIC MONITORING AND FORECASTING
traffic intensity, and to predict its future values by forward forecasting. This provides
the basis for planning a future circuit provision programme.
Forecasting, however, need not be confinedto predicting future traffic intensity. The
measuringandforecasting ofotherparameterscanalsobevaluablepredictors of
network traffic demand. For example, in telephone networks where customers pay for
calls based on the total time used in conversation, the measurement of conversation
time not only allows the preparation of customer bills, but also enables the network
operator to calculate the revenue due and the resulting profits, immediately after the
end of each financial period. Meanwhile, packet data network customers usually pay
for the total volume of data carried (measured in segments). The actual amount of use
and predicted future revenue may well be vital to the network operator on the business
and financial side.
Forecasting allows
also investments to be plannedmeet
to futuredemand,
commensuratewithpredicted levels of profits and financialresources.Forecasting
also helpsto determine when established become unprofitable,and therefore when it
will
may be appropriate to consider run-down or discontinuation of the service. No com-
mercially-minded company can put upwith loss-making products or services for long.
31.2
USAGEMONITORING IN CIRCUIT-SWITCHED NETWORKS
The parameters commonly used to monitor the usage of circuit-switched or point-to-
point (leased) networks are listed below
0 the trafficintensity
0 the total number of minutes of usage
0 the total number of calls completed
0 the total number of calls attempted
The reason for monitoringeach of these parameters, and the methods of measurement,
are described in the following sections.
31.3 TRAFFIC INTENSITY
TrafJic intensity, as we learned in Chapter 30, is the measure of the average number of
simultaneous calls in progress on a route between two exchanges, or across an exchange,
during agiven period of time. It is normally measured during the hourof greatest traffic
activity (or so-called busy hour) and is quoted Erlangs. There are two principal methods
of measuring traffic intensity: either by frequent sampling of the number of circuits
actually in use and calculationthe statistical average, or using a call logging system
of by
which records the start and finish time of each individual call, so allowing the total
circuit usage timeto be measured accurately. Total usage during the period is divided by
the duration of the period to give the average number of circuits in use.
- TRAFFIC INTENSITY 557
A check at least once per month of the busy-hour traffic intensity on each route in a
network is essential to make sure there are enough circuits on each individual route.
Circuit requirements for each route are calculated from these values according to the
Erlang formula, using the forecast traffic and the desired grade of service as inputs to
the formula.
On their sampling days, many network operators are not content to monitor the
traffic intensity during the anticipated busy hour (though this is an accepted practice);
they also monitor the traffic activity profile on each route throughout the whole day.
The profile is invaluable as an indicator of overall calling patterns, revealing short or
long periods during the day when thetraffic on a particular route is either very heavily
congested or is under-utilizing the available capacity. In the latter case, the capacity
may be usefulasameans of relieving acongested route (by transit routing in the
manner discussed in the next chapter). Alternatively, an advertising campaign could be
used to stimulate extra network usage.
Traffic profiles on different routes vary greatly according to the nature of the route
(e.g. local, trunk or international) and of its users, and they may change slowly over a
period of months. Some profiles slowly distort towards a highly peaked, busy-hour
oriented pattern, and others reflect a steadier traffic load throughout the day.
Highly peaked and busy-hour oriented profiles, of which Figure 3 1.1 is an example,
can be problematic for network operators for two reasons. First congestion is highly
likely during the period of peak traffic demand (even if circuits are relatively well
provided), and second over the dayas a whole there will be relatively little network use,
with correspondingly low revenue.
Highly peaked traffic profiles can sometimes be flattened by charging a heavy price
premium for calls made at the peak time. The idea is to persuade customers to make
their calls either a little earlier or later during the day. If it can be achieved, a very flat
profile such as that shown in Figure 31.2, makes highly efficient use of the network.
Charges which depend on the time of daytend to have an effect on domesticcall
demand, but the effect is less marked on business calls, becausefew of the callers worry
about the cost.
Possible
Number of
clrcults
m use
Number of
circuits required
Wasted,c.ircuit
ovallablltty tlme
1 1 l 1 1 1 1 1 l 1 ~
I
0 2 4 6 8 1210 2 L 6 8 10 12 Time o f d a y
Figure 31.1 Busy hour oriented traffic profile
- 558 FORECASTING AND
TRAFFIC MONITORING
in use
Number of
circuits required
1 1 1 1 1 1 1 1
I I I I I I I I I I I I
2 4 6 8 10 12
Higher
Figure 31.2 Traffic redistribution
Figure 31.2 illustrates an extreme redistribution of the traffic previously shown in
Figure 3 1.1. In the course of the day the same number of calls have been completed in
Figure 31.2 as in Figure 31.1 and thus a similar revenue has been earned, but theflatter
traffic profile hasmeant a lower busy-hour traffic value and a reducedcircuit
requirement. The profile has been flattened by stimulating ‘time-shift’ of some of the
peak traffic, by charging a higher price for the peak period 8 a.m.-2 p.m.
There are times when significantly higher busy-hour prices fail to dissuade customers
from calling at the busiest time, and none of the traffic is timeshifted. When this hap-
pens the only way to improve overall network utilization and revenue is by stimulating
new off-peak traffic.
31.4 TOTAL USAGE
MONITORING
Besides coping with its route busy-hour demand, a network must be designed to carry the
total volume of demand. Exchanges maybe limited in their capability record the details
to
of individual calls, so that above certain limit,
a billing or other
call detail information may
be lost. In addition, the amount exchange equipment (switching points, registers, etc.)
of
which is required will depend on the total volume of demand. There are two related
measures of the total usage of a circuit-switched network or a leased point-to-point
connection. The two measures are thepaid time and the holding time.
The paid time equals the total number of usage minutes for which customers have
been charged. In telephone network terms this is the number of conversation minutes.
In a telex, or a circuit-switched data network, the paid time is the connected time, the
duration of which has a direct relationship with the total amount textual information
of
that could be carried.
By contrast, the total holding time is the time for which the network itself has been in
use, and it is always slightly greater than thepaid time, as Figure 3 1.3 shows. It includes
not only the ‘connected’ or ‘conversation’ time, but also the time needed for call set-up
- TOTAL USAGE MONITORING 559
Caller
CalledDestination lifts
handset telephone party Caller Network
and dials rings answers clears disconnected
I 4 4 l 4
1(Network idle)
Holding time 4
Figure 31.3 ‘Holding’ and ‘paid’ time
and cleardown and the ringing time of the destination telephone. The holding time is
thebetterindicator of networkresourcerequirements,butnetwork operators may
choose to monitor either paid or holding time, or both.
31.4.1 Paid
Time
The paid time is a direct measure of customers’ actual usage. This is the time during
which the customer actually communicates, and could be considered to be the best
measure of ‘real’ demand. The more paid-time use of a network is made, the greater
is the volume of communication.Thiscontrasts with the traffic intensity, which is
only a measure of how many people choose to communicate at the same time. It is
paid time that attracts revenue and is therefore most
the important financial
parameter. The paid time accumulated by particular customers or on particular routes
between individual exchanges gives a strong indication to network operators of where
they should concentrate their resources and efforts, and where they are most at risk
from competitors. Operators can ill-afford to lose valuable customers or the traffic on
profitable routes.
There are three methods of either measuring or estimating paid time. The first and
most obvious method is to sum the usage made by individual customers. However,
although summation gives valuable route-by-route paid minutes statistics when derived
from itemized customer bills, in those networks which only record their customers’
usage as a bulk total number of units on simple cyclic meters (as Chapter 35 discusses),
summation only reveals totalnetwork use, withoutany route-specific paidminute
information. However, all is notlost, because thepaid time on each route may be
measured directly by sampling the traffic intensity on the route itself at a number of
regular time intervals, and applyingone of the following conversionformulae for
estimating the paid minutes.
Busy hour paid minutes = Busy hour traffic in Erlangs X 60 X efficiency factor
paid time
where efficiency factor =
holding time
- 560 TRAFFIC MONITORING AND FORECASTING
(the value is estimated from historical information)
daily paid minutes = 5 or 6 times busy hour paid minutes
(again the value is obtained from historical records)
total monthly paid minutes = approx. 22 X daily paid minutes
(It is a good enough assumption to estimate an entire month’s traffic as equivalent to
that of 22 working days).
31.4.2 Holding Time
In contrast to the paidtime, the longer period of holding time represents the total time
when network resources are in use, and the difference between holding and paid time
volumes is the time when the network common equipment (used for call set-up and
cleardown) is in use. This quantity is called the common equipment holding time. The
total holding time, the common equipment holding time, and the traffic intensity values
all help to determine how many of each of the various equipments must be provided in
the network.
The amount of a particular type of common equipment (e.g. signalling receivers)
needed in an exchange is usually calculated by the normal Erlang method, with the
common equipment traffic load calculated by multiplying the expected number of calls
by the average common equipment holding time per call. Typical grades of service
demanded of common equipment might be 0.5% lost calls, 0.1 % or even 0.05%.
31.5 NUMBER OF CALLS ATTEMPTED
The total number calls attempted (also called the numberof bids) is the best measure
of
of unconstrained customer demand, because unlike the paid minute volume or the
traffic intensity actually carried by a network, it is not limited by network congestion.
At atimeofnetworkcongestion,the busy hourcall attempt ( B H C A ) count may
continue to increase (as unsatisfied customer demand continues to grow) whereas paid
minute volumes and traffic intensities may saturate at the maximum capacity of the
network. A very large number of unsatisfied call attempts is an almost certain sign of
congestion, either through underprovision of equipment or resulting from short term
network failure, and is a good means of spotting suppressed traffic within a network.
Unfortunately theexact amount of suppressed traffic is difficult to estimate, because the
persistence of customers in repeating call attempts many times over affects the overall
bid count. Figure 31.4 illustrates an example of the failure of some call attempts.
The number of call attempts can only be measured accurately by monitoring each
individual customer’s line.A value measuredat any point deeperin the network will not
be an accurate measure, as it will have been reduced by the effects of any congestion at
the network fringe.
- COMPLETED
NUMBER OF CALLS 561
Small number of attempts
lost because no out oing
circuits a r e availabi'e,or Attempts lost
because common equipment due to congestion
i s congested
Figure 31.4 Call attempts andcongestion
In cases where the call attempt count suggests that traffic is being suppressed, then
the unexpurgated busy-hour traffic intensity and paid minute demand can estimated
be
from one of the following conversion formulae
busy hour traffic, in Erlangs =No. of call attempts in the busy hour
X (average call holding time)/60
busy hour paid time =No. of call attempts in the busy hour
X average paid time per call
Sometimes exchange monitoring limitations make it impossible for network operators
to measureaccuratelythenumber of call attempts from a given source to a given
destination. If so, the the number of call attempts may be estimated as the number of
outgoing circuit seizures. As shown in Figure 31.4, however, this estimate will be lower
than the actual because a small number of call attempts (bids) fail in the first exchange
due to congestion, and so do not mature into an outgoing circuit seizure.
31.6 NUMBER OF CALLS COMPLETED
The numberof calls completed in a network sense reaching ringing toneor answer),
(i.e.
when compared with the numberof calls attempted, gives another measure of the state
of network congestion. The proportionof busy hour calls completed, when expressed as
a percentage of the number of calls attempted, should equal the design grade service.
of
Hence
(number of busy hour call attempts) - (number of busy hour call completions) X 100%
grade of service =
(number of busy hour call attempts)
The grade of service is a measure of thefrustration that a customer experience when
will
trying to complete a callduring thebusiest hour of the day. We could calculate the aver-
age daily percentage of lost call attempts, but this is not so commonly done, because
psychological analysis of customer behaviour suggests that it is of no relevance.
- 562 FORECASTING AND
TRAFFIC MONITORING
The number of calls completed by the network is a difficult quantity to measure,
because not all signalling systems indicate ‘network-completed’
the state. the
In
American network a peg-count overfIow was used to monitor this value butits absence
in other networks means that it is also common to measure onlythe number and
proportion of answered calls. number
This is clearly
lower than number
the
completed by the network as some calls are bound to encounter either a subscriber
busy state or a ring tone-no reply condition. The proportion of answered to attempted
calls,andthat ofansweredcallsto seizures, areknownasthe answer bid ratio
( A B R ) andthe answer seizureratio ( A S R ) , respectively;they are definedmathe-
matically below. Measured over relatively short periods of time (5-15 minutes), both
aregoodindicators of instantaneousnetworkcongestion.The higherthenetwork
congestion, the lower the ABR or ASR. The converse, that the higher the ABR or
ASRthegreaterthecongestion, is notnecessarily true because calls mayremain
unanswered for a range of other reasons (e.g. people are simply not answering their
phones). These same uncertainties mean that no conclusion can be drawn from the
actual value of the ABR or ASR. A conclusion can only be drawn from the value
relative to its ‘normal’.
no. of answers
answer bid ratio (ABR) =
no. of call attempts
no. of answers
answer seizure ratio (ASR) =
no. of seizures
In Chapter 37 the use of ABR and ASR statistics as tools for short term network
management surveillance of the network will be described further.
31.7 MONITORINGUSAGE OF DATANETWORKS
31.7.1 Packet-,Frame- and Cell-switchedNetworks,LANsandMANS
There are two prime factors importance to data network users. These are the overall
of
throughput capacity and the responsetime (network transactiontime or propagation
time) of the network.
The throughput capacity the amountof data that can sent over the network dur-
is be
ing a given period. Usually a networkis designed to cope with the maximum demand of
thepeakhour,althoughnetworkcostsavingscan be realized by queueingup less
important data, for transmission outside the peak hour. Such queueing leads to more
effective 24-hour use of the network.
Throughput capacity canbe measured in bits per second (bit/s), messages per second,
packets (frames or cells) per second, or segments per hour (1 segment = 512 bits).
In designing and upgrading network capacity,important to consider the practical net-
is it
work throughput rather than just the transmission line speed capacity. The practical
network throughputwill always be less than the line speed capacity,first because not all
messages are received without errors and some have to be re-transmitted, and second
- MONITORING USAGE OF DATA NETWORKS 563
becausesome of theavailablelinecapacity is effectively ‘wasted’ingapsbetween
messages. The two effects are summarized in the formulae below.
correct messages received
successful message throughput ( W ) =
time duration taken
writing it another way
total messages X proportion with no errors
TP =
time taken for message transmission + wasted time between messages
where M = average message length in bits; P = probability of errors in the received
message; R = line speed in bits per second; T = average line time ‘wasted’ between
messages; M / R = time required to transmit average message.
Rearranging the formulae, so that we can relate the required network line speed to
the required user data throughput
R = TP
(1 + TR/M)
l-P
This formula ensures an adequate average throughput capacity of the network, but
another equally important characteristic of a data network is the response time. The
computer systems linkedby data networks will have been designed to carry out a given
set of functions within a given period of time, to maintain for example a database of
share prices updated at least once every hour, or (more onerously in terms of response
time) to control the trains on a metropolitan underground railway.
The overall response timeof a computer and data network includes the time takento
transmit an error-free message along the line and back, plus the computer processing
time at the far end the communication link, and the line turn-around time the link
of (if
is half-duplex). The line transmission time includesnot only the period during which the
line is conveying data, but also any time spent waiting for any previous messages to be
transmitted or acknowledged. If the response time of the network is too long, then the
networkdesignermustincreasethenetwork throughput capacity, which he can do
either by upgrading line speeds (as might be appropriate on a LAN or on a point-to-
point leased circuit network, using say 9600 bit/s modems rather than 4800 bit/s ones)
or by providing a larger number of circuit connections (as might be appropriate in a
circuit- or packet-switched data network).
The dimensioning method of calculating what overall bit throughput is required to
meet a given response time constraint is based upon the Erlang waiting time formula
described in Chapter 30. As might be expected, the faster the required response time,
the greater is the transmission linespeed required (see also Chapter 20).
In conclusion, no matter which dimensioning method is used, if the network is to
have adequate capacity to meet the user’s throughput demand, the transmission line
speed must be chosen with a relatively higher capacity, sufficient to counteract theeffect
of errorsandthe ‘wasted’time between messages. The‘wasted’time,incidentally,
- 564 TRAFFIC MONITORING AND FORECASTING
includes not only periods of instantaneous non-use, but also includes someof the data
overheads which accompany each message (e.g. the header and destination codesof the
network protocol about which we learned in Chapter 9).
Because they have a marked impact on the performance data networks, it is worth
of
digressing for a moment to discuss theeffect of data packet, block or frame lengths. As
we learnedinChapter 9, theprotocolusedtoconveydatamessagesbreaksthese
messages down into a number of frames, blocks or packets, and transmits each with
some other overhead information which is needed to ensure delivery to the correct
destinationandtocontrolthefrequencyoferrors.Duringperiods of linenoise
disturbance, longer packets of data are more likely to be affected by the noise than
shorter ones. The result a lower effective data throughput and a slower response time
is
because of the extra workload imposed the need for large scale data re-transmission.
by
On the other hand, shorter blocks have the disadvantage of decreasing the throughput
at all times (whether the network is busy or not). This is because of the overhead of
protocol headers, one each for the larger number of packets or frames.
31.8 FORECASTING MODELS FORPREDICTING
FUTURE NETWORK USE
The operation of networks, their efficient utilization, carriage of traffic, and their suc-
cessful evolution all depend critically on accurate estimation of future needs.It is vital
for network operators to have an efficient method of forecasting thetraffic the network
must carry so that a network of sufficient size can be maintained to meet users’ needs.
Forecasts need to cover both short and long-term periods that the whole business
so
of providing for the future, the planning and the capital investment, can put in hand
be
in good time. For straight circuit provision a twelve month forecast may be adequate,
but normal
for extensions of major switching and transmissionequipment it is
necessary to look 1-5 years into the future, in line with the ordering lead time. Even
longer forecasts maybe needed when planning new cables, layingnew ducts, and setting
up major new sites. The longer term forecasts are inevitably less accurate than the
shorter, but because there is time for adjustment, this is acceptable.
Forecasting uses historic measurements of network usage, and particularly of growth
in usage, to predict likely future customer demand. A number of different forecasting
models have been developed, but none of the methods are100% reliable. This is hardly
surprising as they all involve attempts at predicting future human behaviour. The reli-
ability of any forecast relies on the accuracyof the historical information and the period
over which it has been collected. The shorter the period historical knowledge and the
of
lower the number of observations on which it is based, the less reliable is the forecast.
In general, the selection of an appropriate forecasting method ensures that the forecast is
accurate for a period into the future approximately equal to one-third the period of
of
the historical information. Thus a one year forward forecast should based on at least
be
three years of historical information. The best to select a particular forecasting met-
way
hod for use in any given circumstance is by experience. If the model appears to give
reliable results in practice and therefore is a ‘good predictor’, then use it, if not; try
another method.
- FORECASTING
PREDICTING
MODELS
FOR FUTURE NETWORK
USA 565
In this chapter we shall not attempt to cover all the available forecasting methods,
butinstead review a few simplemethods,particularlythosedescribedinITU-T’s
recommendation E.506 on ‘forecasting of traffic’.
As we have seen in the earlier part of this chapter the most important parameters
measuring the overall network usage are the maximum instantaneous traffic demand
(i.e. the trajic intensity, or its equivalent) and the maximum volume demand (i.e. the
paid minutes or their equivalent). These parameters more than any others govern how
much capacity is needed in the network and the revenues that result. As we have also
seen, the two values are related by the daily traffic profile, so that paid minutes may be
estimated from the busy-hour traffic intensity or vice-versa. Because of this fact, either
or bothof the parameters canbe forecast, and a forecast for the other parameter can be
calculated using the conversion factors presented earlier. Forecasting the values directly
is called direct forecasting, whereas deriving them from conversion formulae is called
composite forecasting. By using both methods, the forecast can be double-checked.
In preparation for a traffic forecast, the forecaster should identify any regular or
irregular disturbances which have affected past traffic and may affect the future fore-
cast. Thesemayincludediscontinuitiesin traffic growthordecline,resultingfrom
disturbances such as changes in tariffs, modernization of the network, or removal of
congestion. The use of composite forecasting as a double check comes into its own when
a large number of discontinuities or irregular jumpsin volume are present. Figure 3 1.5
shows an example of a leap in traffic growth caused by the stimulation of traffic resulting
from the introduction of automatic service. Such leaps in telephonetraffic growth, even
growth of many times over, are not uncommon.
The process of developing a forecast comprises five steps. First, one of a number of
different types of forecasting models must be chosen. Examples include simple models
whichmightassumelinear, quadratic or exponential growth in traffic as shownin
Figure 31.6. Alternatively,morecomplexforecastingmodelssuch as econometric
models be
may used. Econometric models attemptto predict futuredemand by
assuming that the traffic depends upon a number of socio-economic variables such as
Introduction o f
automatic ervice
s
104 7 I
l Logarithm o f
I - traffic emand
d
Time
Figure 31.5 Discontinuity in traffic growth
- 566 TRAFFIC MONITORING AND FORECASTING
X=At+B X= At’+Bt*C X = A exp ( B t l
[a) Linear ( b ) Quadratic (c) Exponential
Figure 31.6 Simple forecasting models. X, historic values; Q, forecast future values
the retail price index, thepopulation count, etc. The equations an econometric model
of
reflect a relationship between these variables and the traffic parameters.
The second step is to guess exactly which model within the class fits the historical
values most accurately, and the third step is to evaluate the unknowncoefficients in the
appropriate formula (i.e. values A , B and C in Figure 31.6).
The accuracy of themodelshould be confirmedinthe fourth step by a method
knownas diagnosticchecking. Thisensures that there arenomajor discrepancies
between the model and the historical information. If there are discrepancies, it attempts
to remedy them by adding correction factors or by adjusting the formula coefficients.
Finally, the futureforecast can be made. The workprocess is shown in the flow diagram
of Figure 31.7.
Guess a type of forecast model
parabolic, exponential, econometric
l.
Guess exactly which model will be the
r
best fit
l
Adjust the
1
model if the
fit is not formula using the historical information
acceptable
Perform diagnostic checking to ensure that
the model fits the historical information
I Modelconfirmed.Forecastfuturevalues
l
Figure 31.7 Forecastingprocedure
- FITTING THE FORECASTING MODEL 567
31.9 FITTING THE FORECASTING MODEL
Identification of the appropriate model and of the explanatory variables is the most
difficult aspect of forecasting. Even when a suitable form of equation for estimating the
behaviour has been determined, the evaluation of the coefficients is complex and a num-
ber of methods are available. Perhaps the simplest method of estimating the coefficient
values is called regression. This is the only method that we discuss here, and the example
that follows shows how it can be applied to a linear forecasting model. In such an
instance the method is known as linear regression.
In linear regression we assume that the growth in traffic demand (or the change in
value of some other parameter)will change over time in a linear straight line manner.
or
Linear regression therefore attempts to ‘fit’ the best straight line possible onto the
historically recorded values of the parameter. Whenthis has been done, the straightline
is extended into the future to give aforecast of subsequent values, as Figure 31.8
illustrates.
In Figure 31.8 the value of a parameter has been measured every year from 1982 to
1988; based on this information, a forecast of future values corresponding to the years
1989 to 1992 is required. (The forecast is likely to be valid for around two years (one
third of the period of historical information.) Theforecast values are those predicted by
the straight line showninFigure 31.8. This is the best-fit straight line through the
historical values.
The best fit straight line can be determined by minimizing the total of the vertical
distances of each of the historic parameter values from the best-fit line as illustrated
by Figure 31.9. In this example the best-fit line is taken to be that line for which the
+ +
sum of the errors (a + b c d ) is minimized. This is the so-called least-dzfference
regression line.
I
Parameter
value
1982 198L Year 1990
1986
1992 1988
Figure 31.8 Forecasting by linearregression. X, measured historic values; 0,
forecast future
values
- 568 FORECASTING AND
TRAFFIC MONITORING
Y
9 line f a,b,c,d
Best f i tsum ocorresponds to t h e
lowest
X
Figure 31.9 Least-differenceregression
It is more normal, however, when using the regression method of line fitting,to work
on least squares regression rather than least dEfSerence. In least squares regression the
best-Jit line is that corresponding to the minimum value of the squared errors, in other
+ +
words the minimum value of (a2 + b2 c2 d 2 ) .
The method produces an equation for the best-fit or regression line as follows
Y=a+bX
In this book we shall not discuss in detail how to derive the values of a and b, and if
interested the reader should refer to an advanced statistics or mathematics book. How-
ever, for the benefit of those of a mathematical mind the formulae are as shown in the
table of Figure 31.10. Future values of the parameter Y are predicted one at a time by
substituting relevant values of X (the future dates), a and b into the equation shown in
Figure 3 1.10.
Forecasting and regression methods need not be confined to straight line predictions
of future events, and Figure 31 .l 1 demonstratestheapplictionofeitherthe least
difference or least squares regression technique to determine the coefficients of a curved
forecasting model.
~~
B e s t fit line e q u a t i o n Y = a + bX
a=My-bMx
Xiyi- M x M y
b=
S?
where =
Mx m e a n of the x-values
M , = m e a n of the y-values
S = standard deviation of the x-values
,
Figure 31.10 Least square regression equation
- OTHER MODELS 569
Y /
/
/
/
/
X
(date)
Figure 31.11 Using a curve for forecasting
31.10 OTHER
FORECASTING MODELS
Many parameters exhibit a curved growth pattern similar to that already shown in
Figure 3 1.11, and this is typical of a graph of total traffic demand on many different
types of network. Almost any shape of curved forecasting model may be used, and
some examples are given in Figure 31.12. The method remains the same: the forecaster
needs to make a hypothesis as to the shapeof curve that best predict future events, and
then to conduct regression analysis to fix the coefficients of the corresponding formula
which represents the forecast curve.
As we said earlier, the type of curve that is used is entirely up to the forecaster, and
previous success with a given curve and forecasting method is usually the best guide.
There is no ‘correct’ method, and only time will tell whether the prediction is correct.
The only recourse is to try something different next time. The forecaster must check
that any forecast he makes is sensible. Simple questions give a good guide.For example
a 10000 line local exchange is unlikely to generate 3000 Erlangs of traffic; that would
require 0.3 Erlang per exchange line (18 minutes of use each day at the peak time).
I/Exponential
growth
S- curve, or
saturation
growth
Sporadic
growth
I Exponential
decline
Figure 31.12 Possiblepredictioncurves
- 570 TRAFFIC MONITORING AND FORECASTING
Amongthelargepublictelecommunicationscompanies,econometricforecasting
models are becoming the most popular. By using such models, the forecasters assume
that the total volume public networktraffic is related to the overall economics, social
of
conditions and trade of the company or nation. There is no doubt about their value
because the companies continue to use them. However, to end on a gloomy note, it is
important to recognize that forecasts are seldom accurate, and it is no disgrace to be
under or overforecast.What is inexcusable is forplansto be drawnupwithout
sufficient contingency and flexibility to be adapted as time goes on and the real level of
demand makes itself known.
nguon tai.lieu . vn
