5i'
SECTION 2
FORECASTING OF TRAFFIC
Recommendation E.506
FORECASTING INTERNATIONAL TRAFFIC
1 Introduction
This Recommendation is the first in a series of three Recom
mendations that cover international telecommunications forecasting.
In the operation and administration of the international tele
phone network, proper and successful development depends to a large
degree upon estimates for the future. Accordingly, for the planning
of equipment and circuit provision and of telephone plant invest
ments, it is necessary that Administrations forecast the traffic
which the network will carry. In view of the heavy capital invest
ments in the international network, the economic importance of the
most reliable forecast is evident.
The purpose of this Recommendation is to give guidance on some
of the prerequisites for forecasting international telecommunica
tions traffic. Base data, not only traffic and call data but also
economic, social and demographic data, are of vital importance for
forecasting. These data series may be incomplete; strategies are
recommended for dealing with missing data. Different forecasting
approaches are presented including direct and composite methods,
matrix forecasting, and top down and bottom up procedures.
Recommendation E.507 provides guidelines for building fore
casting models and contains an overview of various forecasting
techniques. Recommendation E.508 covers the forecasting of new
international telecommunications services.
2 Base data for forecasting
_________________________
The old Recommendation E.506 which appeared in the
Red Book was split into two Recommendations, revised
E.506 and new E.507 and considerable new material was
added to both.
An output of the international traffic forecasting process is
the estimated number of circuits required for each period in the
forecast horizon. To obtain these values, traffic engineering tech
niques are applied to forecast Erlangs, a measure of traffic.
Figure 1/E.506 outlines two different approaches for determining
forecasted Erlangs.
The two different strategies for forecasting are the direct
strategy and the composite strategy. The first step in either pro
cess is to collect raw data. These raw data, perhaps adjusted, will
be the base data used to generate the traffic forecasts. Base data
may be hourly, daily, monthly, quarterly, or annual. Most Adminis
trations use monthly accounting data for forecasting purposes.
With the direct strategy, the traffic carried in Erlangs, or
measured usage, for each relation would be regarded as the base
data in forecasting traffic growth. These data may be adjusted to
account for such occurrences as regeneration (see
Recommendation E.500).
Figure 1/E.506, p. 1
In both strategies (direct and composite) it is necessary to
convert the carried traffic into offered traffic Erlangs. The
conversion formula can be found in Recommendation E.501 for the
direct strategy and in this Recommendation for the composite stra
tegy.
Composite forecasting uses historical international accounting
data of monthly paid minute traffic as the base data. The data may
be adjusted by a number of factors, either before or after the
forecasting process, that are used for converting paid minutes on
the basis of the accounting data into busyhour Erlang forecasts.
As seen in Figure 1/E.506, the forecasting process is common
to both the direct and composite strategy. However, the actual
methods or models used in the process vary. Forecasts can be gen
erated, for example, using traffic matrix methods (see S 4),
econometric models or autoregressive models (see S 3,
Recommendation E.507). There are various other data that are input
to the forecasting process. Examples of these are explanatory vari
ables, market segmentation information and price elasticities.
Wherever possible, both the direct and composite forecasting
strategies should be used and compared. This comparison may reveal
irregularities not evident from the use of only one method. Where
these are significant, in particular in the case of the busy hour,
the causes for the differences should be identified before the
resulting forecast is adopted.
In econometric modelling especially, explanatory variables are
used to forecast international traffic. Some of the most important
of these variables are the following:
 exports,
 imports,
 degree of automation,
 quality of service,
 time differences between countries,
 tariffs,
 consumer price index, and
 gross national product.
Other explanatory variables, such as foreign business travell
ers and nationals living in other countries, may also be important
to consider. It is recommended that data bases for explanatory
variables should be as comprehensive as possible to provide more
information to the forecasting process.
Forecasts may be based on market segmentation. Base data may
be segmented, for example, along regional lines, by business,
nonbusiness, or by type of service. Price elasticities should also
be examined, if possible, to quantify the impact of tariffs on the
forecasting data.
3 Composite strategy  Conversion method
The monthly paidminutes traffic is converted to busyhour
Erlangs for dimensioning purposes by the application of a number of
traffic related conversion factors for each service category. The
conversion is carried out in accordance with the formula:
A = Mdh /60e
(31)
where
A is the estimated mean traffic in the busy hour,
M is the monthly paidminutes,
d is daytomonth ratio,
h is the busy hourtoday ratio, and
e is the efficiency factor.
The formula is described in detail in Annex A.
4 Procedures for traffic matrix forecasting
4.1 Introduction
To use traffic matrix or pointtopoint forecasts the follow
ing procedures may be used:
 Direct pointtopoint forecasts,
 Kruithof's method,
 Extension of Kruithof's method,
 Weighted least squares method.
It is also possible to develop a Kalman Filter model for
pointtopoint traffic taking into account the aggregated fore
casts. Tu and Pack describe such a model in [16].
The forecasting procedures can be used to produce forecasts of
internal traffic within groups of countries, for example, the
Nordic countries. Another application is to produce forecasts for
national traffic on various levels.
4.2 Direct pointtopoint forecasts
It is possible to produce better forecasts for accumulated
traffic than forecast of traffic on a lower level.
Hence, forecasts of outgoing traffic (row sum) or incoming
traffic (column sum) between one country and a group of countries
will give a relatively higher precision than the separate forecasts
between countries.
In this situation it is possible to adjust the individual
forecasts by taking into account the aggregated forecasts.
On the other hand, if the forecasts of the different elements
in the traffic matrix turn out to be as good as the accumulated
forecasts, then it is not necessary to adjust the forecasts.
Evaluation of the relative precision of forecasts may be car
ried out by comparing the ratios s(X )/X where X is the forecast
and s(X ) the estimated forecast error.
4.3 Kruithof's method
Kruithof's method [11] is well known. The method uses the last
known traffic matrix and forecasts of the row and column sum to
make forecasts of the traffic matrix. This is carried out by an
efficient iteration procedure.
Kruithof's method does not take into account the change over
time in the pointtopoint traffic. Because Kruithof's method only
uses the last known traffic matrix, information on the previous
traffic matrices does not contribute to the forecasts. This would
be disadvantageous. Especially when the growth of the distinct
pointtopoint traffic varies. Also when the traffic matrices
reflect seasonal data, Kruithof's method may give poor forecasts.
4.4 Extension of Kruithof's method
The traditional Kruithof's method is a projection of the
traffic based on the last known traffic matrix and forecasts of the
row and column sums.
It is possible to extend Kruithof's method by taking into
account not only forecasts of the row and column but also forecasts
of pointtopoint traffic. Kruithof's method is then used to adjust
the pointtopoint traffic forecasts to obtain consistency with the
forecasts of row and column sums.
The extended Kruithof's method is superior to the traditional
Kruithof's method and is therefore recommended.
4.5 Weighted least squares method
Weighted least squares method is again an extension of the
last method. Let { fICi\dj } { fICi } and { fIC.j } be forecasts
of pointtopoint traffic, row sums and column sums respectively.
The extended Kruithof's method assumes that the row and column
sums are "true" and adjust { fICi\dj } to obtain consistency.
The weighted least squares method [2] is based on the assump
tion that both the pointtopoint forecasts and the row and column
sum forecasts are uncertain. A reasonable way to solve the problem
is to give the various forecasts different weights.
Let the weighted least squares forecasts be { fIDi\dj } The
square sum Q is defined by:
Q =
ij
~ aij(Cij Dij)2+
i
~
bi(Ci.  Di.)2+
j
~
cj(C.j D.j)2
(41)
where { fIai\dj } { fIbi } { fIcj } are chosen constants or
weights.
The weighted least squares forecast is found by:
DfIij
inQ BOCAD15(DfIij)
subject to Di. = j
~ Dij i = 1, 2, .   (42)
and
D.j= i
~ Dij j = 1, 2, .  
A natural choice of weights is the inverse of the variance of
the forecasts. One way to find an estimate of the standard devia
tion of the forecasts is to perform expost forecasting and then
calculate the root mean square error.
The properties of this method are analyzed in [14].
5 Top down and bottom up methods
5.1 Choice of model
The object is to produce forecasts for the traffic between
countries. For this to be a sensible procedure, it is necessary
that the traffic between the countries should not be too small, so
that the forecasts may be accurate. A method of this type is usu
ally denoted as "bottom up".
Alternatively, when there is a small amount of traffic between
the countries in question, it is better to start out with forecast
ing the traffic for a larger group of countries. These forecasts
are often used as a basis for forecasts for the traffic to each
country. This is done by a correction procedure to be described in
more detail below. Methods of this type are called "top down". The
following comments concern the preference of one method over
another.
Let ~T
2 be the variance of the aggregated forecast, and ~i
2 be
the variance of the local forecast No. i and /i\djbe the covari
ance of the local forecast No. i and j . If the following inequal
ity is true:
then, in general, it is not recommended to use the bottom up
method, but to use the top down method.
In many situations it is possible to use a more advanced fore
casting model on the aggregated level. Also, the data on an aggre
gated level may be more consistent and less influenced by stochas
tic changes compared to data on a lower level. Hence, in most cases
the inequality stated above will be satisfied for small countries.
5.2 Bottom up method
As outlined in S 5.1 the bottom up method is defined as a pro
cedure for making separate forecasts of the traffic between dif
ferent countries directly. If the inequality given in S 5.1 is not
satisfied, which may be the case for large countries, it is suffi
cient to use the bottom up method. Hence, one of the forecasting
models mentioned in Recommendation E.507 can be used to produce
traffic forecasts for different countries.
5.3 Top down procedure
In most cases the top down procedure is recommended for pro
ducing forecasts of international traffic for a small country. In
Annex D a detailed example of such a forecasting procedure is
given.
The first step in the procedure is to find a forecasting model
on the aggregated level, which may be a rather sophisticated model.
Let XTbe the traffic forecasts on the aggregated level and ~Tthe
estimated standard deviation of the forecasts.
The next step is to develop separate forecasting models of
traffic to different countries. Let Xibe the traffic forecast to
the i th country and ~ithe standard deviation. Now, the separate
forecasts [Xi] have to be corrected by taking into account the
aggregated forecasts XT. We know that in general
Let the corrections of [Xi] be [X`i], and the corrected aggre
gated forecast then be X`T= ~" X`i.
The procedure for finding [X`i] is described in Annex C.
6 Forecasting methods when observations are missing
6.1 Introduction
Most forecasting models are based on equally spaced time
series. If one observation or a set of observations are missing, it
is necessary either to use an estimate of missing observations and
then use the forecasting model or to modify the forecasting model.
All smoothing models are applied on equally spaced observa
tions. Also autoregressive integrated moving average (ARIMA)models
operate on equally spaced time series, while regression models work
on irregularly spaced observations without modifications.
In the literature it is shown that most forecasting methods
can be formulated as dynamic linear models (DLM). The Kalman Filter
is a linear method to estimate states in a time series which is
modelled as a dynamic linear model. The Kalman Filter introduces a
recursive procedure to calculate the forecasts in a DLM which is
optimal in the sense of minimizing the mean squared one step ahead
forecast error. The Kalman Filter also gives an optimal solution in
the case of missing data.
6.2 Adjustment procedure based on comparable observations
In situations when some observations are missing, it may be
possible to use related data for estimating the missing observa
tions. For instance, if measurements are carried out on a set of
trunk groups in the same area, then the traffic measurements on
various trunk groups are correlated, which means that traffic meas
urements on a given trunk group to a certain degree explain traffic
measurements on other trunk groups.
When there is high correlation between two time series of
traffic measurements, the relative change in level and trend will
be of the same size.
Suppose that a time series xtof equidistant observations from
1 to n  as an inside gap  (mu  fIxtis, for instance, the yearly
increase. The gap consists of k missing observations between r
and r + k + 1.
A procedure for estimating the missing observations is given
by the following steps:
i) Examine similar time series to the series with
missing observations and calculate the cross correlation.
ii) Identify time series with high cross correla
tion at lag zero.
iii) Calculate the growth factor ?63r\d+ibetween r
and r + k of the similar time series yt:
iv) Estimates of the missing observations are then
given by:
xr\d+i= xr+ ?63r\d+i(x
r +k +1
 xr)
i = 1, 2, .   k
(62)
Example
Suppose we want to forecast the time series xt. The series is
observed from 1 to 10, but the observations at time 6, 7 and 8 are
missing. However a related time series ytis measured. The measure
ments are given in Table 1/506.
H.T. [T1.506]
TABLE 1/E.506
Measurements of two related time series; one with missing
observations
________________________________________________________________
t 1 2 3 4 5 6 7 8 9 10
________________________________________________________________
x 100 112 125 140 152    206 221
y 300 338 380 422 460 496 532 574 622 670
________________________________________________________________




























































Table 1/E.506 [T1.506], p.
The last known observation of xtbefore the gap at time 5 is
152, while the first known observation after the gap at time 9 is
206.
Hence r = 5 and k = 3. The calculation gives:
6.3 Modification of forecasting models
The other possibility for handling missing observations is to
extend the forecasting models with specific procedures. When obser
vations are missing, a modified procedure, instead of the ordinary
forecasting model, is used to estimate the traffic.
To illustrate such a procedure we look at simple exponential
smoothing. The simple exponential smoothing model is expressed by:
mt= (1  a ) yt+ a
mt\d\u(em1
(63)
where
ytis the measured traffic at time t
mtis the estimated level at time t
a  s the discount factor [and (1  a ) is the smoothing
parameter].
Equation (63) is a recursive formula. The recursion starts at
time 1 and ends at n  if no observation is missing. Then a one
step ahead forecast is given by:
yt(1) = mt
(64)
If some observations lying in between 1 and n  are missing,
then it is necessary to modify the recursion procedure. Suppose now
that y1, y2, .   , yr, y r +k +1 , y r +k +2 , .   , ynare
known and yr\d+\d1, yr\d+\d2, yr\d+kare unknown. Then the time
series has a gap consisting of k missing observations.
The following modified forecasting model for simple exponen
tial smoothing is proposed in Aldrin [2].
where
ak=
__________
(66)
By using the (65) and (66) it is possible to skip the recur
sive procedure in the gap between r and r + k + 1.
In Aldrin [2] similar procedures are proposed for the follow
ing forecasting models:
 Holt's method,
 Double exponential smoothing,
 Discounted least squares method with level and
trend,
 HoltWinters seasonal methods.
Wright [17] and [18] also suggests specific procedures to
modify the smoothing models when observations are missing.
As mentioned in the first paragraph, regression models are
invariant of missing observations. When using the least squares
method, all observations are given the same weight. Hence, missing
observations do not affect the estimation procedure and forecast
are made in the usual way.
On the other hand it is necessary to modify ARIMA models when
observations are missing. In the literature several procedures are
suggested in the presence of missing data. The basic idea is to
formulate the ARIMA model as a dynamic linear model. Then the
likelihood function is easy to obtain and the parameters in the
model can be estimated recursively. References to work on this
field are Jones [9] and [10], Harvey and Pierse [8], Ansley and
Kohn [3] and Aldrin [2].
State space models or dynamic linear models and the Kalman
Filter are a large class of models. Smoothing models, ARIMA models
and regression models may be formulated as dynamic linear models.
This is shown, for instance, in Abraham and Ledolter [1]. Using
dynamic linear models and the Kalman Filter the parameters in the
model are estimated in a recursive way. The description is given,
for instance, in Harrison and Stevens [7], Pack and Whitaker [13],
Moreland [12], Szelag [15] and Chemouil and Garnier [6].
In Jones [9] and [10], Barham and Dunstan [4], Harvey and
Pierse [8], Aldrin [2] and B/lviken [5] it is shown how the
dynamic linear models and the Kalman Filter handle missing observa
tions.
ANNEX A
(to Recommendation E.506)
Composite strategy
A.1 Introduction
This annex describes a method for estimating international
traffic based on monthly paidminutes and a number of conversion
factors. It demonstrates the method by examining the factors and
showing their utility.
The method is seen to have two main features:
1) Monthly paidminutes exchanged continuously
between Administrations for accounting purposes provide a large and
continuous volume of data.
2) Traffic conversion factors are relatively
stable, when compared with traffic growth and change slowly since
they are governed by customers' habits and network performance. By
separately considering the paid minutes and the traffic conversion
factors, we gain an insight into the nature of traffic growth which
cannot be obtained by circuit occupancy measurements alone.
Because of the stability of the conversion factors, these may be
measured using relatively small samples, thus contributing to the
economy of the procedure.
A.2 Basic procedure
A.2.1 General
The composite strategy is carried out for each stream, for
each direction and generally for each service category.
The estimated mean offered busyhour traffic (in Erlangs) is
derived from the monthly paidminutes using the formula:
A = Mdh /60e
(A1)
where
A is the estimated mean traffic in Erlangs offered in
the busy hour,
M is the total monthly paidminutes,
d is the day/month ratio, i.e. the ratio of average week
day paidtime to monthly paidtime,
h is the busyhour/day ratio, i.e. the ratio of the
busyhour paidtime to the average daily paidtime,
e is the efficiency factor, i.e. the ratio of
busyhour paidtime to busyhour occupiedtime.
A.2.2 Monthly paidminutes (M)
The starting point for the composite strategy is paid minutes.
Sudden changes in subscriber demand, for example, resulting from
improvements in transmission quality, have a time constant of the
order of several months, and on this basis paid minutes accumulated
over monthly intervals appear to be optimum in terms of monitoring
traffic growth. A longer period (e.g. annually) tends to mask sig
nificant changes, whereas a shorter period (e.g. daily) not only
increases the amount of data, but also increases the magnitude of
fluctuations from one period to the next. A further advantage
of the onemonth period is that monthly paidminute figures are
exchanged between Administrations for accounting purposes and con
sequently historical records covering many years are normally
readily available.
It should be recognized, however, that accounting information
exchanges between Administrations often take place after the event,
and it may take some time to reach full adjustments (e.g. collect
call traffic).
A.2.3 Day/month ratio (d)
This ratio is related to the amount of traffic carried on a
typical weekday compared with the total amount of traffic carried
in a month.
As the number of weekdays and nonweekdays (weekends and holi
days) varies month by month, it is not convenient to refer to a
typical month, but it should be possible to compute the ratio for
the month for which the busy hour traffic is relevant.
_________________________
In a situation where only yearly paidminutes are
available, this may be converted to M by a suitable
factor.
Hence if:
X denotes the number of weekdays in the related month
Y denotes the number of nonweekdays (weekend days
and holidays) in the selected month, then
[ Formula deleted ]
(A2)
where
r =
verage weekday traffic
__________________________
The relative amount of nonweekday traffic is very sensitive
to the relative amount of social contact between origin and desti
nation. (Social calls, are, in general, made more frequently on
weekends.) Since changes in such social contact would be very slow,
r or d are expected to be the most stable conversion factors, which
in general vary only within relatively narrow limits. However, tar
iff policies such as reduced weekend rates can have a significant
effect on r and d .
When r is in the region of 1, the Sunday traffic may exceed
the typical weekday level. If this is the case, consideration
should be given to dimensioning the route to cater for the addi
tional weekend (Sunday) traffic or adopting a suitable overflow
routing arrangement.
A.2.4 Busyhour/day ratio (h)
The relative amount of average weekday traffic in the busy
hour primarily depends on the difference between the local time at
origin and destination. Moderately successful attempts have been
made to predict the diurnal distribution of traffic based on this
information together with supposed "degree of convenience" at ori
gin and destination. However, sufficient discrepancies exist to
warrant measuring the diurnal distribution, from which the
busyhour/day ratio may be calculated.
Where measurement data is not available, a good starting point
is Recommendation E.523. From the theoretical distributions found
in Recommendation E.523, one finds variations in the busyhour/day
ratio from 10% for 0 to 2 hours time difference and up to 13.5% for
7 hours time difference.
As described above, the composite strategy is implemented as
an accountingbased procedure. However, it may be more practical
for some Administrations to measure d and h based on occupied time,
derived from available call recording equipment.
A.2.5 Efficiency factor (e)
The efficiency factor (ratio of busyhour paid time to
busyhour occupied time, e ) converts the paid time into a measure
of total circuit occupancy. It is therefore necessary to include
all occupied circuit time in the measurement of this ratio, and not
merely circuit time taken up in establishing paid calls. For exam
ple, the measurement of total circuit occupied time should include
the occupied time for paid calls (time from circuit seizure to cir
cuit clearance) and, in addition, the occupied time for directory
inquiry calls, test calls, service calls, ineffective attempts and
other classes of unpaid traffic handled during the busy hour.
There is a tendency for the efficiency to change with time. In
this regard, efficiency is mainly a function of operating method
(manual, semiautomatic, international subscriber dialling), the
Bsubscriber's availability, and the quality of the distant net
work.
Forecasts of the efficiency can be made on the basis of extra
polation of past trends together with adjustments for planned
improvements.
The detailed consideration of efficiency is also an advantage
from an operational viewpoint in that it may be possible to iden
tify improvements that may be made, and quantify the benefits
deriving from such improvements.
It should be noted that the practical limit for e is generally
about 0.8 to 0.9 for automatic working.
A.2.6 Mean offered busy hour traffic (A)
It should be noted that A  s the mean offered busyhour
traffic expressed in Erlangs.
A.2.7 Use of composite strategy
In the case of countries with lower traffic volumes and manual
operation, the paidtime factors (d and h ) would be available
from analysis of call vouchers (dockets). For derivation of the
efficiency e , the manual operator would have to log the busyhour
occupied time as well as the paid time during the sampling period.
In countries using storedprogram controlled exchanges with
associated manual assistance positions, computer analysis may aid
the composite forecasting procedure.
One consequence of the procedure is that the factors d  nd h
 ive a picture of subscriber behaviour, in that unpaid time
(inquiry calls, test calls, service calls, etc.) are not included
in the measurement of these factors. The importance of deriving the
efficiency, e , during the busy hour, should also be emphasized.
ANNEX B
(to Recommendation E.506)
Example using weighted least squares method
B.1 Telex data
The telex traffic between the following countries has been
analyzed:
 Germany (D)
 Denmark (DNK)
 USA (USA)
 Finland (FIN)
 Norway (NOR)
 Sweden (S)
The data consists of yearly observations from 1973 to
1984 [19].
B.2 Forecasting
Before using the weighted least squares method, separate fore
casts for the traffic matrix have to be made. In this example a
simple ARIMA (0,2,1) model with logarithmic transformed observa
tions without explanatory variables is used for forecasting. It may
be possible to develop better forecasting models for the telex
traffic between the various countries. However the main point in
this example only is to illustrate the use of the weighted least
squares technique.
Forecasts for 1984 based on observations from 1973 to 1983 are
given in Table B1/E.506.
H.T. [T2.506]
TABLE B1/E.506
Forecasts for telex traffic between Germany
(D),
Denmark
(DNK),
USA
(USA), Finland
(FIN), Norway
(NOR) and
Sweden
(S) in 1984
______________________________________________________________________________________________
From To D DNK USA FIN NOR S Sum Forecasted sum
______________________________________________________________________________________________
D  4869 12  30 2879 2397 5230 28  05 27  88
DNK 5196  1655 751 1270 1959 10  31 10  05
USA 11  03 1313  719 1657 2401 17  93 17  09
FIN 2655 715 741  489 1896 6496 6458
NOR 2415 1255 1821 541  1548 7580 7597
S 4828 1821 2283 1798 1333  12  63 12  53
______________________________________________________________________________________________
Sum 26  97 9973 19  30 6688 7146 13  34
______________________________________________________________________________________________
Forecasted sum 26  97 9967 19  53 6659 7110 12  14
______________________________________________________________________________________________


































































































































Tableau B1/E.506 [T2.506], p.
It should be noticed that there is no consistency between row
and column sum forecasts and forecasts of the elements in the
traffic matrix. For instance, the sum of forecasted outgoing telex
traffic from Germany is 28  05, while the forecasted row sum is 27
 88.
To adjust the forecasts to get consistency and to utilize both
row/column forecasts and forecasts of the traffic elements the
weighted least squares method is used.
B.3 Adjustment of the traffic matrix forecasts
To be able to use the weighted least squares method, the
weights and the separate forecasts are needed as input. The
separate forecasts are found in Table B2/E.506, while the weights
are based on the mean squared one step ahead forecasting errors.
Let yt  be the traffic at time t . The ARIMA (0,2,1) model
with logarithmic transformed data is given by:
zt= (1  B)2  n yt= (1  B)
at
or
zt= at at\d\u(em1
where
zt= ln yt 2 ln yt\d\u(em1+ ln yt\d\u(em2
at is white noise,
 is a parameter,
B is the backwards shift operator.
The mean squared one step ahead forecasting error of zt  is:
MSQ =
[Formula Deleted]
where
/*^zt\d\u(em1(1) is the one step ahead forecast.
The results of using the weighted least squares method is
found in Table B3/E.506 and show that the factors in
Table B1/E.506 have been adjusted. In this example only minor
changes have been performed because of the high conformity in the
forecasts of row/column sums and traffic elements.
H.T. [T3.506]
TABLE B2/E.506
Inverse weights as mean as squared one step ahead forecasting
errors
of telex traffic (100^^4) between
Germany
(D), Denmark
(DNK),
USA
(USA), Finland
(FIN),
Norway
(NOR) and Sweden
(S) in 1984
___________________________________________________________________
From To D DNK USA FIN NOR S Sum
___________________________________________________________________
D  28.72 13.18 11.40 8.29 44.61 7.77
DNK 5.91  43.14 18.28 39.99 18.40 10.61
USA 23.76 39.19  42.07 50.72 51.55 21.27
FIN 23.05 12.15 99.08  34.41 19.96 17.46
NOR 21.47 40.16 132.57 24.64  17.15 20.56
S 6.38 12.95 28.60 28.08 8.76  6.48
___________________________________________________________________
Sum 6.15 3.85 14.27 9.55 12.94 8.53
___________________________________________________________________



































































































Tableau B2/E.506 [T3.506], p.4
H.T. [T4.506]
TABLE B3/E.506
Adjusted telex forecasts using the weighted least
squares method
_______________________________________________________________________
From To D DNK USA FIN NOR S Sum
_______________________________________________________________________
D  4850 12  84 2858 2383 5090 27  65
DNK 5185  1674 750 1257 1959 10  25
USA 11  01 1321  717 1644 2407 17  90
FIN 2633 715 745  487 1891 6471
NOR 2402 1258 1870 540  1547 7617
S 4823 1817 2307 1788 1331  12  66
_______________________________________________________________________
Sum 26  44 9961 19  80 6653 7102 12  94
_______________________________________________________________________



































































































Tableau B3/E.506 [T4.506], p.5
ANNEX C
(to Recommendation E.506)
Description of a top down procedure
Let
XT be the traffic forecast on an aggregated level,
Xi be the traffic forecast to country i ,
sT the estimated standard deviation of the aggregated
forecast,
si the estimated standard deviation of the forecast to
country i .
Usually
X
T /
i
~ Xi,
(C1)
so that it is necessary to find a correction
[X `
i] of [Xi]
and [X `
T] of [XT]
by minimizing the expression
subject to
where ( and [(i] are chosen to be
The solution of the optimization problem gives the values [X `
i]:
A closer inspection of the data base may result in other
expressions for the coefficients [(i], i = 0, 1, .   On some
occasions, it will also be reasonable to use other criteria for
finding the corrected forecasting values [X ` i]. This is shown in
the top down example in Annex D.
If, on the other hand, the variance of the top forecast XTis
fairly small, the following procedure may be chosen:
The corrections [Xi] are found by minimizing the expression
subject to
If (i, i = 1, 2, .   is chosen to be the inverse of the
estimated variances, the solution of the optimization problem is
given by
ANNEX D
(to Recommendation E.506)
Example of a top down modelling method
The model for forecasting telephone traffic from Norway to the
European countries is divided into two separate parts. The first
step is an econometric model for the total traffic from Norway to
Europe. Thereafter, we apply a model for the breakdown of the total
traffic on each country.
D.1 Econometric model of the total traffic from Norway to
Europe
With an econometric model we try to explain the development in
telephone traffic, measured in charged minutes, as a function of
the main explanatory variables. Because of the lack of data for
some variables, such as tourism, these variables have had to be
omitted in the model.
The general model may be written:
X
t = e
K x GNP $$Ei:a :t _ x P $$Ei:b
:t _
x A $$Ei:c :t _ x e
u
t
(t = 1, 2, .   ,
N )
(D1)
where:
Xt is the demand for telephone traffic from Norway to
Europe at time t (charged minutes).
GNPt is the gross national product in Norway at time t
(real prices).
Pt is the index of charges for traffic from Norway to
Europe at time t (real prices).
At is the percentage directdialled telephone traffic from
Norway to Europe (to take account of the effect of automation). For
statistical reasons (i.e. impossibility of taking logarithm of
zero) Atgoes from 1 to 2 instead of from 0 to 1.
K is the constant.
a is the elasticity with respect to GNP .
b is the price elasticity.
c is the elasticity with respect to automation.
ut is the stochastic variable, summarizing the impact
of those variables that are not explicitly introduced in the model
and whose effects tend to compensate each other (expectation of
ut = 0 and var ut = ~2).
By applying regression analysis (OLSQ) we have arrived at the
coefficients (elasticities) in the forecasting model for telephone
traffic from Norway to Europe given in Table D1/E.506 (in our cal
culations we have used data for the period 19511980).
The t  tatistics should be compared with the Student's Dis
tribution with N  (em  fId degrees of freedom, where N is the
number of observations and d is the number of estimated parameters.
In this example, N = 30 and d = 4.
The model "explains" 99.7% of the variation in the demand for
telephone traffic from Norway to Europe in the period 19511980.
From this logarithmic model it can be seen that:
 an increase in GNP of 1% causes an increase in
the telephone traffic of 2.80%,
 an increase of 1% in the charges, measured in
real prices, causes a decrease in the telephone traffic of 0.26%,
and
 an increase of 1% in Atcauses an increase in the
traffic of 0.29%.
We now use the expected future development in charges to
Europe, in GNP, and in the future automation of traffic to Europe
to forecast the development in telephone traffic from Norway to
Europe from the equation:
X
t = e
t
16.095
x GNP
t
2.80
x
P
t u
0.26
x A
t
0.29
(D2)
H.T. [T5.506]
TABLE D1/E.506
____________________________________________________
Coefficients Estimated values t  statistics
____________________________________________________
K 16.095 4.2
a  2.799  8.2
b  0.264 1.0
c  0.290  2.1
____________________________________________________




























Table D1/E.506 [T5.506], p.
D.2 Model for breakdown of the total traffic from Norway to
Europe
The method of breakdown is first to apply the trend to fore
cast the traffic to each country. However, we let the trend become
less important the further into the period of forecast we are, i.e.
we let the trend for each country converge to the increase in the
total traffic to Europe. Secondly, the traffic to each country is
adjusted up or down, by a percentage that is equal to all coun
tries, so that the sum of the traffic to each country equals the
forecasted total traffic to Europe from equation (D2).
Mathematically, the breakdown model can be expressed as fol
lows:
Calculation of the trend for country i:
R
it
= b
i + a
i x t , i = 1, .   ,
34 t = 1, .   , N
(D3)
where
R it = fIX tfR
_________ , i.e country i 's share of the total
traffic to Europe.
X it is the traffic to country i at time t
Xt is the traffic to Europe at time t
t is the trend variable
aiand biare two coefficients specific to country i ; i.e. aiis
country i 's trend. The coefficients are estimated by using regres
sion analysis, and we have based calculations on observed traffic
for the period 19661980.
The forecasted shares  or country i  s then calculated by
R it = R iN + a
i
x (t  N ) x
e

0
______
(D4)
where N  s the last year of observation, and e is the exponential
function.
The factor e  0
______ is a correcting factor which ensures
that the growth in the telephone traffic v'5p' to each country
will converge towards the growth of total traffic to Europe after
the adjustment made in Equation (D6).
To have the sum of the countries' shares equal one, it is
necessary that
i
~ R
it
= 1
(D5)
This we obtain by setting the adjusted share, R it , equal to
Each country's forecast traffic is then calculated by multi
plying the total traffic to Europe, Xt, by each country's share of
the total traffic:
X
it
= R
it
x X
t
(D7)
D.3 Econometric model for telephone traffic from Norway to
Central and South America, Africa, Asia, and Oceania .
For telephone traffic from Norway to these continents we have
used the same explanatory variables and estimated coefficients.
Instead of gross national product, our analysis has shown that for
the traffic to these continents the number of telephone stations
within each continent are a better and more significant explanatory
variable.
After using crosssection/timeseries simultaneous estimation
we have arrived at the coefficients in Table D2/E.506 for the
forecasting model for telephone traffic from Norway to these con
tinents (for each continent we have based our calculations on data
for the period 19611980):
H.T. [T6.506]
TABLE D2/E.506
__________________________________________________________
Coefficients Estimated values t  statistics
__________________________________________________________
Charges 1.930 5.5
Telephone stations  2.009  4.2
Automation  0.5  
__________________________________________________________
























Table D2/E.506 [T6.506], p.
We then have R 2 = 0.96. The model may be written:
X
k
t = e
K x
(TS
k
t )
2.009
x
(P
k
t )
1.930
x
(A
k
t )
0.5
(D8)
where
X k
t is the telephone traffic to continent k (k =
Central America, .   , Oceania) at time t ,
e K is the constant specific to each continent. For
telephone traffic from Norway to:
Central America: K 1 = 11.025
South America: K 2 = 12.62
Africa: K 3 = 11.395
Asia: K 4 = 15.02
Oceania: K 5 = 13.194
TS k
t is the number of telephone stations within
continent k at time t ,
P k
t is the index of charges, measured in real prices, to
continent k at time t , and
A k
t is the percentage directdialled telephone
traffic to continent k .
Equation (D8) is now used  together with the expected future
development in charges to each continent, future development in
telephone stations on each continent and future development in
automation of telephone traffic from Norway to the continent  to
forecast the future development in telephone traffic from Norway to
the continent.
References
[1] ABRAHAM (A.) and LEDOLTER (J.): Statistical methods for
forecasting. J. Wiley , New York, 1983.
[2] ALDRIN (M.): Forecasting time series with missing
observations. Stat 15/86 Norwegian Computing Center , 1986.
[3] ANSLEY (C.  .) and KOHN (R.): Estimation, filtering
and smoothing in state space models with incomplete specified ini
tial conditions. The Annals of Statistics , 13, pp. 12861316,
1985.
[4] BARHAM (S.  .) and DUNSTAN (F.  .  .): Missing
values in time series. Time Series Analysis: Theory and Practice 2
: Anderson, O.  ., ed., pp. 2541, North Holland, Amsterdam, 1982.
[5] B/LVIKEN (E.): Forecasting telephone traffic using Kal
man Filtering: Theoretical considerations. Stat 5/86 Norwegian Com
puting Center , 1986.
[6] CHEMOUIL (P.) and GARNIER (B.): An adaptive shortterm
traffic forecasting procedure using Kalman Filtering.
XI International Teletraffic Congress , Kyoto, 1985.
[7] HARRISON (P.  .) and STEVENS (C.  .): Bayesian fore
casting. Journal of Royal Statistical Society . Ser B 37,
pp. 205228, 1976.
[8] HARVEY (A.  .) and PIERSE (R.  .): Estimating missing
observations in econometric time series. Journal of American Sta
tistical As. , 79, pp. 125131, 1984.
[9] JONES (R.  .): Maximum likelihood fitting of ARMA
models to time series with missing observations. Technometrics ,
22, No. 3, pp. 389396, 1980.
[10] JONES (R.  .): Time series with unequally spaced
data. Handbook of Statistics 5. ed. Hannah, E.  ., et al.,
pp. 157177, North Holland, Amsterdam, 1985.
[11] KRUITHOF (J.): Telefoonverkeersrekening. De Ingenieur
, 52, No. 8, 1937.
[12] MORELAND (J.  .): A robust sequential projection
algorithm for traffic load forecasting. The Bell Technical Journal
, 61, pp. 1538, 1982.
[13] PACK (C.  .) and WHITAKER (B.  .): Kalman Filter
models for network forecasting. The Bell Technical Journal , 61,
pp. 114, 1982.
[14] STORDAHL (K.) and HOLDEN (L.): Traffic forecasting
models based on top down and bottom up models. ITC 11 , Kyoto,
1985.
[15] SZELAG (C.  .): A shortterm forecasting algorithm
for trunk demand servicing. The Bell Technical Journal , 61,
pp. 6796, 1982.
[16] TU (M.) and PACK (D.): Improved forecasts for local
telecommunications network. 6th International Forecasting Sympo
sium, Paris, 1986.
[17] WRIGHT (D.  .): Forecasting irregularly spaced data:
An extension of double exponential smoothing. Computer and
Engineering , 10, pp. 135147, 1986.
[18] WRIGHT (D.  .): Forecasting data published at irreg
ular time intervals using an extension of Holt's method. Management
science , 32, pp. 499510, 1986.
[19] Table of international telex relations and traffic ,
ITU, Geneva, 19731984.
Recommendation E.507
MODELS FOR FORECASTING INTERNATIONAL TRAFFIC
1 Introduction
Econometric and time series model development and forecasting
requires familiarity with methods and techniques to deal with a
range of different situations. Thus, the purpose of this Recommen
dation is to present some of the basic ideas and leave the explana
tion of the details to the publications cited in the reference
list. As such, this Recommendation is not intended to be a complete
guide to econometric and time series modelling and forecasting.
The Recommendation also gives guidelines for building various
forecasting models: identification of the model, inclusion of
explanatory variables, adjustment for irregularities, estimation of
parameters, diagnostic checks, etc.
In addition the Recommendation describes various methods for
evaluation of forecasting models and choice of model.
2 Building the forecasting model
This procedure can conveniently be described as four consecu
tive steps. The first step consists in finding a useful class of
models to describe the actual situation. Examples of such classes
are simple models, smoothing models, autoregressive models, autore
gressive integrated moving average (ARIMA) models or econometric
models. Before choosing the class of models, the influence of
external variables should be analyzed. If special external vari
ables have significant impact on the traffic demand, one ought to
_________________________
The old Recommendation E.506 which appeared in the Red
Book was split into two Recommendations, revised E.506
and new E.507, and considerable new material was added
to both.
include them in the forecasting models, provided enough historical
data are available.
The next step is to identify one tentative model in the class
of models which have been chosen. If the class is too extensive to
be conveniently fitted directly to data, rough methods for identi
fying subclasses can be used. Such methods of model identification
employ data and knowledge of the system to suggest an appropriate
parsimonious subclass of models. The identification procedure may
also, in some occasions, be used to yield rough preliminary esti
mates of the parameters in the model. Then the tentative model is
fitted to data by estimating the parameters. Usually, maximum
likelihood estimators or least square estimators are used.
The next step is to check the model. This procedure is often
called diagnostic checking. The object is to find out how well the
model fits the data and, in case the discrepancy is judged to be
too severe, to indicate possible remedies. The outcome of this step
may thus be acceptance of the model if the fit is acceptable. If on
the other hand it is inadequate, it is an indication that new ten
tative models may in turn be estimated and subjected to diagnostic
checking.
In Figure 1/E.507 the steps in the model building procedure
are illustrated.
Figure 1/E.507, p.
3 Various forecasting models
The objective of S 3 is to give a brief overview of the most
important forecasting models. In the GAS 10 Manual on planning data
and forecasting methods [5], a more detailed description of the
models is given.
3.1 Curve fitting models
In curve fitting models the traffic trend is extrapolated by
calculating the values of the parameters of some function that is
expected to characterize the growth of international traffic over
time. The numerical calculations of some curve fitting models can
be performed by using the least squares method.
The following are examples of common curve fitting models used
for forecasting international traffic:
Linear: Yt = a + bt (31)
Parabolic: Yt = a + bt + ct 2 (32)
Exponential: Yt = ae t (33)
Logistic: Yt = [Formula Deleted]
Gompertz: Yt = M (a ) t (35)
where
Ytis the traffic at time t ,
a, b, c  are parameters,
M  is a parameter describing the saturation level.
The various trend curves are shown in Figures 2/E.507 and
3/E.507.
The logistic and Gompertz curves differ from the linear, para
bolic and exponential curves by having saturation or ceiling level.
For further study see [10].
FIGURE 2/E.507, p.9
FIGURE 3/E.507 DIMINUER LA FIGURE, p.10
3.2 Smoothing models
By using a smooth process in curve fitting, it is possible to
calculate the parameters of the models to fit current data very
well but not necessarily the data obtained from the distant past.
The best known smoothing process is that of the moving aver
age. The degree of smoothing is controlled by the number of most
recent observations included in the average. All observations
included in the average have the same weight.
In addition to moving average models, there exists another
group of smoothing models based on weighting the observations. The
most common models are:
 simple exponential smoothing,
 double exponential smoothing,
 discounted regression,
 Holt's method, and
 HoltWinters' seasonal models.
For example, in the method of exponential smoothing the weight
given to previous observations decreases geometrically with age
according to the following equation:
ut = (1  a )Yt+ a
*^m
t 1
(36)
where:
Ytis the measured traffic at time t ,
utis the estimated level at time t , and
a  is the discount factor [and (1  a ) is the smoothing
parameter].
The impact of past observations on the forecasts is controlled
by the magnitude of the discount factor.
Use of smoothing models is especially appropriate for
shortterm forecasts. For further studies see [1], [5] and [9].
3.3 Autoregressive models
If the traffic demand, Xt, at time t can be expressed as a
linear combination of earlier equidistant observations of the past
traffic demand, the process is an autoregressive process. Then the
model is defined by the expression:
X
t = ?71
1X
t 1
+ ?71
2X
t 2
+
pX
t p
+ a
t
(37)
where
at is white noise at time t ;
?71k, k = 1, .   p are the autoregressive parameters.
The model is denoted by AR (p ) since the order of the model is p .
By use of regression analysis the estimates of the parameters
can be found. Because of common trends the exogenous variables (X t
1 , X t 2 , .   X t p ) are usually strongly correlated.
Hence the parameter estimates will be correlated. Furthermore, sig
nificance tests of the estimates are somewhat difficult to perform.
Another possibility is to compute the empirical autocorrela
tion coefficients and then use the YuleWalker equations to esti
mate the parameters [?71k]. This procedure can be performed when
the time series [Xt] are stationary. If, on the other hand, the
time series are non stationary, the series can often be transformed
to stationarity e.g., by differencing the series. The estimation
procedure is given in Annex A, S A.1.
3.4 Autoregressive integrated moving average (ARIMA) models
An extention of the class of autoregressive models which
include the moving average models is called autoregressive moving
average models (ARMA models). A moving average model of order q is
given by:
X
t = a
t  
1a
t 1
 
2a
t 2
qa
t q
(38)
where
at is white noise at time t ;
[k] are the moving average parameters
Assuming that the white noise term in the autoregressive
models in S 3.3 is described by a moving average model, one
obtains the socalled ARMA (p , q ) model:
X
t = ?71
1X
t 1
+ ?71
2X
t 2
+
pX
t p
+
a
t  
1a
t 1
 
2a
t 2
.    
qa
t q
(39)
The ARMA model describes a stationary time series. If the time
series is nonstationary, it is necessary to difference the series.
This is done as follow:
Let Ytbe the time series and B the backwards shift operator,
then
Xt= (1  B )t
(310)
where
d is the number of differences to have stationarity.
The new model ARIMA ( p, d, q ) is found by inserting equa
tion (310) into equation (39).
The method for analyzing such time series was developed by G.
 .  . Box and G.  . Jenkins [3]. To analyze and forecast such
time series it is usually necessary to use a time series program
package.
As indicated in Figure 1/E.507 a tentative model is identi
fied. This is carried out by determination of necessary transforma
tions and number of autoregressive and moving average parameters.
The identification is based on the structure of the autocorrela
tions and partial autocorrelations.
The next step as indicated in Figure 1/E.507 is the estimation
procedure. The maximum likelihood estimates are used. Unfor
tunately, it is difficult to find these estimates because of the
necessity to solve a nonlinear system of equations. For practical
purposes, a computer program is necessary for these calculations.
The forecasting model is based on equation (39) and the process of
making forecasts l time units ahead is shown in S A.2.
The forecasting models described so far are univariate fore
casting models. It is also possible to introduce explanatory vari
ables. In this case the system will be described by a transfer
function model. The methods for analyzing the time series in a
transfer function model are rather similar to the methods described
above.
Detailed descriptions of ARIMA models are given in [1], [2],
[3], [5], [11], [15] and [17].
3.5 State space models with Kalman Filtering
State space models are a way to represent discretetime pro
cess by means of difference equations. The state space modelling
approach allows the conversion of any general linear model into a
form suitable for recursive estimation and forecasting. A more
detailed description of ARIMA state space models can be found
in [1].
For a stochastic process such a representation may be of the
following form:
Xt\d+\d1= ?71Xt+ Zt+
wt
(311)
and
Yt= HXt+ vt
(312)
where
Xtis an svector of state variables in period t ,
Ztis an svector of deterministic events,
?71 is an s xs  ransition matrix that may, in general,
depend on t ,
wtis an svector of random modelling errors,
Ytis a dvector of measurements in period t ,
H  s a d xs  atrix called the observation matrix, and
vtis a dvector of measurement errors.
Both wtin equation (311) and vtin equation (312) are addi
tive random sequences with known statistics. The expected value of
each sequence is the zero vector and wtand vtsatisfy the condi
tions:
E 
wfIt(*w $$Ei:T:j_
 = Qt`tj
for all t , j ,
(313)
E 
vfIt(*n $$Ei:T:j_
 = Rt`tjfor all t , j ,
where
Qtand Rtare nonnegative definite matrices,
and
_________________________
A matrix A is nonnegative definite, if and only if, for
all vectors z, z
 (>="  .
`t\djis the Kronecker delta.
Qtis the covariance matrix of the modelling errors and Rtis the
covariance matrix of the measurement errors; the wtand the vtare
assumed to be uncorrelated and are referred to as white noise. In
other words:
E 
vfIt(*w $$Ei:T:j_
 = 0 for all t ,
j ,
(314)
and
E 
vfItfIX $$Ei:T: 0_
 = 0 for all
t .
(315)
Under the assumptions formulated above, determine Xt\d,\dtsuch
that:
E 
(XfIt,t(em XfIt) fIT(XfIt,t(em XfIt)

= minimum,
(316)
where
Xt\d,\dtis an estimate of the state vector at time t , and
Xtis the vector of true state variables.
The Kalman Filtering technique allows the estimation of state
variables recursively for online applications. This is done in the
following manner. Assuming that there is no explanatory variable Z
t, once a new data point becomes available it is used to update the
model:
Xt\d,\dt= X
t,t 1
+ Kt(Yt HX t,t 1
)
(317)
where
Ktis the Kalman Gain matrix that can be computed recur
sively [18].
Intuitively, the gain matrix determines how much relative
weight will be given to the last observed forecast error to correct
it. To create a kstep ahead projection the following formula is
used:
X
t +k,t
= ?71
kXt\d,\dt
(318)
where
X t +k,t is an estimate of X t +k given observations Y1,
Y2,    , Yt.
Equations (317) and (318) show that the Kalman Filtering
technique leads to a convenient forecasting procedure that is
recursive in nature and provides an unbiased, minimum variance
estimate of the discrete time process of interest.
For further studies see [4], [5], [16], [18], [19] and [22].
The Kalman Filtering works well when the data under examina
tion are seasonal. The seasonal traffic load data can be
represented by a periodic time series. In this way, a seasonal Kal
man Filter can be obtained by superimposing a linear growth model
with a seasonal model. For further discussion of seasonal Kalman
Filter techniques see [6] and [20].
3.6 Regression models
The equations (31) and (32) are typical regression models.
In the equations the traffic, Yt, is the dependent (or explanatory)
variable, while time t is the independent variable.
A regression model describes a linear relation between the
dependent and the independent variables. Given certain assumptions
ordinary least squares (OLS) can be used to estimate the parame
ters.
A model with several independent variables is called a multi
ple regression model. The model is given by:
Yt= 0+ 1X1t+
2X2t+    +
k\dt+ ut
(319)
where
Ytis the traffic at time t ,
i, i = 0, 1, .   , k are the parameters,
Xi\dt, ie= 1, 2, .   , k is the value of the indepen
dent variables at time t ,
utis the error term at time t .
Independent or explanatory variables which can be used in the
regression model are, for instance, tariffs, exports, imports,
degree of automation. Other explanatory variables are given in S 2
"Base data for forecasting" in Recommendation E.506.
Detailed descriptions of regression models are given in [1],
[5], [7], [15] and [23].
3.7 Econometric models
Econometric models involve equations which relate a variable
which we wish to forecast (the dependent or endogenous variable) to
a number of socioeconomic variables (called independent or expla
natory variables). The form of the equations should reflect an
expected
casual relationship between the variables. Given an assumed
model form, historical or cross sectional data are used to estimate
coefficients in the equation. Assuming the model remains valid over
time, estimates of future values of the independent variables can
be used to give forecasts of the variables of interest. An example
of a typical econometric model is given in Annex C.
There is a wide spectrum of possible models and a number of
methods of estimating the coefficients (e.g., least squares, vary
ing parameter methods, nonlinear regression, etc.). In many
respects the family of econometric models available is far more
flexible than other models. For example, lagged effects can be
incorporated, observations weighted, ARIMA residual models sub
sumed, information from separate sections pooled and parameters
allowed to vary in econometric models, to mention a few.
One of the major benefits of building an econometric model to
be used in forecasting is that the structure or the process that
generates the data must be properly identified and appropriate
causal paths must be determined. Explicit structure identification
makes the source of errors in the forecast easier to identify in
econometric models than in other types of models.
Changes in structures can be detected through the use of
econometric models and outliers in the historical data are easily
eliminated or their influence properly weighted. Also, changes in
the factors affecting the variables in question can easily be
incorporated in the forecast generated from an econometric model.
Often, fairly reliable econometric models may be constructed
with less observations than that required for time series models.
In the case of pooled regression models, just a few observations
for several crosssections are sufficient to support a model used
for predictions.
However, care must be taken in estimating the model to satisfy
the underlying assumptions of the techniques which are described in
many of the reference works listed at the end of this
Recommendation. For example the number of independent variables
which can be used is limited by the amount of data available to
estimate the model. Also, independent variables which are corre
lated to one another should be avoided. Sometimes correlation
between the variables can be avoided by using differenced or
detrended data or by transformation of the variables. For further
studies see [8], [12], [13], [14] and [21].
4 Discontinuities in traffic growth
4.1 Examples of discontinuities
It may be difficult to assess in advance the magnitude of a
discontinuity. Often the influence of the factors which cause
discontinuties is spread over a transitional period, and the
discontinuity is not so obvious. Furthermore, discontinuities aris
ing, for example, from the introduction of international subscriber
dialling are difficult to identify accurately, because changes in
the method of working are usually associated with other changes
(e.g. tariff reductions).
An illustration of the bearing of discontinuities on traffic
growth can be observed in the graph of Figure 4/E.507.
Discontinuities representing the doubling  and even more  of
traffic flow are known. It may also be noted that changes could
occur in the growth trend after discontinuities.
In shortterm forecasts it may be desirable to use the trend
of the traffic between discontinuities, but for longterm forecasts
it may be desirable to use a trend estimate which is based on
longterm observations, including previous discontinuities.
In addition to random fluctuations due to unpredictable
traffic surges, faults, etc., traffic measurements are also subject
to systematic fluctuations, due to daily or weekly traffic flow
cycles, influence of time differences, etc.
4.2 Introduction of explanatory variables
Identification of explanatory variables for an econometric
model is probably the most difficult aspect of econometric model
building. The explanatory variables used in an econometric model
identify the main sources of influence on the variable one is con
cerned with. A list of explanatory variables is given in
Recommendation E.506, S 2.
Figure 4/E.507, p.
Economic theory is the starting point for variable selection.
More specifically, demand theory provides the basic framework for
building the general model. However, the description of the struc
ture or the process generating the data often dictate what vari
ables enter the set of explanatory variables. For instance, techno
logical relationships may need to be incorporated in the model in
order to appropriately define the structure.
Although there are some criteria used in selecting explanatory
variables [e.g., R  2, DurbinWatson (DW) statistic, root mean
square error (RMSE), expost forecast performance, explained in the
references], statistical problems and/or availability of data
(either historical or forecasted) limit the set of potential expla
natory variables and one often has to revert to proxy variables.
Unlike pure statistical models, econometric models admit explana
tory variables, not on the basis of statistical criteria alone but,
also, on the premise that causality is, indeed, present.
A completely specified econometric model will capture turning
points. Discontinuities in the dependent variable will not be
present unless the parameters of the model change drastically in a
very short time period. Discontinuities in the growth of telephone
traffic are indications that the underlying market or technological
structure have undergone large changes.
Sustained changes in the growth of telephone demand can either
be captured through varying parameter regression or through the
introduction of a variable that appears to explain the discon
tinuity (e.g., the introduction of an advertising variable if
advertising is judged to be the cause of the structural change).
Onceandforall, or stepwise discontinuities, cannot be handled
by the introduction of explanatory changes: dummy variables can
resolve this problem.
4.3 Introduction of dummy variables
In econometric models, qualitative variables are often
relevant; to measure the impact of qualitative variables , dummy
variables are used. The dummy variable technique uses the value 1
for the presence of the qualitative attribute that has an impact on
the dependent variable and 0 for the absence of the given attri
bute.
Thus, dummy variables are appropriate to use in the case where
a discontinuity in the dependent variable has taken place. A dummy
variable, for example, would take the value of zero during the his
torical period when calls were operator handled and one for the
period for which direct dial service is available.
Dummy variables are often used to capture seasonal effects in
the dependent variable or when one needs to eliminate the effect of
an outlier on the parameters of a model, such as a large jump in
telephone demand due to a postal strike or a sharp decline due to
facility outages associated with severe weather conditions.
Indiscriminate use of dummy variables should be discouraged
for two reasons:
1) dummy variables tend to absorb all the explana
tory power during discontinuties, and
2) they result in a reduction in the degrees of
freedom.
5 Assessing model specification
5.1 General
In this section methods for testing the significance of the
parameters and also methods for calculating confidence intervals
are presented for some of the forecasting models given in S 3. In
particular the methods relating to regression analysis and time
series analysis will be discussed.
All econometric forecasting models presented here are
described as regression models. Also the curve fitting models given
in S 3.1 can be described as regression models.
An exponential model given by
Z
t = ae
bt
 (mu  fIut
(51)
may be transformed to a linear form
ln Z
t = ln a + bt + ln ut
(52)
or
Y
t = 
0 + 
1Xt+ at
(53)
where
Yt = ln Zt
0 = ln a
1 = b
Xt = t
at = ln ut(white noise).
5.2 Autocorrelation
A good forecasting model should lead to small autocorrelated
residuals. If the residuals are significantly correlated, the
estimated parameters and also the forecasts may be poor. To check
whether the errors are correlated, the autocorrelation function rk,
k = 1, 2, .   is calculated. rkis the estimated autocorrela
tion of residuals at lag k . A way to detect autocorrelation among
the residuals is to plot the autocorrelation function and to per
form a DurbinWatson test. The DurbinWatson statistic is:
where
et is the estimated residual at time t ,
N is the number of observations.
5.3 Test of significance of the parameters
One way to evaluate the forecasting model is to analyse the
impact of different exogenous variables. After estimating the
parameters in the regression model, the significance of the parame
ters has to be tested.
In the example of an econometric model in Annex C, the
estimated values of the parameters are given. Below these values
the estimated standard deviation is given in parentheses. As a rule
of thumb, the parameters are considered as significant if the abso
lute value of the estimates exceeds twice the estimated standard
deviation. A more accurate way of testing the significance of the
parameters is to take into account the distributions of their esti
mators.
The multiple correlation coefficient (or coefficient of deter
mination ) may be used as a criterion for the fitting of the equa
tion.
The multiple correlation coefficient, R 2, is given by:
If the multiple correlation coefficient is close to 1 the fit
ting is satisfactory. However, a high R 2 does not imply an accu
rate forecast.
In time series analysis, the discussion of the model is car
ried out in another way. As pointed out in S 3.4, the number of
autoregressive and moving average parameters in an ARIMA model is
determined by an identification procedure based on the structure of
the autocorrelation and partial autocorrelation function.
The estimation of the parameters and their standard deviations
is performed by an iterative nonlinear estimation procedure. Hence,
by using a time series analysis computer program, the estimates of
the parameters can be evaluated by studying the estimated standard
deviations in the same way as in regression analysis.
An overall test of the fitting is based on the statistic
where riis the estimated autocorrelation at lag i and d is the
number of parameters in the model. When the model is adequate, Q N
d is approximately chisquare distributed with N  d degrees of
freedom. To test the fitting, the value Q N d can be compared
with fractiles of the chisquare distribution.
5.4 Validity of exogenous variables
Econometric forecasting models are based on a set of exogenous
variables which explain the development of the endogenous variable
(the traffic demand). To make forecasts of the traffic demand, it
is necessary to make forecasts of each of the exogenous variables.
It is very important to point out that an exogenous variable should
not be included in the forecasting model if the prediction of the
variable is less confident than the prediction of the traffic
demand.
Suppose that the exact development of the exogenous variable
is known which, for example, is the case for the simple models
where time is the explanatory variables. If the model fitting is
good and the white noise is normally distributed with expectation
equal to zero, it is possible to calculate confidence limits for
the forecasts. This is easily done by a computer program.
On the other hand, the values of most of the explanatory vari
ables cannot be predicted exactly. The confidence of the prediction
will then decrease with the number of periods. Hence, the explana
tory variables will cause the confidence interval of the forecasts
to increase with the number of the forecast periods. In these
situations it is difficult to calculate a confidence interval
around the forecasted values.
If the traffic demand can be described by an autoregressive
moving average model, no explanatory variables are included in the
model. Hence, if there are no explanatory variable in the model,
the confidence limits of the forecasting values can be calculated.
This is done by a time series analysis program package.
5.5 Confidence intervals
Confidence intervals, in the context of forecasts, refer to
statistical constructs of forecast bounds or limits of prediction.
Because statistical models have errors associated with them, param
eter estimates have some variability associated with their values.
In other words, even if one has identified the correct forecasting
model, the influence of endogenous factors will cause errors in the
parameter estimates and the forecast. Confidence intervals take
into account the uncertainty associated with the parameter esti
mates.
In causal models, another source of uncertainty in the fore
cast of the series under study are the predictions of the explana
tory variables. This type of uncertainty cannot be handled by con
fidence intervals and is usually ignored, even though it may be
more significant than the uncertainty associated with coefficient
estimates. Also, uncertainty due to possible outside shocks is not
reflected in the confidence intervals.
For a linear, static regression model, the confidence interval
of the forecast depends on the reliability of the regression coef
ficients, the size of the residual variance, and the values of the
explanatory variables. The 95% confidence interval for a forecasted
value Y N +1 is given by:
/*^YN(1)  2*^s YN+1
/*^YN(1) +
2*^s
(57)
where /*^YN(1) is the forecast one step ahead and *^s is the stan
dard error of the forecast.
This says that we expect, with a 95% probability, that the
actual value of the series at time N + 1 will fall within the lim
its given by the confidence interval, assuming that there are no
errors associated with the forecast of the explanatory variables.
6 Comparison of alternative forecasting models
6.1 Diagnostic check  Model evaluation
Tests and diagnostic checks are important elements in the
model building procedure. The quality of the model is characterized
by the residuals. Good forecasting models should lead to small
autocorrelated residuals, the variance of the residuals should not
decrease or increase and the expectation of the residuals should be
zero or close to zero. The precision of the forecast is affected by
the size of the residuals which should be small.
In addition the confidence limits of the parameter estimates
and the forecasts should be relatively small. And in the same way,
the mean square error should be small compared with results from
other models.
6.2 Forecasts of levels versus forecasts of changes
Many econometric models are estimated using levels of the
dependent and independent variables. Since economic variables move
together over time, high coefficients of determination are
obtained. The collinearity among the levels of the explanatory
variables does not present a problem when a model is used for fore
casting purposes alone, given that the collinearity pattern in the
past continues to exist in the future. However, when one attempts
to measure structural coefficients (e.g., price and income elasti
cities) the collinearity of the explanatory variables (known as
multicollinearity) renders the results of the estimated coeffi
cients unreliable.
To avoid the multicollinearity problem and generate benchmark
coefficient estimates and forecasts, one may use changes of the
variables (first difference or first log difference which is
equivalent to a percent change) to estimate a model and forecast
from that model. Using changes of variables to estimate a model
tends to remove the effect of multicollinearity and produce more
reliable coefficient estimates by removing the common effect of
economic influences on the explanatory variables.
By generating forecasts through levels of and changes in the
explanatory variables, one may be able to produce a better forecast
through a reconciliation process. That is, the models are adjusted
so that the two sets of forecasts give equivalent results.
6.3 Expost forecasting
Expost forecasting is the generation of a forecast from a
model estimated over a subsample of the data beginning with the
first observation and ending several periods prior to the last
observation. In expost forecasting, actual values of the explana
tory variables are used to generate the forecast. Also, if fore
casted values of the explanatory variables are used to produce an
expost forecast, one can then measure the error associated with
incorrectly forecasted explanatory variables.
The purpose of expost forecasting is to evaluate the fore
casting performance of the model by comparing the forecasted values
with the actuals of the period after the end of the subsample to
the last observation. With expost forecasting, one is able to
assess forecast accuracy in terms of:
1) percent deviations of forecasted values from
actual values,
2) turning point performance,
3) systematic behaviour of deviations.
Deviations of forecasted values from actual values give a gen
eral idea of the accuracy of the model. Systematic drifts in devia
tions may provide information for either respecifying the model or
adjusting the forecast to account for the drift in deviations. Of
equal importance in evaluating forecast accuracy is turning point
performance, that is, how well the model is able to forecast
changes in the movement of the dependent variable. More criteria
for evaluating forecast accuracy are discussed below.
6.4 Forecast performance criteria
A model might fit the historical data very well. However, when
the forecasts are compared with future data that are not used for
estimation of parameters, the fit might not be so good. Hence com
parison of forecasts with actual observations may give additional
information about the quality of the model. Suppose we have the
time series, Y1, Y2,     , YN, YN\d+\d1,     ,
YN\d+M.
The M last observations are removed from the time series and
the model building procedure. The onestepahead forecasting error
is given by:
eN\d+t= YN\d+t
/*^YN\d+t\d\u(em1(1)
t = 1, 2,     M
(61)
where
/*^YN\d+t\d\u(em1(1) is the onestepahead forecast.
Mean error
The mean error, ME, is defined by
ME is a criterium for forecast bias. Since the expectation of
the residuals should be zero, a large deviation from zero indicates
bias in the forecasts.
Mean percent error
The mean percent error, MPE, is defined by
This statistic also indicates possible bias in the forecasts.
The criterium measures percentage deviation in the bias. It is not
recommended to use MPE when the observations are small.
Root mean square error
The root mean square error, RMSE, of the forecast is defined
as
RMSE is the most commonly used measure for forecasting preci
sion.
Mean absolute error
The mean absolute error, MAE, is given by
Theil's inequality coefficient
Theil's inequality coefficient is defined as follows:
Theil's U is preferred as a measure of forecast accuracy
because the error between forecasted and actual values can be bro
ken down to errors due to:
1) central tendency,
2) unequal variation between predicted and realized
changes, and
3) incomplete covariation of predicted and actual
changes.
This decomposition of prediction errors can be used to adjust
the model so that the accuracy of the model can be improved.
Another quality that a forecasting model must possess is abil
ity to capture turning points. That is, a forecast must be able to
change direction in the same time period that the actual series
under study changes direction. If a model is estimated over a long
period of time which contains several turning points, expost fore
cast analysis can generally detect a model's inability to trace
closely actuals that display turning points.
7 Choice of forecasting model
7.1 Forecasting performance
Although the choice of a forecasting model is usually guided
by its forecasting performance, other considerations must receive
attention. Thus, the length of the forecast period, the functional
form, and the forecast accuracy of the explanatory variables of an
econometric model must be considered.
The length of the forecast period affects the decision to use
one type of a model versus another, along with historical data lim
itations and the purpose of the forecasting model. For instance,
ARIMA models may be appropriate forecasting models for shortterm
forecasts when stability is not an issue, when sufficient histori
cal data are available, and when causality is not of interest.
Also, when the structure that generates the data is difficult to
identify, one has no choice but to use a forecasting model which is
based on historical data of the variable of interest.
The functional form of the model must also be considered in a
forecasting model. While it is true that a more complex model may
reduce the model specification error, it is also true that it will,
in general, considerably increase the effect of data errors. The
model form should be chosen to recognize the tradeoff between
these sources of error.
Availability of forecasts for explanatory variables and their
reliability record is another issue affecting the choice of a fore
casting model. A superior model using explanatory variables which
may not be forecasted accurately can be inferior to an average
model whose explanatory variables are forecasted accurately.
When market stability is an issue, econometric models which
can handle structural changes should be used to forecast. When
causality matters, simple models or ARIMA models cannot be used as
forecasting tools. Nor can they be used when insufficient histori
cal data exist. Finally, when the purpose of the model is to fore
cast the effects associated with changes in the factors that influ
ence the variable in question, time series models may not be
appropriate (with the exception, of course, of transfer function
and multiple time series models).
7.2 Length of forecast period
For normal extensions of switching equipment and additions of
circuits, a forecast period of about six years is necessary. How
ever, a longer forecast period may be necessary for the planning of
new cables or other transmission media or for major plant installa
tions. Estimates in the long term would necessarily be less accu
rate than shortterm forecasts but that would be acceptable.
In forecasting with a statistical model, the length of the
forecast period is entirely determined by:
a) the historical data available,
b) the purpose or use of the forecast,
c) the market structure that generates the data,
d) the forecasting model used,
e) the frequency of the data.
The historical data available depends upon the period over
which it has been collected and the frequency of collection (or the
length of the period over which data is aggregated). A small his
torical data base can only support a short prediction interval. For
example, with 10 or 20 observations
a model can be used to forecast 45 periods past the sample
(i.e. into the future). On the other hand, with 150200 observa
tions, potentially reliable forecasts can be obtained for 30 to
50 periods past the sample  other things being equal.
Certainly, the purpose of the forecast affects the number of
predicted periods. Long range facility planning requires forecasts
extending 1520 or more years into the future. Rate change evalua
tions may only require forecasts for 23 years. Alteration of rout
ing arrangements could only require forecasts extending a few
months past the sample.
Stability of a market, or lack thereof, also affect the length
of the forecast period. With a stable market structure one could
conceivably extend the forecast period to equal the historical
period. However, a volatile market does not afford the same luxury
to the forecaster; the forecast period can only consist of a few
periods into the future.
The forecasting models used to generate forecasts do, by their
nature, influence the decision on how far into the future one can
reasonably forecast. Structural models tend to perform better than
other models in the long run, while for shortrun predictions all
models seem to perform equally well.
It should be noted that while the purpose of the forecast and
the forecasting model affect the length of the forecast, the number
of periods to be forecasted play a crucial role in the choice of
the forecasting model and the use to which a forecast is put.
ANNEX A
(to Recommendation E.507)
Description of forecasting procedures
A.1 Estimation of autoregressive parameters
The empirical autocorrelation at lag k is given by:
r
k =
fIv 0
_______
(A1)
where
and
N being the total number of observations.
The relation between [rk] and the estimates [/k] of [/k] is
given by the YuleWalker equations :
r 1 = / 1 + / 2r 1 + .   + / pr p 1
r 2 = / 1r 1 + / 2r 2 + .   + / pr p 2 x (A4)
x
x
r p = / 1r p 1 + / 2r p 2 + .   + / p
Hence the estimators [/k] can be found by solving this system
of equations.
For computations, an alternative to directly solving the equa
tions is the following recursive procedure. Let [ / k ,  fIj ] j
be estimators of the parameters at lag j = 1, 2, .   , k given
that the total number of parameters are k . The estimators [/ k +1,
 fIj ] j are then found by
Defining / p ,  fIj = / j ,  fIj = 1, 2, .   , p , the
forecast of the traffic demand at time t +1 is expressed by:
X
t +1
= /
1X
t + /
2X
t 1
+ .   +
/
pX
t p
(A7)
A.2 Forecasting with ARIMA models
The forecast l time units ahead is given by:
which means that [Xj] is defined as a forecast when j > t and oth
erwise as an actual observation and that [aj] is defined as 0 when
j > t since white noise has expectation 0. If the observations are
known ( j t ), then [aj] is equal to the residual.
ANNEX B
(to Recommendation E.507)
Kalman Filter for a linear trend model
To model telephone traffic, it is assumed that there are no
deterministic changes in the demand pattern. This situation can be
modelled by setting the deterministic component Ztto zero. Then the
general state space model is:
Xt\d+\d1= Xt+ wt
(B1)
Yt= HXt+ vt
where
Xt is an svector of state variables in period t ,
Yt is a vector of measurements in year t ,
 is an s xs transition matrix that may, in gen
eral, depend on t ,
and
wt is an svector of random modelling errors,
vt is the measurement error in year t .
For modelling telephone traffic demand, adapt a simple
twostate, onedata variable model defined by:
and
yt= xt+ vt
(B3)
where
xt is the true load in year t ,
xt is the true incremental growth in year t ,
yt is the measured load in year t ,
vt is the measurement error in year t .
Thus, in our model
The onestepahead projection is written as follows:
where
X t +1,t is the projection of the state variable in period t
+ 1 given observations through year t .
The (tand tcoefficients are the Kalman gain matrices in year
t . Rewriting the above equation yields:
xt\d,\dt= (1(t)x
t,t 1
+
(t
(B6)
and
xt\d,\dt= (1t)x
t,t 1
+ t(yt x
t 1,t 1
)
(B7)
The Kalman Filter creates a linear trend for each time series
being forecast based on the current observation or measurement of
traffic demand and the previous year's forecast of that demand. The
observation and forecasted traffic load are combined to produce a
smoothed load that corresponds to the level of the process, and a
smoothed growth increment. The Kalman gain values (tand tcan be
either fixed or adaptive. In [16] Moreland presents a method for
selecting fixed, robust parameters that provide adequate perfor
mance independent of system noise, measurement error, and initial
conditions. For further details on the proper selection of these
parameters see [6], [20] and [22].
ANNEX C
(to Recommendation E.507)
Example of an econometric model
To illustrate the workings of an econometric model, we have
chosen the model of United States billed minutes to Brazil. This
model was selected among alternative models for three reasons:
a) to demonstrate the introduction of explanatory
variables,
b) to point out difficulties associated with models
used for both the estimation of the structure and forecasting pur
poses, and
c) to show how transformations may affect the
results.
The demand of United States billed minutes to Brazil (MIN ) is
estimated by a loglinear equation which includes United States
billed messages to Brazil (MSG ), a real telephone price index (RPI
), United States personal income in 1972 prices (YP 72), and real
bilateral trade between the United States and Brazil (RTR ) as
explanatory variables. This model is represented as:
ln(MIN )t= 0 + 1ln(MSG )t
+ 2ln(RPI )t + 3ln(YP 72)t
+ 4ln(RTR )t + ut
(C1)
where utis the error term of the regression and where, 1 > 0,
2 < 0, 3 > 0 and 4 > 0 are expected values.
Using ridge regression to deal with severe multicollinearity
problems, we estimate the equation over the 1971   (i.e. first
quarter of 1971) to 1979   interval and obtain the following
results:
ln(MIN )t= 3.489 + ( 0.619) ln(MSG )t ( 0.447) ln(RPI )t+ (
1.166) ln(YP 72)t+ ( 0.281) ln(RTR )t In(MIN )t= 3.489 + (0.035)
ln(MSG )t (0.095) ln(RPI )t+ (0.269) ln(YP 72)t+ (0.084)
(C2)
.sp 1
R 2 = 0.985, SER = 0.083, DW = 0.922,
k = 0.10
(C3)
where R 2 is the adjusted coefficient of determination, SER is the
standard error of the regression, DW is the DurbinWatson statis
tic, and k is the ridge regression constant. The values in
parentheses under the equation are the estimated standard deviation
of the estimated parameters 1, 3/4d2, 3, 4.
The introduction of messages as an explanatory variable in
this model was necessitated by the fact that since the
midseventies transmission quality has improved and completion
rates have risen while, at the same time, the strong growth in this
market has begun to dissipate. Also, the growth rates for some
periods could not have been explained by rate activity on either
side or real United States personal income. The behaviour of the
message variable in the minute equation was able to account for all
these factors.
Because the model serves a dual purpose  namely, structure
estimation and forecasting  at least one more variable is intro
duced than if the model were to be used for forecasting purposes
alone. The introduction of additional explanatory variables results
in severe multicollinearity and necessitates employing ridge
regression which lowers R 2 and the DurbinWatson statistic. Con
sequently, the predictive power of the model is reduced somewhat.
The effect of transforming the variables of a model are shown
in the expost forecast analysis performed on the model of United
States billed minutes to Brazil. The deviations using levels of
the variables are larger than those of the logarithms of the vari
ables which were used to obtain a better fit (the estimated RMSE
for the loglinear regression model is 0.119  27). The forecast
results in level and logarithmic form are shown in Table C1/E.507.
H.T. [T1.507]
TABLE C1/E.507
________________________________________
Logarithms Levels
________________________________________








_____________________________________________________________________________________
Forecast Actual % deviation Forecast Actual % deviation
_____________________________________________________________________________________
1980: 1 14.858 14.938 0.540 2  36  69 3  73  97  7.725
2 14.842 14.972 0.872 2  91  50 3  80  34 12.234
3 14.916 15.111 1.296 3  05  37 3  54  92 17.746
4 14.959 15.077 0.778 3  37  98 3  29  16 11.089
1981: 1 15.022 15.102 0.535 3  41  33 3  21  35  7.731
2 14.971 15.141 1.123 3  75  77 3  62  92 15.601
3 15.395 15.261  0.879 4  52  78 4  44  78 14.333
4 15.405 15.302  0.674 4  01  46 4  21  55  10.844
1982: 1 15.365 15.348  0.110 4  09  65 4  30  38  1.702
2 15.326 15.386 0.387 4  28  47 4  07  01  5.802
_____________________________________________________________________________________








































































































Table C1/E.507 [T1.507] p.
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John Wiley & Sons , New York, 1981.
Bibliography
PINDYCK (R.  .) and RUBINFELD (D.  .): Econometric Models and
Econometric Forecasts, McGrawHill , New York, 1981.
SASTRI, (T.): A state space modelling approach for time series
forecasting. Management Science , Vol. 31, No. 11, pp. 14511470,
1985.
Recommendation E.508
FORECASTING NEW INTERNATIONAL SERVICES
1 Introduction
The operation and administration of an international telecom
munications network should include the consideration of subscriber
demands for new services which may have different characteristics
than the traditional traffic (i.e. peak busy hours, bandwidth
requirements, and average call durations may be different). By
addressing these new demands, Administrations can be more respon
sive to customer requirements for innovative telecommunications
services. Based on the type of service and estimated demand for a
service, network facilities and capacity may have to be augmented.
An augmentation of the international network could require large
capital investments and additional administrative functions and
responsibilities. Therefore, it is appropriate that Administra
tions forecast new international services within their planning
process.
This Recommendation presents methods for forecasting new ser
vices. The definitions of some of the characteristics of these ser
vices, together with their requirements, are covered in S 2, fol
lowed by base data requirements in S 3. S 4 discusses research to
identify the potential market. Presentation of forecasting methods
are contained in S 5. S 6 concludes with forecast tests and adjust
ments.
2 New service definitions
2.1 A distinction exists between those services which are
enhancements of existing services carried on the existing network
and those services which are novel.
Many of the services in this latter category will be carried
on the Integrated Services Digital Network (ISDN). It is not the
purpose of this section to provide an exhaustive list of services
but rather to establish a framework for their classification. This
framework is required because different base data and forecasting
strategies may be necessary in each case.
2.2 enhanced services offered over the existing network
These are services which are offered over the existing net
work, and which offer an enhancement of the original use for which
the network was intended. Services such as the international free
phone service, credit card calling and closed user groups are exam
ples of enhancements of voice services; while facsimile, telefax
and videotex are examples of nonvoice services. These services may
be carried over the existing network and, therefore, data will con
cern usage or offered load specific to the enhancement. Arrange
ments can be established for the measurement of this traffic, such
as the use of special network access codes for nonvoice applica
tions or by sampling outgoing circuits for the proportion of
nonvoice to voice traffic.
2.3 novel services
Novel services are defined as totally new service offerings
many of which may be carried over the ISDN. In the case of ISDN,
Recommendation I.210 divides telecommunications services into two
broad categories: bearer services and teleservices.
Recommendation I.210 further defines supplementary services which
modify or supplement a basic telecommunications service. The defin
ition of bearer services supported by the
ISDN is contained in Recommendations I.210 and I.211, while
that for teleservices is found in Recommendations I.210 and I.212.
Bearer services may include circuit switched services from
64 kbit/s to 2 Mbit/s and packet services. Circuit switched ser
vices above 2 Mbit/s are for further study.
Teleservices may include Group 4 facsimile, mixed mode text
and facsimile, 64 kbit/s Teletex and Videotex, videophone, video
conferencing, electronic funds transfer and point of sale transac
tion services. These lists are not exhaustive but indicate the
nature and scope of bearer services and teleservices. Examples of
new services are diagrammatically presented in Table 1/E.508.
H.T. [T1.508]
TABLE 1/E.508
Examples of enhanced and novel services
___________________________________________________________________________________
"Novel" services
{
Bearer services Teleservices
___________________________________________________________________________________
Teletex Packet Group 4 facsimile
Facsimile Mixed mode
Videotex Videophone
Message handling systems Circuit switched services Videoconferencing
International freephone  64 kbit/s Electronic funds transfer
Credit cards  2 Mbit/s Point of sale transactions
Closed user groups {
Teletex (64 kbit/s)
Videotex (64 kbit/s)
}
___________________________________________________________________________________



































































Table 1/E.508 [T1.508], p.
3 Base data for forecasting
3.1 Measurement of enhanced services
Measurements for existing services are available in terms of
calls, minutes, Erlangs, etc. These procedures are covered in
Recommendation E.506, S 2. In order to measure/identify enhanced
service data
from other traffic data on the same network it may be neces
sary to establish sampling or other procedures to aid in the esti
mation of this traffic, as described in S 4 and S 5.
3.2 Novel services
Novel services, as defined in S 2, may be carried on the ISDN.
In the case of the ISDN, circuit switched bearer services and their
associated teleservices will be measured in 64 kbit/s increments.
Packet switched bearer services and associated teleservices will be
measured by a unit of throughput, for example, kilocharacters or
kilopackets per second. Other characteristics needed will reflect
service quality measurements such as: noise, echo, postdialing
delay, clipping, biterror rate, holding time, setup time,
errorfree seconds, etc.
4 Market research
Market research is conducted to test consumer response and
behaviour. This research employs the methods of questionnaires,
market analysis, focus groups and interviews. Its purpose is to
determine consumers' intentions to purchase a service, attitudes
towards new and existing services, price sensitivity and cross ser
vice elasticities. Market research helps make
decisions concerning which new services should be developed. A
combination of the qualitative and quantitative phases of market
research can be used in the initial stages of forecasting the
demand for a new service.
The design of market research considers a sampling frame,
customer/market stratification, the selection of a statistically
random sample and the correction of results for nonresponse bias.
The sample can be drawn from the entire market or from subsegments
of the market. In sampling different market segments, factors which
characterize the segments must be alike with respect to consumer
behaviour (small intragroup variance) and should differ as much as
possible from other segments (large intergroup variance); each seg
ment is homogeneous while different segments are heterogeneous.
The market research may be useful in forecasting existing ser
vices or the penetration of new services. The research may be used
in forecasting novel services or any service which has no histori
cal series of demand data. It is important that potential consumers
be given a complete description of the new service, including the
terms and conditions which would accompany its provisioning. It is
also important to ask the surveyees whether they would purchase the
new service under a variety of illustrative tariff structures
and levels. This aspect of market research will aid in redi
mensioning the demand upon final determination of the tariff struc
ture and determining the customers' initial price sensitivity.
5 Forecasting procedures
5.1 General
The absence of historical data is the fundamental difference
between forecasting new services and forecasting existing services.
The forecast methodology is dependent on the base data. For exam
ple, for a service that is planned but has not been introduced,
market research survey data can be used. If the service is already
in existence in some countries, forecasting procedures for its
introduction to a new country will involve historical data on other
countries, its application to the new country and comparison of
characteristics between countries.
5.2 Sampling and questionnaire design
The forecasting procedure for novel services based on market
research is made up of five consecutive steps. The first of these
consists in defining the scope of the study.
The second step involves the definition and selection of a
sample from the population, where the population includes all
potential customers which can be identified by qualitative market
research developed through interviews at focus groups. The research
can use stratified samples which involves grouping the population
into homogeneous segments (or strata) and then sampling within each
strata. Stratification prevents the disproportionate representation
of some parts of the population that can result by chance with sim
ple random sampling. The sample can be structured to include speci
fied numbers of respondents having characteristics that are known,
or believed, to affect the subject of the research. Examples of
customer characteristics would be socioeconomic background and
type of business.
The third step is the questionnaire design. A tradeoff exists
between obtaining as much information as practical and limiting the
questionnaire to a reasonable length, as determined by the sur
veyor. Most questionnaires have three basic sections:
1) qualifying questions to determine if a
knowledgeable person has been contacted;
2) basic questions including all questions which
constitute the body of the questionnaire;
3) classification questions collecting background
on demographic information.
The fourth step involves the implementation of the research 
the actual surveying portion. Professional interviewers, or firms
specializing in market research should be employed for interview
ing.
The fifth and final step is the tabulation and analysis of the
survey data. S 5.35.7 describe this process in detail.
5.3 Conversion ratios for the sample
Conversion ratios are used in estimating the proportion of
respondents expressing an interest in the service who will eventu
ally subscribe.
The analysis of the market research data based on a sample
survey, where a stratified sample is drawn across market segments,
for a service that is newly introduced or is planned, is discussed
below:
Let
X1i = the proportion of firms in market segment
i  hat are very interested in the service.
X2i = the proportion of firms in market segment
i  hat are interested in the service.
X3i = the proportion of firms in market segment
i  hat are not interested in the service.
X4i = the proportion of firms in market segment
i  hat cannot decide whether they are interested or not.
The above example has 4 categories of responses. Greater of
fewer categories may be used depending on the design of the ques
tionnaire.
Notice that
where j = the index of categories of responses.
Market research firms sometimes determine conversion ratios
for selected product/service types. Conversion ratios depend on the
nature of the service, the type of respondents, and the question
naire and its implementation. Conversion ratios applied to the sam
ple will estimate the expected proportion of firms in the survey 
hat will eventually subscribe, over the planning period. For stu
dies related to the estimation of conversion ratios, refer to [1],
[3] and [5].
Then,
c1X1i = the proportion of firms in market segment
i  hat expressed a strong interest and are expected to subscribe.
c2X2i = the proportion of firms in market segment
i  hat expressed an interest and are expected to subscribe.
c3X3i = the proportion of firms in market segment
i  hat expressed no interest but are expected to subscribe.
c4X4i = the proportion of undecided firms in
market segment i  hat are expected to subscribe.
where cj = conversion ratio for response j .
The proportion of firms in market segment i , Pi, that are
expected to subscribe to the service, equals
The conversion ratio is based on the assumption that there is
a 100% market awareness. That is, all surveyees are fully informed
of the service availability, use, tariffs, technical
parameters, etc. Pi,
therefore, can be interpreted as the longrun proportion of
firms in market segment i  hat are expected to subscribe to the
service at some future time period, T .
Two issues arise in the estimation of the proportion of custo
mers that subscribe to the service:
1) while Pirefers to the sample surveyed, the
results need to be extrapolated to represent the population.
2) Piis the longrun (maximum) proportion of firms
expected to subscribe. We are interested in predicting no just the
eventual number of subscribers but, also, those at intermediate
time periods before the service reaches a saturation point.
5.4 Extrapolation from sample to population
To extrapolate the data from the sample to represent the popu
lation, let
Ni = size of market segment i  measured for exam
ple, by the number of firms in market segment i )
Then Si, the expected number of subscribers in the planning
horizon, equals:
Si= PiNi
(52)
5.5 Market penetration over time
To determine the expected number of subscribers at various
points in time before the service reaches maturity, let
pi\dt = the proportion of firms in market segment i
 hat are expected to subscribe at time t .
Clearly,
pi\dt< Pi
and pi\dt Pi as t  fIT
The relation between pi\dtand Pican be explicitly defined as:
pi\dt= ai\dtx Pi
(53)
ai\dtis a penetration function, reflecting changing market aware
ness and acceptance of the service over time, in market segment i
. An appropriate functional form for ai\dtshould be bounded in the
interval (0,1).
As an example, let ai\dtbe a logistic function:
ai\dt=
[Formula Deleted]
(54)
bi 0 is the speed with which pi\dtapproaches Piin market segment i
, as illustrated in Figure 1/E.508.
For other examples of nonlinear penetration functions, refer
to the Annex A.
Figure 1/E.508, p.
The introduction of a new service will usually differ accord
ing to the market segment. The rate of penetration may be expressed
as a function of time, and the speed of adjustment (bi) may vary
across segments. Lower absolute values of bi, for the logistic
function will imply faster rates of penetration.
While the form of the penetration function relating the rate
of penetration to time is the same for all segments, the
parameter bivaries across segments, being greater in segments with
a later introduction of the new service.
Let t0i = time period of introduction of service in
market segment i .
Then, t  t0i = time period elapsed since service
was introduced in market segment i .
In the diagramatic illustration, of Figure 2/E.508, the ser
vice has achieved the same level of market penetration a0, in tC
periods after its introduction in market C as it did in tAperiods
after its introduction in market segment A . Later introductions
may not necessarily lead to faster rates of penetration across seg
ments. However, within the same market segment, across countries
with similar characteristics, such an expectation is reasonable.
Figure 2/E.508, p.
5.6 Growth of market segment over time
The above discussion has accounted for gradual market penetra
tion of the new service, by allowing pi\dtto adjust to Piover time.
The same argument can be extended to the size of market segment i
 ver time.
Let ni\dt= size of market segment i at time t .
Then, the expected number of subscribers at time t  n market
segment i , equals:
si\dt= ai\dtx pi\dtx
ni\dt
(55)
and
St = i
~sit= expected number of subscribers across all
market segments at time t .
5.7 Quantities forecasted
The above procedure forecasts the expected number of customers
for a new service within a country. Other quantities of interest
may include lines, minutes, messages, revenue, packets,
kilobits, etc. The most straight forward
forecasting method for some of these quantities is to assume
constant relationships such as:
expected access lines = (average access
lines) x expected number of subscribers
expected minutes = (average use per line) x
expected access lines
expected messages = expected
minutes/(average length of conversation)
expected revenue = (average rate per
minute) x expected minutes
The constants, appearing in parentheses, above, can be deter
mined through 1) the process of market research, or 2) past trends
in similar services.
5.8 Forecasting with historical data: application analysis
After a new service has been introduced, historical data can
be analyzed to forecast demand for expanded availability to other
countries. Development of a new service will follow trends based
on applications, such as data transmission, travel reservations,
intracompany communications, and
supplier contact. Applications of a service vary widely and no
single variable may be an adequate indicator of total demand.
The following procedure links demand to country characteris
tics for forecasting expanded availability of a new service to
other countries.
Let D = (Di, D2,       , Dn)`
represent a vector of countryspecific annual demand for the ser
vice across n  ountries, where the service currently exists. Let
C = matrix of m
 haracteristics relating to each of the n  ountries that are
reasonable explanatory variables of demand. The components of m 
ould vary depending on the nature of the service and its applica
tion.
Some essential components of m  ould be the price of the
service (or an index representing its price) and some proxy for
market awareness. As discussed in earlier sections, market aware
ness is one of the key determinants of the rate of market penetra
tion of the service. Reasonable proxies would be advertising expen
ditures and time (measured as t * = t  t0) where t * would meas
ure time elapsed since the service was first introduced at time t0.
Market
awareness can be characterized as some nonlinear function
of t *, as presented in S 5.5. Other components of m  ay include
socioeconomic characteristics of the customers, market size and
location of customers.
The model that is estimated is:
D = C  + u
(56)
where
C is a (n x m ) matrix of country characteristics
D is a (n x 1) vector of demand
 is a (m x 1) vector of coefficients corresponding to each
of the m  haracteristics
u = (n x 1) vector of error terms
The estimated regression is:
D = C 
(57)
Traditional methods of estimating regressions will be applied.
Equation (57) can be used for predicting demand for any country
where the service is being newly introduced, as long as elements of
the matrix C
 re available.
5.9 Forecasting with limited information
In the extreme case where no market research data is available
(or is uneconomical given resource constraints), or country charac
teristics that affect demand are not easily available or quantifi
able, other methods of forecasting need to be devised.
For example, to forecast the demand for a new international
private line service using digital technology, the following
elements should be taken into account in the development of reason
able estimates of the expected number of lines:
a) discussions with foreign telephone companies,
b) discussions with very large potential customers
regarding their future needs,
c) service inquiries from customers,
d) customer letters of intent, and
e) any other similar qualitative information.
6 Forecast tests and adjustments
6.1 General
Forecast tests and adjustments are dependent on the methodol
ogy applied. For example, in the case of a market research based
forecast, it is important to track the forecast of market size,
awareness and rate of penetration over time and to adjust forecasts
accordingly. However, for an applicationbased methodology, tradi
tional tests and adjustments applicable to regression methods will
be employed, as discussed below.
6.2 Market research based analysis
This section discusses adjustments to forecasts based on the
methodology described in SS 5.2 to 5.8. The methodology was based
on quantification of responses from a sample survey.
The forecast was done in two parts:
a) extrapolating the sample to the population,
using market size, Ni;
b) allowing for gradual market penetration (aware
ness), ai\dtof the new service over time.
The values attributed to ni\dt(which represents the size of
market segment i at time t ) and ai\dtcan be tracked over time and
forecast adjustments made in the following manner:
a) As an example for ni\di, the segments could be
categorized as travel or financial services. The size of the seg
ment would be the number of tourists, and the number of large
banks. Historical data, where available, on these units of measure
ment can be used to forecast their sizes at any point of time in
the future. Where history is not available, reasonable growth
factors can be developed through subject matter experts and past
experiences. The forecast of ni\dtshould be tracked against actual
measured values and adjusted for large deviations.
b) For ai\dt, testing with only a few observations
since the introduction of the service is more difficult.
Given that,
ai\dt=
fIPfIi
________
(61)
and Piis assumed fixed (in the long run), testing ai\dtis
equivalent to testing pi\dt. pi\dtcan be tracked by observing the
proportion of respondents that actually subscribe to the service at
time t . This assumes the need to track the same individuals who
were originally in the survey, as is customary in a panel survey.
Panel data is collected through sample surveys of crosssections of
the same individuals, over time. This method is commonly used for
household socioeconomic surveys. Having observed pi\dtfor a new
period, values of ai\dtcan be plotted against time to study the
nature of the penetration function, ai\dt, and the most appropriate
functional
form that fits the data should be chosen. At very early stages of
service introduction, traditional functional forms for market pene
tration, such as a logistic function (as illustrated in the example
in S 5.5), will be a reasonable form to assume. Other variations of
the functional form depicting market penetration would be the Gom
pertz or Gauss growth curves. The restriction is that the penetra
tion function should be bounded in the interval (0,1). See Annex A
for an algebraic depiction of functional forms.
There are various statistical forms that may be chosen as
representations for the penetration function. The appropriate func
tional form should be based on some theoretical based information
such as the expected nature of penetration of the specific service
over time.
Continuous tracking of ni\dt, pi\dtand ai\dtover time will
enable adjustments to these values whenever necessary and enable
greater confidence in the forecasts.
6.3 Application based analysis
The application based analysis is a regression based approach
and traditional forecast tests for a regression model will apply.
For instance, hypothesis tests on each of the explanatory variables
included in the model will be necessary. Corrections may be needed
for heteroelasticity, serial correlation and multicollinearity,
when suspect. The methodology for performing such tests are
described in most econometrics text books. In particular, refer
ences [2] and [4] can be used as guidelines. Recommendation E.507
also discusses these corrections.
Adjustments need to be made for variables that should be
included in the regression model but are not easily quantifiable.
For example, market
awareness that results from advertising and promotional cam
paigns plays an important role in the growth of a new service, but
data on such expenditures or the associated awareness may not be
readily available. Some international services are targeted towards
international travelers, and fluctuations in exchange rates will be
a determining factor. Such variables, while not impossible to meas
ure, may be expensive to acquire. However, expectations of future
trends in such variables can enable the forecaster to arrive at
some reasonable estimates of their impact on demand. Unexpected
occurrences such as political turmoil and natural disasters in par
ticular countries will also necessitate post forecast adjustments
based upon managerial judgement.
Another important adjustment that may be necessary is the
expected competition from other carriers offering similar or sub
stitutable services. Competitor prices, if available, may be used
as explanatory variables within the model and allow the measurement
of a crossprice impact. In most situations, it is difficult to
obtain competitor prices. In such cases, other methods of calculat
ing competitor market shares need to be developed.
Regardless of forecasting methodology, the final forecasts
will have to be reviewed by management responsible for planning the
service as well as
by network engineers in order to assess the feasibility both
from a planning implementation and from a technical point of view.
ANNEX A
(to Recommendation E.508)
Penetration functions (growth curves)
Some examples of nonlinear penetration functions are illus
trated below:
A.1 Logistic curve
ai\dt= ( / { + eDlF261
t }
(A1)
For ( = 1, the curve is bounded in the interval (0,1).
Changing b will alter the steepness of the curve. The higher the
value of b , the faster the rate of penetration. This curve is
Sshaped and is symmetrical about its point of inflection, the
latter being where;
t 2
_________ = 0
(A2)
A.2 Gompertz curve
ai\dt= ( exp
[Formula Deleted]
(A3)
As t oo ai\dt (, the limiting growth.
Holding k = 1 and ( = 1, higher values of b will imply slower
rates of penetration. This curve is also Sshaped like the logistic
curve, but is not symmetrical about its inflection point.
When t = 0, then ai\dt= (eDlF261 b, which is the initial rate
of penetration.
A.3 Gauss curve
ai\dt= (

1  eDlF261 fIbt 2 

(A4)
As t oo, then a it (
As t 0, then a it 0.
Choosing ( = 1, the curve is bounded in the interval (0,1).
References
[1] AXELROD (J.  .): Attitude measures that predict pur
chase, Journal of Advertising Research , Vol. 8, No. 1, pp. 317,
New York, March 1968.
[2] JOHNSTON (J.): Econometric methods, Second Edition,
McGrawHill , New York, 1972.
[3] KALWANI (M.  .), SILK, (A.  .): On the reliability
and predictive validity of purchase intention measures, Marketing
Science , Vol. 1, No. 3, pp. 243286, Providence, RI, Summer 1982.
[4] KMENTA (J.): Elements of econometrics, Macmillan Pub
lishing Co. , New York, 1971.
[5] MORRISON (D.  .): Purchase intentions and purchase
behavior, Journal of Marketing , Vol. 43, pp. 6574, Chicago, Ill.,
Spring 1979.
Bibliography
BENAKIVA (M.) and LERMAN (S.  .): Discrete choice analysis.
DRAPER (N.) and SMITH (H.): Applied regression analysis, Second
Edition, John Wiley & Sons , New York, 1981.
SECTION 3
DETERMINATION OF THE NUMBER OF CIRCUITS IN
MANUAL OPERATION
Recommendation E.510
DETERMINATION OF THE NUMBER OF CIRCUITS
IN MANUAL OPERATION
1 The quality of an international manual demand service should
be defined as the percentage of call requests which, during the
average busy hour (as defined later under S 3) cannot be satisfied
immediately because no circuit is free in the relation considered.
By call requests satisfied immediately are meant those for
which the call is established by the same operator who received the
call, and within a period of two minutes from receipt of that call,
whether the operator (when she does not immediately find a free
circuit) continues observation of the group of circuits, or whether
she makes several attempts in the course of this period.
Ultimately, it will be desirable to evolve a corresponding
definition based on the average speed of establishing calls in the
busy hour, i.e. the average time which elapses between the moment
when the operator has completed the recording of the call request
and the moment when the called subscriber is on the line, or the
caller receives the advice subscriber engaged , no reply , etc. But
for the moment, in the absence of information about the operating
time in the European international service, such a definition
_________________________
This Recommendation dates from the XIIIth Plenary As
sembly of the CCIF (London, 1946) and has not been fun
damentally revised since. It was studied under
Question 13/II in the Study Period 19681972 and was
found to be still valid.
cannot be established.
2 The number of circuits it is necessary to allocate to an
international relation, in order to obtain a given grade of ser
vice, should be determined as a function of the total holding time
of the group in the busy hour.
The total holding time is the product of the number of calls
in the busy hour and a factor which is the sum of the average call
duration and the average operating time
These durations will be obtained by means of a large number of
observations made during the busy hours, by agreement between the
Administrations concerned. If necessary, the particulars entered on
the tickets could also serve to determine the average duration of
the calls.
The average call duration will be obtained by dividing the
total number of minutes of conversation recorded by the recorded
number of effective calls.
The average operating time will be obtained by dividing the
total number of minutes given to operating (including ineffective
calls) by the number of effective calls recorded.
3 The number of calls in the busy hour will be determined from
the average of returns taken during the busy hours on a certain
number of busy days in the year.
Exceptionally busy days, such as those which occur around cer
tain holidays, etc., will be eliminated from these returns. The
Administrations concerned should plan, whenever possible, to put
additional circuits into service for these days.
In principle, these returns will be taken during the working
days of two consecutive weeks, or during ten consecutive working
days. If the monthly traffic curve shows only small variations,
they will be repeated twice a year only. They will be taken three
or four times a year or more if there are material seasonal varia
tions, so that the average established is in accordance with all
the characteristic periods of traffic flow.
4 The total occupied time thus determined should be increased
by a certain amount determined by agreement between the Administra
tions concerned according to the statistics of traffic growth dur
ing earlier years, to take account of the probable growth in
traffic and the fact that putting new circuits into service takes
place some time after they are first found to be necessary.
5 The total holding time of the circuits thus obtained, in
conjunction with a suitable table (see Table 1/E.510), will enable
the required number of circuits to be ascertained.
6 In the international manual telephone service, the following
Tables A and B should be used as a basis of minimum allocation:
Table A corresponds to about 30% of calls failing at the first
attempt because of all circuits being engaged and to about 20% of
the calls being deferred.
Table B, corresponding to about 7% of calls deferred, will be
used whenever possible.
These tables do not take account of the fact that the possi
bility of using secondary routes permits, particularly for small
groups, an increase in the permissible occupation time.
H.T. [T1.510]
TABLE 1/E.510
Capacity of circuit groups
(See Supplement No. 2 at the end of this fascicle)
_________________________________________________________________________________
Table A Table B
Percentage of circuit usage {
Percentage of circuit usage {
Number of circuits
_________________________________________________________________________________
1 65.0 39  
2 76.7 92 46.6 56
3 83.3 150 56.7 102
4 86.7 208 63.3 152
5 88.6 266 68.3 205
6 90.0 324 72.0 259
7 91.0 382 74.5 313
8 91.7 440 76.5 367
9 92.2 498 78.0 421
10 92.6 556 79.2 475
11 93.0 614 80.1 529
12 93.4 672 81.0 583
13 93.6 730 81.7 637
14 93.9 788 82.3 691
15 94.1 846 82.8 745
16 94.2 904 83.2 799
17 94.3 962 83.6 853
18 94.4 1020 83.9 907
19 94.5 1078 84.2 961
20 94.6 1136 84.6 1015
_________________________________________________________________________________


















































































































































































Note  Tables A and B can be extended for groups comprising more
than 20 circuits by using the values given for 20 circuits.
Tableau 1/E.510 [T1.510], p.16