.rs
.\" Troff code generated by TPS Convert from ITU Original Files
.\"                 Not Copyright ( c) 1991 
.\"
.\" Assumes tbl, eqn, MS macros, and lots of luck.
.TA 1c 2c 3c 4c 5c 6c 7c 8c
.ds CH
.ds CF
.EQ
delim @@
.EN
.nr LL 40.5P
.nr ll 40.5P
.nr HM 3P
.nr FM 6P
.nr PO 4P
.nr PD 9p
.po 4P

.rs
\v | 5i'
.sp 1P
.ce 1000
\v'3P'
SECTION\ 2
.ce 0
.sp 1P
.ce 1000
\fBFORECASTING\ OF\ TRAFFIC\fR 
.ce 0
.sp 1P
.sp 2P
.LP
\fBRecommendation E.506\fR 
.RT
.sp 2P
.sp 1P
.ce 1000
\fBFORECASTING\ INTERNATIONAL\ TRAFFIC\fR 
.FS
The old
Recommendation\ E.506 which appeared in the \fIRed\ Book\fR was split into two
Recommendations, revised E.506 and new E.507 and considerable new
material was added to both.
.FE
.EF '%	Fascicle\ II.3\ \(em\ Rec.\ E.506''
.OF '''Fascicle\ II.3\ \(em\ Rec.\ E.506	%'
.ce 0
.sp 1P
.LP
\fB1\fR 	\fBIntroduction\fR 
.sp 1P
.RT
.PP
This Recommendation is the first in a series of three
Recommendations that cover international telecommunications forecasting.
.PP
In the operation and administration of the international telephone
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 investments, it is necessary that
Administrations forecast the traffic which the network will carry. In view 
of the heavy capital investments in the international network, the economic 
importance of the most reliable forecast is evident.
.PP
The purpose of this Recommendation is to give guidance on some of the prerequisites 
for forecasting international telecommunications 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.
.PP
Recommendation E.507 provides guidelines for building forecasting
models and contains an overview of various forecasting techniques.
Recommendation\ E.508 covers the forecasting of new international
telecommunications services.
.RT
.LP
.sp 2P
.LP
\fB2\fR 	\fBBase data for forecasting\fR 
.sp 1P
.RT
.PP
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 techniques are applied to forecast 
Erlangs, a measure of traffic. Figure\ 1/E.506 outlines two different approaches 
for determining forecasted Erlangs. 
.PP
The two different strategies for forecasting are the direct strategy and 
the composite strategy. The first step in either process 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 Administrations use monthly accounting data 
for forecasting 
purposes.
.PP
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). 
.bp
.RT
.LP
.rs
.sp 32P
.ad r
\fBFigure 1/E.506, p. 1 
.sp 1P
.RT
.ad b
.RT
.PP
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 strategy.
.PP
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
busy\(hyhour Erlang forecasts.
.PP
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 generated, for example, using traffic 
matrix methods (see \(sc\ 4), econometric models or autoregressive models 
(see \(sc\ 3, Recommendation\ E.507). There are various other data that 
are input to the 
forecasting process. Examples of these are explanatory variables, market
segmentation information and price elasticities.
.PP
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.
.bp
.PP
In econometric modelling especially, explanatory variables are used to
forecast international traffic. Some of the most important of these variables 
are the following: 
.RT
.LP
	\(em
	exports,
.LP
	\(em
	imports,
.LP
	\(em
	degree of automation,
.LP
	\(em
	quality of service,
.LP
	\(em
	time differences between countries,
.LP
	\(em
	tariffs,
.LP
	\(em
	consumer price index, and
.LP
	\(em
	gross national product.
.PP
Other explanatory variables, such as foreign business travellers 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.
.PP
Forecasts may be based on market segmentation. Base data may be
segmented, for example, along regional lines, by business, non\(hybusiness, 
or by type of service. Price elasticities should also be examined, if possible, 
to 
quantify the impact of tariffs on the forecasting data.
.RT
.LP
.sp 2P
.LP
\fB3\fR 	\fBComposite strategy \(em Conversion method\fR 
.sp 1P
.RT
.PP
The monthly paid\(hyminutes traffic is converted to busy\(hyhour 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: 
\v'6p'
.RT
.ce 1000
\fIA\fR = \fIMdh\fR /60\fIe\fR 
.ce 0
.ad r
(3\(hy1)
.ad b
.RT
.LP
.sp 1
.LP
where
.LP
	\fIA\fR 	is the estimated mean traffic in the busy hour,
.LP
	\fIM\fR 	is the monthly paid\(hyminutes,
.LP
	\fId\fR 	is day\(hyto\(hymonth ratio,
.LP
	\fIh\fR 	is the busy hour\(hyto\(hyday ratio, and
.LP
	\fIe\fR 	is the efficiency factor.
.PP
The formula is described in detail in Annex A.
.LP
.sp 2P
.LP
\fB4\fR 	\fBProcedures for \fR \fBtraffic matrix forecasting\fR 
.sp 1P
.RT
.sp 1P
.LP
\fB
4.1
	\fIIntroduction\fR 
.sp 9p
.RT
.PP
To use traffic matrix or point\(hyto\(hypoint forecasts the following
procedures may be used:
.RT
.LP
	\(em
	Direct point\(hyto\(hypoint forecasts,
.LP
	\(em
	Kruithof's method,
.LP
	\(em
	Extension of Kruithof's method,
.LP
	\(em
	Weighted least squares method.
.PP
It is also possible to develop a Kalman Filter model for
point\(hyto\(hypoint traffic taking into account the aggregated forecasts. 
Tu and 
Pack describe such a model in\ [16].
.PP
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.
.RT
.LP
.sp 1P
.LP
4.2
	\fIDirect point\(hyto\(hypoint forecasts\fR 
.sp 9p
.RT
.PP
It is possible to produce better forecasts for accumulated traffic than 
forecast of traffic on a lower level. 
.PP
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.
.bp
.PP
In this situation it is possible to adjust the individual forecasts by 
taking into account the aggregated forecasts. 
.PP
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. 
.PP
Evaluation of the relative precision of forecasts may be carried out by 
comparing the ratios \(*\*^s(\fIX\fR )/\fIX\fR where \fIX\fR is the forecast 
and \(*\*^s(\fIX\fR ) the estimated forecast error. 
.RT
.sp 1P
.LP
4.3
	\fIKruithof's method\fR 
.sp 9p
.RT
.PP
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.
.PP
Kruithof's method does not take into account the change over time in the 
point\(hyto\(hypoint 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 point\(hyto\(hypoint traffic varies. 
Also when the traffic matrices reflect
seasonal data, Kruithof's method may give poor forecasts.
.RT
.sp 1P
.LP
4.4
	\fIExtension of Kruithof's method\fR 
.sp 9p
.RT
.PP
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.
.PP
It is possible to extend Kruithof's method by taking into account not only 
forecasts of the row and column but also forecasts of point\(hyto\(hypoint 
traffic. Kruithof's method is then used to adjust the point\(hyto\(hypoint 
traffic forecasts to obtain consistency with the forecasts of row and column 
sums. 
.PP
The extended Kruithof's method is superior to the traditional
Kruithof's method and is therefore recommended.
.RT
.LP
.sp 1P
.LP
4.5
	\fIWeighted least squares method\fR 
.sp 9p
.RT
.PP
Weighted least squares method is again an extension of the last
method. Let { fIC\fR\d\fIi\fR\\d\fIj\fR\ }  { fIC\fR\d\fIi\fR\u } and { fIC.\fI\d\fIj\fR\ } 
be 
forecasts of point\(hyto\(hypoint traffic, row sums and column sums respectively. 
.PP
The extended Kruithof's method assumes that the row and column sums
are \*Qtrue\*U and adjust { fIC\fR\d\fIi\fR\\d\fIj\fR\ } to obtain consistency.
.PP
The weighted least squares method [2] is based on the assumption that both 
the point\(hyto\(hypoint 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.
.PP
Let the weighted least squares forecasts be { fID\fR\d\fIi\fR\\d\fIj\fR\ }  
The square sum \fIQ\fR is defined by: 
\v'6p'
.RT
.ce 1000
\fIQ\fR = 
@ pile { sum above \fIij\fR } @\fIa\fR\d\fIi\fR\\d\fIj\fR\u(\fIC\fR\d\fIi\fR\\d\fIj\fR\u\(em \fID\fR\d\fIi\fR\\d\fIj\fR\u)\u2\d+
@ pile { sum above \fIi\fR } @
\fIb\fR\d\fIi\fR\u(\fIC\fR\d\fIi\fR\u. \(em \fID\fR\d\fIi\fR\u.)\u2\d+
@ pile { sum above \fIj\fR } @
\fIc\fR\d\fIj\fR\u(\fIC.\fI\d\fIj\fR\u\(em \fID.\fI\d\fIj\fR\u)\u2\d
.ce 0
.ad r
(4\(hy1)
.ad b
.RT
.LP
.sp 1
where { fIa\fR\d\fIi\fR\\d\fIj\fR\ }  { fIb\fR\d\fIi\fR\u\ }  { fIc\fR\d\fIj\fR\ } 
are chosen constants or weights. 
.LP
.PP
The weighted least squares forecast is found by:
\v'6p'
.LP
	@ pile { { in\fIQ\fR~BOCAD15\fR (\fID\fR\d\fIi\fR\\d\fIj\fR\u)  } above { \fID\fR\d\fIi\fR\\d\fIj\fR\ } } @ 
.LP
.sp 1
subject to
\v'6p'
.ad r
	\fID\fR\d\fIi\fR\u. = 
@ pile { sum above \fIj\fR } @ \fID\fR\d\fIi\fR\\d\fIj\fR\u\ \ \fIi\fR = 1, 2, . |  | 
(4\(hy2)
.ad b
.RT
.LP
.sp 1
and
\v'6p'
.LP
	\fID.\fI\d\fIj\fR\u= 
@ pile { sum above \fIi\fR } @ \fID\fR\d\fIi\fR\\d\fIj\fR\u\ \ \fIj\fR = 1, 2, . |  | 
.LP
.sp 1
.bp
.PP
A natural choice of weights is the inverse of the variance of the forecasts. 
One way to find an estimate of the standard deviation of the 
forecasts is to perform ex\(hypost forecasting and then calculate the root mean
square error.
.PP
The properties of this method are analyzed in [14].
.RT
.LP
.sp 2P
.LP
\fB5\fR 	\fBTop down and bottom up methods\fR 
.sp 1P
.RT
.sp 1P
.LP
5.1
	\fIChoice of model\fR 
.sp 9p
.RT
.PP
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 usually denoted as \*Qbottom up\*U.
.PP
Alternatively, when there is a small amount of traffic between the
countries in question, it is better to start out with forecasting 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 \*Qtop down\*U. The following comments concern the preference of 
one method over 
another.
.PP
Let \(*s\fI\fI\d\fIT\fR\u\u2\d be the variance of the aggregated forecast, and
\(*s\fI\fI\d\fIi\fR\u\u2\d be the variance of the local forecast\ No.\ 
\fIi\fR and \(*g\fI\fI\d\fIi\fR\\d\fIj\fR\ube 
the covariance of the local forecast\ No.\ \fIi\fR and\ \fIj\fR . If the 
following 
inequality is true:
\v'6p'
.RT
.ad r
.ad b
.RT
.LP
then, in general, it is not recommended to use the bottom up method, but 
to use the top down method. 
.PP
In many situations it is possible to use a more advanced
forecasting model on the aggregated level. Also, the data on an aggregated
level may be more consistent and less influenced by stochastic changes 
compared to data on a lower level. Hence, in most cases the inequality 
stated above will be satisfied for small countries. 
.sp 1P
.LP
5.2
	\fIBottom up method\fR 
.sp 9p
.RT
.PP
As outlined in \(sc 5.1 the bottom up method is defined as a
procedure for making separate forecasts of the traffic between different
countries directly. If the inequality given in \(sc\ 5.1 is not satisfied,
which may be the case for large countries, it is sufficient 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.
.RT
.sp 1P
.LP
5.3
	\fITop down procedure\fR 
.sp 9p
.RT
.PP
In most cases the top down procedure is recommended for producing forecasts 
of international traffic for a small country. In Annex\ D a 
detailed example of such a forecasting procedure is given.
.PP
The first step in the procedure is to find a forecasting model on the aggregated 
level, which may be a rather sophisticated model. Let \fIX\fR\d\fIT\fR\ube 
the traffic forecasts on the aggregated level and \(*s\fI\fI\d\fIT\fR\uthe 
estimated standard 
deviation of the forecasts.
.PP
The next step is to develop separate forecasting models of traffic to different 
countries. Let \fIX\fR\d\fIi\fR\ube the traffic forecast to the 
\fIi\fR \ut\d\uh\d country and \(*\*"s\fI\fI\d\fIi\fR\uthe standard deviation. 
Now, the 
separate
forecasts [\fIX\fR\d\fIi\fR\u] have to be corrected by taking into account the
aggregated forecasts \fIX\fR\d\fIT\fR\u. We know that in general
\v'6p'
.RT
.ad r
.ad b
.RT
.PP
Let the corrections of [\fIX\fR\d\fIi\fR\u] be [\fIX`\fI\d\fIi\fR\u],
and the corrected aggregated forecast then be
\fIX`\fI\d\fIT\fR\u= \(*s" \fIX`\fI\d\fIi\fR\u.
.PP
The procedure for finding [\fIX`\fI\d\fIi\fR\u] is described in
Annex C.
.bp
.RT
.sp 2P
.LP
\fB6\fR 	\fBForecasting methods when observations are missing\fR 
.sp 1P
.RT
.sp 1P
.LP
6.1
	\fIIntroduction\fR 
.sp 9p
.RT
.PP
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. 
.PP
All smoothing models are applied on equally spaced observations. Also autoregressive 
integrated moving average (ARIMA)\(hymodels operate on equally 
spaced time series, while regression
models work on irregularly spaced observations without modifications.
.PP
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. 
.RT
.LP
.sp 1P
.LP
6.2
	\fIAdjustment procedure based on comparable observations\fR 
.sp 9p
.RT
.PP
In situations when some observations are missing, it may be
possible to use related data for estimating the missing observations. 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 measurements on a given trunk group to a certain
degree explain traffic measurements on other trunk groups.
.PP
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.
.PP
Suppose that a time series \fIx\fR\d\fIt\fR\uof equidistant observations 
from 1 to \fIn\fR  | as an inside gap | (mu | fIx\fR\d\fIt\fR\uis, for 
instance, the yearly increase. The gap consists of \fIk\fR missing observations 
between\ \fIr\fR and\ \fIr\fR + \fIk\fR + 1. 
.PP
A procedure for estimating the missing observations is given by the
following steps:
.RT
.LP
	i)
	Examine similar time series to the series with missing
observations and calculate the cross correlation.
.LP
	ii)
	Identify time series with high cross correlation at lag
zero.
.LP
	iii)
	 Calculate the growth factor ?63\fI\fI\d\fIr\fR\\d+\u\fI\fI\d\fIi\fR\ubetween 
\fIr\fR and \fIr\fR + \fIk\fR of the similar time series\ \fIy\fR\d\fIt\fR\u: 
\v'6p'
.ad r
.ad b
.RT
.LP
	iv)
	Estimates of the missing observations are then given
by:
\v'6p'
.ce 1000
\fI\*^x\fI\fI\d\fIr\fR\\d+\u\fI\fI\d\fIi\fR\u= \fIx\fR\d\fIr\fR\u+ ?63\fI\fI\d\fIr\fR\\d+\u\fI\fI\d\fIi\fR\u(\fIx 
\dr\fR +\fIk\fR +1
\u \(em \fIx\fR\d\fIr\fR\u)\ \ 
\fIi\fR = 1, 2, . |  |  \fIk\fR 
.ce 0
.ad r
(6\(hy2)
.ad b
.RT
.LP
.sp 1
.LP
	\fIExample\fR 
.PP
Suppose we want to forecast the time series \fIx\fR\d\fIt\fR\u. The
series is observed from 1\ to 10, but the observations 
at time\ 6, 7 and\ 8 are missing. However a related time series\ \fIy\fR\d\fIt\fR\uis 
measured. The measurements 
are given 
in Table\ 1/506.
.RT
.ce
\fBH.T. [T1.506]\fR 
.ce
TABLE\ 1/E.506
.ce
\fBMeasurements of two related time series; one with missing
.ce
\fBobservations\fR 
.ps 9
.vs 11
.nr VS 11
.nr PS 9
.TS
center box;
cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) .
\fIt\fR	1	2	3	4	5	6	7	8	9	10
_
.T&
cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) .
\fIx\fI	100	112	125	140	152	\(em	\(em	\(em	206	221
.T&
cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) .
\fIy\fI	300	338	380	422	460	496	532	574	622	670
_
.TE
.nr PS 9
.RT
.ad r
\fBTable 1/E.506 [T1.506], p. 
.sp 1P
.RT
.ad b
.RT
.LP
.bp
.PP
The last known observation of \fIx\fR\d\fIt\fR\ubefore the gap at time 
5 is 152, while the first known observation after the gap at time\ 9 is\ 
206. 
.PP
Hence \fIr\fR = 5 and \fIk\fR = 3. The calculation gives:
\v'6p'
.RT
.ad r
.ad b
.RT
.sp 1P
.LP
6.3
	\fIModification of forecasting models\fR 
.sp 9p
.RT
.PP
The other possibility for handling missing observations is to
extend the forecasting models with specific procedures. When observations 
are missing, a modified procedure, instead of the ordinary forecasting 
model, is 
used to estimate the traffic.
.PP
To illustrate such a procedure we look at simple exponential
smoothing. The simple exponential smoothing model is expressed by:
\v'6p'
.RT
.ce 1000
\(*\*^m\fI\fI\d\fIt\fR\u= (1 \(em \fIa\fR ) \fIy\fR\d\fIt\fR\u+ \fIa\fR 
\(*\*^m\fI\fI\d\fIt\fR\\d\\u(em\d1\u 
.ce 0
.ad r
(6\(hy3)
.ad b
.RT
.LP
.sp 1
where
.LP
	\fIy\fR\d\fIt\fR\uis the measured traffic at time \fIt\fR 
.LP
	\(*\*^m\fI\fI\d\fIt\fR\uis the estimated level at time \fIt\fR 
.LP
	\fIa\fR  | s the discount factor [and (1 \(em \fIa\fR ) is the smoothing
parameter].
.PP
Equation (6\(hy3) is a recursive formula. The recursion starts at
time 1 and ends at \fIn\fR  | if no observation is missing. Then a one 
step ahead 
forecast is given by:
\v'6p'
.ce 1000
\fI\*^y\fI\d\fIt\fR\u(1) = \(*\*^m\fI\fI\d\fIt\fR\u
.ce 0
.ad r
(6\(hy4)
.ad b
.RT
.PP
.sp 1
If some observations lying in between 1 and \fIn\fR  | are missing, then it
is necessary to modify the recursion procedure. Suppose now that \fIy\fR\d1\u, 
\fIy\fR\d2\u, . |  | , \fIy\fR\d\fIr\fR\u, \fIy 
\dr\fR +\fIk\fR +1
\u, \fIy
\dr\fR +\fIk\fR +2
\u,  . |  | ,
\fIy\fR\d\fIn\fR\uare known and \fIy\fR\d\fIr\fR\\d+\\d1\u, \fIy\fR\d\fIr\fR\\d+\\d2\u, 
. |  | , 
\fIy\fR\d\fIr\fR\\d+\u\fI\fI\d\fIk\fR\uare unknown. Then the time series 
has a gap consisting of \fIk\fR missing observations. 
.PP
The following modified forecasting model for simple exponential
smoothing is proposed in Aldrin\ [2].
\v'6p'
.RT
.ad r
.ad b
.RT
.LP
where
\v'6p'
.ce 1000
\fIa\fR\d\fIk\fR\u= 
@ { fIa\fR } over { ~+~\fIk\fR (1\(em\fIa\fR )\u2\d } @ 
.ce 0
.ad r
(6\(hy6)
.ad b
.RT
.LP
.sp 1
.bp
.PP
By using the (6\(hy5) and (6\(hy6) it is possible to skip the recursive 
procedure in the gap between \fIr\fR and 
\fIr\fR \ +\ \fIk\fR \ +\ 1.
.PP
In Aldrin [2] similar procedures are proposed for the following
forecasting models:
.RT
.LP
	\(em
	Holt's method,
.LP
	\(em
	Double exponential smoothing,
.LP
	\(em
	Discounted least squares method with level and trend,
.LP
	\(em
	Holt\(hyWinters seasonal methods.
.PP
Wright [17] and [18] also suggests specific procedures to modify the smoothing 
models when observations are missing. 
.PP
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.
.PP
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]. 
.PP
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].
.PP
In Jones [9] and [10], Barham and Dunstan [4], Harvey and Pierse [8], Aldrin\ 
[2] and B\o"\(*s/"lviken\ [5] it is shown how the dynamic linear models 
and the Kalman Filter handle missing observations. 
\v'1P'
.RT
.ce 1000
ANNEX\ A
.ce 0
.ce 1000
(to Recommendation E.506)
.sp 9p
.RT
.ce 0
.ce 1000
\fBComposite strategy\fR 
.sp 1P
.RT
.ce 0
.LP
A.1
	\fIIntroduction\fR 
.sp 1P
.RT
.PP
This annex describes a method for estimating international
traffic based on monthly paid\(hyminutes and a number of conversion factors.
It demonstrates the method by examining the factors and showing their utility.
.PP
The method is seen to have two main features:
.RT
.LP
	1)
	Monthly paid\(hyminutes exchanged continuously between
Administrations for accounting purposes provide a large and
continuous volume of data.
.LP
	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.
.sp 2P
.LP
A.2
	\fIBasic procedure\fR 
.sp 1P
.RT
.sp 1P
.LP
A.2.1
	\fIGeneral\fR 
.sp 9p
.RT
.PP
The composite strategy is carried out for each stream,
for each direction and generally for each service category.
.bp
.PP
The estimated mean offered busy\(hyhour traffic (in Erlangs) is derived 
from the monthly paid\(hyminutes using the formula: 
\v'6p'
.RT
.ce 1000
\fIA\fR = \fIMdh\fR /60\fIe\fR 
.ce 0
.ad r
(A\(hy1)
.ad b
.RT
.LP
.sp 1
.LP
where
.LP
	\fIA\fR 	is the estimated mean traffic in Erlangs offered in the
busy hour,
.LP
	\fIM\fR 	is the total monthly paid\(hyminutes,
.LP
	 \fId\fR is the day/month ratio, i.e.\ the ratio of average weekday paid\(hytime 
to monthly paid\(hytime, 
.LP
	\fIh\fR 	is the busy\(hyhour/day ratio, i.e.\ the ratio of the
busy\(hyhour paid\(hytime to the average daily paid\(hytime,
.LP
	\fIe\fR 	is the efficiency factor, i.e.\ the ratio of busy\(hyhour
paid\(hytime to busy\(hyhour occupied\(hytime.
.sp 1P
.LP
A.2.2
	\fIMonthly paid\(hyminutes (M)\fR 
.FS
In a situation
where only yearly paid\(hyminutes are available, this may be converted 
to \fIM\fR by a suitable factor. 
.FE
.sp 9p
.RT
.PP
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 significant changes, whereas a shorter period (e.g.\ 
daily) not 
only increases the amount of data, but also increases the magnitude of
.PP
fluctuations from one period to the next. A further advantage of the one\(hymonth 
period is that monthly paid\(hyminute figures are exchanged between 
Administrations for accounting purposes and consequently historical records
covering many years are normally readily available.
.PP
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). 
.RT
.sp 1P
.LP
A.2.3
	\fIDay/month ratio (d)\fR 
.sp 9p
.RT
.PP
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. 
.PP
As the number of weekdays and non\(hyweekdays (weekends and holidays)
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. 
.PP
Hence if:
.RT
.LP
	\fIX\fR 	denotes the number of weekdays in the related month
.LP
	\fIY\fR 	denotes the number of non\(hyweekdays (weekend days and
holidays) in the selected month, then
\v'6p'
.ce 1000
[ Formula deleted ]
.ce 0
.ad r
(A\(hy2)
.ad b
.RT
.LP
.sp 1
.LP
where
\v'6p'
.sp 1P
.ce 1000
\fIr\fR = 
@ { verage~non\(hyweekday~traffic } over { verage~weekday~traffic } @ 
.ce 0
.sp 1P
.PP
.sp 1
The relative amount of non\(hyweekday traffic is very sensitive to
the relative amount of social contact between origin and destination. (Social 
calls, are, in general, made more frequently on weekends.) Since changes 
in 
such social contact would be very slow, \fIr\fR or \fId\fR are expected 
to be the most stable conversion factors, which in general vary only within 
relatively narrow limits. However, tariff policies such as reduced weekend 
rates can have a 
significant effect on \fIr\fR and\ \fId\fR .
.bp
.PP
When \fIr\fR 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 additional weekend (Sunday) traffic 
or adopting a suitable overflow routing arrangement. 
.RT
.sp 1P
.LP
A.2.4
	\fIBusy\(hyhour/day ratio (h)\fR 
.sp 9p
.RT
.PP
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 \*Qdegree of convenience\*U at origin and destination. However, 
sufficient discrepancies exist to warrant measuring the diurnal distribution, 
from which the busy\(hyhour/day ratio may be calculated. 
.PP
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 busy\(hyhour/day ratio 
from 10% for 0 to 2 hours time difference and up to 13.5% for 7 hours time 
difference. 
.PP
As described above, the composite strategy is
implemented as an accounting\(hybased procedure. However, it may be more 
practical for some Administrations to measure \fId\fR and \fIh\fR based 
on occupied time, derived from available call recording equipment. 
.RT
.LP
.sp 1P
.LP
A.2.5
	\fIEfficiency factor (e)\fR 
.sp 9p
.RT
.PP
The efficiency factor (ratio of busy\(hyhour paid time to busy\(hyhour
occupied time, \fIe\fR ) 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 example, the measurement of total circuit 
occupied time should include the occupied time for paid calls (time from 
circuit seizure to circuit 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.
.PP
There is a tendency for the efficiency to change with time. In this
regard, efficiency is mainly a function of operating method (manual,
semi\(hyautomatic, international subscriber dialling), the B\(hysubscriber's
availability, and the quality of the distant network.
.PP
Forecasts of the efficiency can be made on the basis of extrapolation of 
past trends together with adjustments for planned improvements. 
.PP
The detailed consideration of efficiency is also an advantage from an operational 
viewpoint in that it may be possible to identify improvements that may 
be made, and quantify the benefits deriving from such improvements. 
.PP
It should be noted that the practical limit for \fIe\fR is generally about 
0.8 to 0.9 for automatic working. 
.RT
.sp 1P
.LP
A.2.6
	\fIMean offered busy hour traffic (A)\fR 
.sp 9p
.RT
.PP
It should be noted that \fIA\fR  | s the mean offered busy\(hyhour traffic 
expressed in Erlangs. 
.RT
.sp 1P
.LP
A.2.7
	\fIUse of composite strategy\fR 
.sp 9p
.RT
.PP
In the case of countries with lower traffic volumes and manual
operation, the paid\(hytime factors (\fId\fR \ and\ \fIh\fR ) would be 
available from analysis of call vouchers (dockets). For derivation of the 
efficiency\ \fIe\fR , the manual 
operator would have to log the busy\(hyhour occupied time as well as the 
paid time during the sampling period. 
.PP
In countries using stored\(hyprogram controlled exchanges with
associated manual assistance positions, computer analysis may aid the composite 
forecasting procedure. 
.PP
One consequence of the procedure is that the factors \fId\fR  | nd \fIh\fR 
 | 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, \fIe\fR , during 
the busy hour, should also be emphasized. 
\v'1P'
.bp
.RT
.ce 1000
ANNEX\ B
.ce 0
.ce 1000
(to Recommendation E.506)
.sp 9p
.RT
.ce 0
.ce 1000
\fBExample using weighted least squares method\fR 
.sp 1P
.RT
.ce 0
.LP
B.1
	\fITelex data\fR 
.sp 1P
.RT
.PP
The telex traffic between the following countries has been
analyzed:
.RT
.LP
	\(em
	Germany (D)
.LP
	\(em
	Denmark (DNK)
.LP
	\(em
	USA (USA)
.LP
	\(em
	Finland (FIN)
.LP
	\(em
	Norway (NOR)
.LP
	\(em
	Sweden (S)
.PP
The data consists of yearly observations from 1973 to 1984\ [19].
.LP
.sp 1P
.LP
B.2
	\fIForecasting\fR 
.sp 9p
.RT
.PP
Before using the weighted least squares method, separate forecasts for 
the traffic matrix have to be made. In this example a simple ARIMA (0,2,1) 
model with logarithmic transformed observations 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.
.PP
Forecasts for 1984 based on observations from 1973 to 1983 are given in 
Table\ B\(hy1/E.506. 
.RT
.ce
\fBH.T. [T2.506]\fR 
.ce
TABLE\ B\(hy1/E.506
.ce
\fBForecasts for telex traffic between Germany\fR 
.ce
\fB(D),
.ce
Denmark\fR 
.ce
\fB(DNK),\fR 
.ce
\fBUSA\fR 
.ce
\fB(USA), Finland\fR 
.ce
\fB(FIN), Norway\fR 
.ce
\fB(NOR) and
.ce
Sweden\fR 
.ce
\fB(S) in 1984\fR 
.ps 9
.vs 11
.nr VS 11
.nr PS 9
.TS
center box;
lw(36p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(30p) | cw(30p) .
From  To	D	DNK	USA	FIN	NOR	S	Sum	Forecasted sum
_
.T&
cw(36p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(30p) | cw(30p) .
D	\(em	4869	12 | 30	2879	2397	\ 5230	28 | 05	27 | 88
.T&
cw(36p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(30p) | cw(30p) .
DNK	\ 5196	\(em	\ 1655	\ 751	1270	\ 1959	10 | 31	10 | 05
.T&
cw(36p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(30p) | cw(30p) .
USA	11 | 03	1313	\(em	\ 719	1657	\ 2401	17 | 93	17 | 09
.T&
cw(36p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(30p) | cw(30p) .
FIN	\ 2655	\ 715	\ \ 741	\(em	\ 489	\ 1896	\ 6496	\ 6458
.T&
cw(36p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(30p) | cw(30p) .
NOR	\ 2415	1255	\ 1821	\ 541	\(em	\ 1548	\ 7580	\ 7597
.T&
cw(36p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(30p) | cw(30p) .
S	\ 4828	1821	\ 2283	1798	1333	\(em	12 | 63	12 | 53
_
.T&
cw(36p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(30p) | cw(30p) .
Sum	26 | 97	9973	19 | 30	6688	7146	13 | 34		
_
.T&
cw(36p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(18p) | cw(30p) | cw(30p) .
Forecasted sum	26 | 97	9967	19 | 53	6659	7110	12 | 14		
_
.TE
.nr PS 9
.RT
.ad r
\fBTableau B\(hy1/E.506 [T2.506], p.\fR 
.sp 1P
.RT
.ad b
.RT
.PP
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. 
.PP
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. 
.bp
.RT
.sp 1P
.LP
B.3
	\fIAdjustment of the traffic matrix forecasts\fR 
.sp 9p
.RT
.PP
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\ B\(hy2/E.506, while the weights are based on the mean squared 
one step ahead forecasting errors. 
.PP
Let \fIy\fR\d\fIt\fR\u | be the traffic at time \fIt\fR . The ARIMA (0,2,1) 
model with logarithmic transformed data is given by: 
\v'6p'
.RT
.sp 1P
.ce 1000
\fIz\fR\d\fIt\fR\u= (1 \(em B)\u2\d | n \fIy\fR\d\fIt\fR\u= (1 \(em \(*hB) 
\fIa\fR\d\fIt\fR\u 
.ce 0
.sp 1P
.LP
.sp 1
or
\v'6p'
.sp 1P
.ce 1000
\fIz\fR\d\fIt\fR\u= \fIa\fR\d\fIt\fR\u\(em \(*h\fIa\fR\d\fIt\fR\\d\\u(em\d1\u
.ce 0
.sp 1P
.LP
.sp 1
.LP
where
.LP
	 \fIz\fR\d\fIt\fR\u= ln \fIy\fR\d\fIt\fR\u\(em 2 ln \fIy\fR\d\fIt\fR\\d\\u(em\d1\u+ 
ln \fIy\fR\d\fIt\fR\\d\\u(em\d2\u 
.LP
	\fIa\fR\d\fIt\fR\u	is white noise,
.LP
	\(*h 
	is a parameter,
.LP
	\fIB\fR 	is the backwards shift operator.
.PP
The mean squared one step ahead forecasting error of \fIz\fR\d\fIt\fR\u |
is:
\v'6p'
.sp 1P
.ce 1000
\fIMSQ\fR = 
[Formula Deleted]
.sp 1P
.LP
.sp 1
where
.LP
	\fI/*^z\fI\d\fIt\fR\\d\\u(em\d1\u(1) is the one step ahead forecast.
.PP
The results of using the weighted least squares method is found in Table 
B\(hy3/E.506 and show that the factors in Table\ B\(hy1/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. 
.LP
.sp 1
.ce
\fBH.T. [T3.506]\fR 
.ce
TABLE\ B\(hy2/E.506
.ce
\fBInverse weights as mean as squared one step ahead forecasting\fR 
.ce

.ce
\fBerrors\fR 
.ce
\fBof telex traffic (100\fR\(ua\fB\(em\fB\(ua\fB4) between\fR 
.ce
\fBGermany\fR 
.ce
\fB(D), Denmark\fR 
.ce
\fB(DNK),\fR 
.ce
\fBUSA\fR 
.ce
\fB(USA), Finland\fR 
.ce
\fB(FIN),\fR 
.ce
\fBNorway\fR 
.ce
\fB(NOR) and Sweden\fR 
.ce
\fB(S) in 1984\fR 
.ps 9
.vs 11
.nr VS 11
.nr PS 9
.TS
center box;
lw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
From  To	D	DNK	USA	FIN	NOR	S	Sum
_
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
D	\(em	28.72	\ 13.18	11.40	\ 8.29	44.61	\ 7.77
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
DNK	\ 5.91	\(em	\ 43.14	18.28	39.99	18.40	10.61
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
USA	23.76	39.19	\(em	42.07	50.72	51.55	21.27
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
FIN	23.05	12.15	\ 99.08	\(em	34.41	19.96	17.46
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
NOR	21.47	40.16	132.57	24.64	\(em	17.15	20.56
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
S	\ 6.38	12.95	\ 28.60	28.08	\ 8.76	\(em	\ 6.48
_
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
Sum	\ 6.15	\ 3.85	\ 14.27	\ 9.55	12.94	\ 8.53	
_
.TE
.nr PS 9
.RT
.ad r
\fBTableau B\(hy2/E.506 [T3.506], p.4 
.sp 1P
.RT
.ad b
.RT
.LP
.bp
.ce
\fBH.T. [T4.506]\fR 
.ce
TABLE\ B\(hy3/E.506
.ce
\fBAdjusted telex forecasts using the weighted least
.ce
squares method\fR 
.ps 9
.vs 11
.nr VS 11
.nr PS 9
.TS
center box;
lw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
From  To	D	DNK	USA	FIN	NOR	S	Sum
_
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
D	\(em	4850	12 | 84	2858	2383	\ 5090	27 | 65
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
DNK	\ 5185	\(em	\ 1674	\ 750	1257	\ 1959	10 | 25
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
USA	11 | 01	1321	\(em	\ 717	1644	\ 2407	17 | 90
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
FIN	\ 2633	\ 715	\ \ 745	\(em	\ 487	\ 1891	\ 6471
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
NOR	\ 2402	1258	\ 1870	\ 540	\(em	\ 1547	\ 7617
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
S	\ 4823	1817	\ 2307	1788	1331	\(em	12 | 66
_
.T&
cw(30p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(24p) | cw(30p) .
Sum	26 | 44	9961	19 | 80	6653	7102	12 | 94	
_
.TE
.nr PS 9
.RT
.ad r
\fBTableau B\(hy3/E.506 [T4.506], p.5 
.sp 1P
.RT
.ad b
.RT
.ce 1000
ANNEX\ C
.ce 0
.ce 1000
(to Recommendation E.506)
.sp 9p
.RT
.ce 0
.ce 1000
\fBDescription of a \fR \fBtop down procedure\fR 
.sp 1P
.RT
.ce 0
.PP
Let
.sp 1P
.RT
.LP
	\fIX\fR\d\fIT\fR\u	be the traffic forecast on an aggregated level,
.LP
	\fIX\fR\d\fIi\fR\u	be the traffic forecast to country \fIi\fR ,
.LP
	\(*\*^s\fI\fI\d\fIT\fR\u	the estimated standard deviation of the aggregated
forecast,
.LP
	\(*\*^s\fI\fI\d\fIi\fR\u	the estimated standard deviation of the forecast
to country \fIi\fR .
.LP
Usually
\v'6p'
.ce 1000
\fIX
\dT\u\fR \(!= 
@ pile { sum above \fIi\fR } @ \fIX\fR\d\fIi\fR\u,
.ce 0
.ad r
(C\(hy1)
.ad b
.RT
.LP
.sp 1
so that it is necessary to find a correction
\v'6p'
.sp 1P
.ce 1000
[\fIX\fR `
\fI\fI\d\fIi\fR\u] of [\fIX\fR\d\fIi\fR\u]
and [\fIX\fR `
\fI\fI\d\fIT\fR\u] of [\fIX\fR\d\fIT\fR\u]
.ce 0
.sp 1P
.LP
.sp 1
by minimizing the expression
\v'6p'
.ad r
.ad b
.RT
.LP
subject to
\v'6p'
.ad r
.ad b
.RT
.LP
where \(*a and [\(*a\fI\fI\d\fIi\fR\u] are chosen to be
\v'6p'
.ad r
.ad b
.RT
.LP
.bp
.PP
The solution of the optimization problem gives the values
[\fIX\fR `
\fI\fI\d\fIi\fR\u]:
\v'6p'
.ad r
.ad b
.RT
.PP
A closer inspection of the data base may result in other
expressions for the coefficients [\(*a\fI\fI\d\fIi\fR\u], \fIi\fR \ =\ 
0,\ 1,\ . |  |  On some 
occasions, it will also be reasonable to use other criteria for finding the
corrected forecasting values [\fIX\fR `
\fI\fI\d\fIi\fR\u]. This is shown in   the top down example in Annex\ D.
.PP
If, on the other hand, the variance of the top forecast \fIX\fR\d\fIT\fR\uis
fairly small, the following procedure may be chosen:
.PP
The corrections [\fIX\fR\d\fIi\fR\u] are found by minimizing the
expression
\v'6p'
.RT
.ad r
.ad b
.RT
.LP
subject to
\v'6p'
.ad r
.ad b
.RT
.PP
If \(*a\fI\fI\d\fIi\fR\u, \fIi\fR \ =\ 1,\ 2,\ . |  |  is chosen to be 
the inverse of the estimated variances, the solution of the optimization 
problem is given 
by
\v'6p'
.ad r
\v'6p'
.ad b
.RT
.ce 1000
ANNEX\ D
.ce 0
.ce 1000
(to Recommendation\ E.506)
.sp 9p
.RT
.ce 0
.ce 1000
\fBExample of a \fR \fBtop down modelling method\fR 
.sp 1P
.RT
.ce 0
.PP
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. 
.sp 1P
.RT
.LP
.sp 1P
.LP
D.1
	\fIEconometric model\fR \fIof the total traffic from Norway to\fR 
\fIEurope\fR 
.sp 9p
.RT
.PP
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. 
.PP
The general model may be written:
\v'6p'
.RT
.ce 1000
\fIX
\dt\u\fR = e\fI
\uK\d\fR \(mu \fIGNP\fR $$Ei:\fIa\fR :\fIt\fR _ \(mu \fIP\fR $$Ei:\fIb\fR 
:\fIt\fR _ 
\(mu \fIA\fR $$Ei:\fIc\fR :\fIt\fR _ \(mu e\fI
\uu
t
\d\fR \ \ \ (\fIt\fR \ =\ 1,\ 2, . |  | ,
\fIN\fR )
.ce 0
.ad r
(D\(hy1)
.ad b
.RT
.LP
.sp 1
where:
.LP
	 \fIX\fR\d\fIt\fR\u is the demand for telephone traffic from Norway to 
Europe at 
time \fIt\fR (charged minutes).
.LP
	\fIGNP\fI\d\fIt\fR\u	is the gross national product in Norway at
time \fIt\fR (real prices).
.LP
	 \fIP\fR\d\fIt\fR\u is the index of charges for traffic from Norway to 
Europe at 
time \fIt\fR (real prices).
.bp
.LP
	 \fIA\fR\d\fIt\fR\u is the percentage direct\(hydialled 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)
\fIA\fR\d\fIt\fR\ugoes from 1 to 2 instead of from 0 to 1.
.LP
	\fIK\fR 	is the constant.
.LP
	\fIa\fR 	is the elasticity with respect to \fIGNP\fR .
.LP
	\fIb\fR 	is the price elasticity.
.LP
	\fIc\fR 	is the elasticity with respect to automation.
.LP
	\fIu\fR\d\fIt\fR\u	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 \fIu\fR\d\fIt\fR\u\ =\ 0
and var\ \fIu\fR\d\fIt\fR\u\ =\ \(*s\u2\d).
.PP
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\ D\(hy1/E.506 (in our calculations 
we have used data for the period 1951\(hy1980). 
.PP
The \fIt\fR  | tatistics should be compared with the Student's
Distribution with \fIN\fR  | (em | fId\fR degrees of freedom, where \fIN\fR 
is the number of 
observations and \fId\fR is the number of estimated parameters. In this 
example, 
\fIN\fR \ =\ 30 and\ \fId\fR \ =\ 4.
.PP
The model \*Qexplains\*U 99.7% of the variation in the demand for
telephone traffic from Norway to Europe in the period 1951\(hy1980.
.PP
From this logarithmic model it can be seen that:
.RT
.LP
	\(em
	an increase in GNP of 1% causes an increase in the
telephone traffic of 2.80%,
.LP
	\(em
	an increase of 1% in the charges, measured in real
prices, causes a decrease in the telephone traffic of 0.26%,
and 
.LP
	\(em
	an increase of 1% in \fIA\fR\d\fIt\fR\ucauses an increase in
the traffic of 0.29%.
.PP
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:
\v'6p'
.ce 1000
\fIX
\dt\u\fR = e\fI
\dt\u\fR \u\(em16.095
\d \(mu \fIGNP
\dt\u\fR \u2.80
\d \(mu
\fIP
\dt\u\fR u
\u\(em0.26
\d \(mu \fIA
\dt\u\fR \u0.29
\d
.ce 0
.ad r
(D\(hy2)
.ad b
.RT
.LP
.sp 1
.ce
\fBH.T. [T5.506]\fR 
.ce
TABLE\ D\(hy1/E.506
.ps 9
.vs 11
.nr VS 11
.nr PS 9
.TS
center box;
cw(60p) | cw(60p) | cw(60p) .
Coefficients	Estimated  values	\fIt\fR  | statistics
_
.T&
cw(60p) | cw(60p) | cw(60p) .
K	\(em16.095	\(em4.2
.T&
cw(60p) | cw(60p) | cw(60p) .
a	\(em \ 2.799	\(em 8.2
.T&
cw(60p) | cw(60p) | cw(60p) .
b	\(em\ 0.264	\(em1.0
.T&
cw(60p) | cw(60p) | cw(60p) .
c	\(em \ 0.290	\(em 2.1
_
.TE
.nr PS 9
.RT
.ad r
\fBTable D\(hy1/E.506  [T5.506], p. 
.sp 1P
.RT
.ad b
.RT
.LP
.sp 1
.sp 1P
.LP
D.2
	\fIModel for breakdown of the total traffic from Norway to\fR 
\fIEurope\fR 
.sp 9p
.RT
.PP
The method of breakdown is first to apply the trend to forecast 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 countries, so that the sum of the traffic to each country 
equals the 
forecasted total traffic to Europe from equation\ (D\(hy2).
.bp
.PP
Mathematically, the breakdown model can be expressed as follows:
.PP
\fICalculation of the trend for country i:\fR \v'6p'
.RT
.ce 1000
\fIR
\dit
\u\fR = \fIb
\di\u\fR + \fIa
\di\u\fR \(mu \fIt\fR ,\ \ \fIi\fR = 1, . |  | ,
34\ \ \fIt\fR = 1, . |  | , \fIN\fR 
.ce 0
.ad r
(D\(hy3)
.ad b
.RT
.LP
.sp 1
where
.LP
	\fIR
\dit
\u\fR = 
@ { fIX~\dit~\u\fR } over { fIX~\dt\u\fR } @ ,
i.e country \fIi\fR 's share of the total traffic to Europe.
.LP
	\fIX
\dit
\u\fR 	is the traffic to country \fIi\fR at time \fIt\fR 
.LP
	\fIX\fR\d\fIt\fR\u	is the traffic to Europe at time \fIt\fR 
.LP
	\fIt\fR 	is the trend variable
.LP
\fIa\fR\d\fIi\fR\uand \fIb\fR\d\fIi\fR\uare two coefficients specific to 
country \fIi\fR ; i.e. \fIa\fR\d\fIi\fR\uis country \fIi\fR 's trend. The 
coefficients are estimated by using regression 
analysis, and we have based calculations on observed traffic for the period
1966\(hy1980.
.PP
\fIThe forecasted shares\fR  | or country \fIi\fR  | s then calculated
by
\v'6p'
.ce 1000
\fIR\fR \d\fIit\fR \u = \fIR\fR \d\fIiN\fR \u + \fIa
\di
\u\fR \(mu (\fIt\fR \(em \fIN\fR ) \(mu
e
\s7\(em
@ { fIt\fR~\(em5 } over { 0 } @ 
\s9
.RT
.ce 0
.ad r
(D\(hy4)
\v'5p'
.ad b
.RT
.LP
.sp 1
where \fIN\fR  | s the last year of observation, and e is the exponential
function.
\v'4p'
.PP
The factor e
\s7\(em
@ { fIt\fR~\(em5 } over { 0 } @ 
\s9is
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\ 
(D\(hy6). 
.RT
.PP
To have the sum of the countries' shares equal one, it is necessary
that
\v'6p'
.RT
.ce 1000
@ pile { sum above \fIi\fR } @ \fIR
\dit
\u\fR = 1
.ce 0
.ad r
(D\(hy5)
\v'8p'
.ad b
.RT
.LP
.sp 1
.PP
This we obtain by setting the adjusted share, \fI\*~R
\dit
\u\fR ,
equal to
\v'6p'
.ad r
.ad b
.RT
.PP
Each country's forecast traffic is then calculated by multiplying the total 
traffic to Europe, \fIX\fR\d\fIt\fR\u, by each country's share of the 
total traffic:
\v'6p'
.ce 1000
\fIX
\dit
\u\fR = \fI\*~R
\dit
\u\fR \(mu \fIX
\dt\u\fR 
.ce 0
.ad r
(D\(hy7)
.ad b
.RT
.LP
.sp 1
.sp 1P
.LP
D.3
	\fIEconometric model for telephone traffic from Norway to Central\fR 
\fIand South America, Africa, Asia, and Oceania\fR .
.sp 9p
.RT
.PP
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. 
.PP
After using cross\(hysection/time\(hyseries simultaneous estimation we 
have arrived at the coefficients in Table\ D\(hy2/E.506 for the forecasting 
model for 
telephone traffic from Norway to these continents (for each continent we 
have based our calculations on data for the period 1961\(hy1980): 
.bp
.RT
.LP
.ce
\fBH.T. [T6.506]\fR 
.ce
TABLE\ D\(hy2/E.506
.ps 9
.vs 11
.nr VS 11
.nr PS 9
.TS
center box;
cw(60p) | cw(60p) | cw(60p) .
Coefficients	Estimated  values	\fIt\fR  | statistics
_
.T&
lw(60p) | cw(60p) | cw(60p) .
Charges	\(em1.930	\(em5.5
.T&
lw(60p) | cw(60p) | cw(60p) .
Telephone stations	\(em 2.009	\(em 4.2
.T&
lw(60p) | cw(60p) | cw(60p) .
Automation	\(em 0.5\ \ 	\(em \ \(em\ 
_
.TE
.nr PS 9
.RT
.ad r
\fBTable D\(hy2/E.506 [T6.506], p. 
.sp 1P
.RT
.ad b
.RT
.LP
We then have \fIR\fR \u2\d = 0.96. The model may be written:
\v'6p'
.ce 1000
\fIX\fR \fI
\uk\d\fR \fI
\dt\u\fR = e\fI
\uK\d\fR \(mu
(\fITS\fR \fI
\uk\d\fR \fI
\dt\u\fR )
\u2.009
\d \(mu
(\fIP\fR \fI
\uk\d\fR \fI
\dt\u\fR )
\u1.930
\d \(mu
(\fIA
\uk\d\fR \fI
\dt\u\fR )
\u0.5
\d
.ce 0
.ad r
(D\(hy8)
.ad b
.RT
.LP
.sp 1
where
.LP
	\fIX\fR \fI
\uk\d\fR \fI
\dt\u\fR 	is the telephone traffic to
continent \fIk\fR (\fIk\fR = Central America, . |  | , Oceania) at time 
\fIt\fR , 
.LP
	e\fI\fI
\u\fIK\fR\d	is the constant specific to each continent. For
telephone traffic from Norway to:
.LP
	Central\ America:
	\fIK\fR \u1\d\ =\ \(em11.025
.LP
	South\ America:
	\fIK\fR \u2\d\ =\ \(em12.62
.LP
	Africa:
	\fIK\fR \u3\d\ =\ \(em11.395
.LP
	Asia:
	\fIK\fR \u4\d\ =\ \(em15.02
.LP
	Oceania:
	\fIK\fR \u5\d\ =\ \(em13.194
.LP
	\fITS
\uk\d\fR \fI
\dt\u\fR 	is the number of telephone stations
within continent\ \fIk\fR at time\ \fIt\fR ,
.LP
	\fIP\fR \fI
\uk\d\fR \fI
\dt\u\fR is the index of charges, measured in real prices, to continent\ 
\fIk\fR at time\ \fIt\fR , and 
.LP
	\fIA
\uk\d\fR \fI
\dt\u\fR 	is the percentage direct\(hydialled telephone
traffic to continent\ \fIk\fR .
.PP
Equation (D\(hy8) is now used \(em 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 \(em to forecast the future development in
telephone traffic from Norway to the continent.
.LP
.sp 2P
.LP
	\fBReferences\fR 
.sp 1P
.RT
.LP
[1]
	ABRAHAM (A.) and LEDOLTER (J.): Statistical methods for forecasting.
\fIJ. Wiley\fR , New York, 1983.
.LP
[2]
	ALDRIN (M.): Forecasting time series with missing observations.
\fIStat 15/86 Norwegian Computing Center\fR , 1986.
.LP
[3]
	ANSLEY (C. | .) and KOHN (R.): Estimation, filtering and smoothing in
state space models with incomplete specified initial conditions.
\fIThe Annals of Statistics\fR , 13, pp.\ 1286\(hy1316, 1985.
.LP
[4]
	BARHAM (S. | .) and DUNSTAN (F. | . | .): Missing values in time series.
\fITime Series Analysis: Theory and Practice\ 2\fR : Anderson,\ O. | ., ed.,
pp.\ 25\(hy41, North Holland, Amsterdam, 1982.
.LP
[5]
	B\o"\(*s/"LVIKEN (E.): Forecasting telephone traffic using Kalman Filtering:
Theoretical considerations. \fIStat 5/86 Norwegian Computing Center\fR , 1986.
.LP
[6]
	CHEMOUIL (P.) and GARNIER (B.): An adaptive short\(hyterm traffic
forecasting procedure using Kalman Filtering.
\fIXI\ International Teletraffic Congress\fR , Kyoto, 1985.
.LP
[7]
	HARRISON (P. | .) and STEVENS (C. | .): Bayesian forecasting.
\fIJournal of Royal Statistical Society\fR . Ser\ B\ 37, pp.\ 205\(hy228, 1976.
.LP
[8]
	HARVEY (A. | .) and PIERSE (R. | .): Estimating missing observations in
econometric time series. \fIJournal of American Statistical As.\fR ,
79, pp.\ 125\(hy131, 1984.
.LP
[9]
	 JONES (R. | .): Maximum likelihood fitting of ARMA models to time series 
with missing observations. \fITechnometrics\fR , 22, No.\ 3, pp.\ 389\(hy396, 
1980. 
.LP
[10]
	JONES (R. | .): Time series with unequally spaced data.
\fIHandbook of Statistics 5.\fR ed.\ Hannah,\ E. | ., et\ al., pp.\ 157\(hy177,
North Holland, Amsterdam, 1985.
.LP
[11]
	KRUITHOF (J.): Telefoonverkeersrekening. \fIDe Ingenieur\fR , 52, No.\ 8,
1937.
.bp
.LP
[12]
	 MORELAND (J. | .): A robust sequential projection algorithm for traffic 
load forecasting. \fIThe Bell Technical Journal\fR , 61, pp.\ 15\(hy38, 
1982. 
.LP
[13]
	PACK (C. | .) and WHITAKER (B. | .): Kalman Filter models for network
forecasting. \fIThe Bell Technical Journal\fR , 61, pp.\ 1\(hy14, 1982.
.LP
[14]
	STORDAHL (K.) and HOLDEN (L.): Traffic forecasting models based on
top down and bottom up models. \fIITC\ 11\fR , Kyoto, 1985.
.LP
[15]
	SZELAG (C. | .): A short\(hyterm forecasting algorithm for trunk demand
servicing. \fIThe Bell Technical Journal\fR , 61, pp.\ 67\(hy96, 1982.
.LP
[16]
	TU (M.) and PACK (D.): Improved forecasts for local
telecommunications network. 6th\ International Forecasting Symposium,
Paris, 1986.
.LP
[17]
	WRIGHT (D. | .): Forecasting irregularly spaced data: An extension of
double exponential smoothing. \fIComputer and Engineering\fR ,
10, pp.\ 135\(hy147, 1986.
.LP
[18]
	 WRIGHT (D. | .): Forecasting data published at irregular time intervals 
using an extension of Holt's method. \fIManagement science\fR , 32, 
pp.\ 499\(hy510, 1986.
.LP
[19]
	\fITable of international telex relations and traffic\fR , ITU,
Geneva, 1973\(hy1984.
.sp 2P
.LP
\fBRecommendation E.507\fR 
.FS
The old Recommendation E.506 which appeared  in the \fIRed Book \fR was
split into two Recommendations, revised E.506 and new E.507,
and considerable new material was added to both.
.FE
.RT
.sp 2P
.sp 1P
.ce 1000
\fBMODELS\ FOR\ \fR \fBFORECASTING\ INTERNATIONAL\ TRAFFIC\fR 
.EF '%	Fascicle\ II.3\ \(em\ Rec.\ E.507''
.OF '''Fascicle\ II.3\ \(em\ Rec.\ E.507	%'
.ce 0
.sp 1P
.LP
\fB1\fR 	\fBIntroduction\fR 
.sp 1P
.RT
.PP
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 Recommendation is to present
some of the basic ideas and leave the explanation 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.
.PP
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. 
.PP
In addition the Recommendation describes various methods for
evaluation of forecasting models and choice of model.
.RT
.sp 2P
.LP
\fB2\fR 	\fBBuilding the forecasting model\fR 
.sp 1P
.RT
.PP
This procedure can conveniently be described as four consecutive
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, autoregressive 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 variables have significant impact on the 
traffic demand, one ought to include them in the forecasting models, provided 
enough 
historical data are available.
.PP
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 identifying 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
estimates 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. 
.PP
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 tentative models may in turn be estimated 
and subjected to diagnostic checking. 
.bp
.PP
In Figure\ 1/E.507 the steps in the model building procedure are
illustrated.
.RT
.LP
.rs
.sp 18P
.ad r
\fBFigure 1/E.507, p.\fR 
.sp 1P
.RT
.ad b
.RT
.sp 2P
.LP
\fB3\fR 	\fBVarious forecasting models\fR 
.sp 1P
.RT
.PP
The objective of \(sc\ 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.
.RT
.sp 1P
.LP
3.1
	\fICurve fitting models\fR 
.sp 9p
.RT
.PP
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. 
.PP
The following are examples of common curve fitting models used for
forecasting international traffic:
\v'6p'
.RT
.ad r
	Linear:
	\fIY\fR\d\fIt\fR\u\ =\ \fIa\fR + \fIbt\fR 	(3\(hy1)
.ad b
.RT
.ad r
	Parabolic:
	\fIY\fR\d\fIt\fR\u\ =\ \fIa\fR + \fIbt\fR +
\fIct\fR \u2\d
	(3\(hy2)
.ad b
.RT
.ad r
	Exponential:
	\fIY\fR\d\fIt\fR\u\ =\ \fIae\fR \fI\fI
\u\fIt\fR\d	(3\(hy3)
.ad b
.RT
.ad r
	Logistic:
	\fIY\fR\d\fIt\fR\u\ =\ \fI
[Formula Deleted]
\v'7p'
.ad b
.RT
.ad r
	Gompertz:
	\fIY\fR\d\fIt\fR\u\ =\ \fIM\fR (\fIa\fR )\fI\fI
\u\fIt\fR\d	(3\(hy5)
.ad b
.RT
.LP
.sp 1
.LP
where
.LP
	\fIY\fR\d\fIt\fR\uis the traffic at time \fIt\fR ,
.LP
	\fIa, b, c\fR  | are parameters,
.LP
	\fIM\fR  | is a parameter describing the saturation level.
.PP
The various trend curves are shown in Figures 2/E.507 and
3/E.507.
.PP
The logistic and Gompertz curves differ from the linear, parabolic
and exponential curves by having saturation or ceiling level. For further 
study see\ [10]. 
.bp
.RT
.LP
.rs
.sp 18P
.ad r
\fBFIGURE 2/E.507, p.9\fR 
.sp 1P
.RT
.ad b
.RT
.LP
.rs
.sp 30P
.ad r
\fBFIGURE 3/E.507\ \ \ \ DIMINUER LA FIGURE, p.10\fR 
.sp 1P
.RT
.ad b
.RT
.LP
.bp
.sp 1P
.LP
3.2
	\fISmoothing models\fR 
.sp 9p
.RT
.PP
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.
.PP
The best known smoothing process is that of the moving average. 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. 
.PP
In addition to moving average models, there exists another group of
smoothing models based on weighting the observations. The most common models
are:
.RT
.LP
	\(em
	simple exponential smoothing,
.LP
	\(em
	double exponential smoothing,
.LP
	\(em
	discounted regression,
.LP
	\(em
	Holt's method, and
.LP
	\(em
	Holt\(hyWinters' seasonal models.
.PP
For example, in the method of exponential smoothing the weight
given to previous observations decreases geometrically with age according to
the following equation:
\v'6p'
.ce 1000
\(*m\fI\fI\d\fIt\fR\u\ =\ (1 \(em \fIa\fR )\fIY\fR\d\fIt\fR\u+ \fIa\fR 
\(*/*^m\fI 
\dt\fR \(em1\fI
\u\fR 
.ce 0
.ad r
(3\(hy6)
.ad b
.RT
.LP
.sp 1
where:
.LP
	\fIY\fR\d\fIt\fR\uis the measured traffic at time \fIt\fR ,
.LP
	\(*m\fI\fI\d\fIt\fR\uis the estimated level at time \fIt\fR , and
.LP
	 \fIa\fR  | is the discount factor [and (1 \(em \fIa\fR ) is the smoothing 
parameter]. 
.PP
The impact of past observations on the forecasts is controlled by the magnitude 
of the discount factor. 
.PP
Use of smoothing models is especially appropriate for short\(hyterm
forecasts. For further studies see [1], [5] and\ [9].
.RT
.sp 1P
.LP
3.3
	\fIAutoregressive models\fR 
.sp 9p
.RT
.PP
If the traffic demand, \fIX\fR\d\fIt\fR\u, at time \fIt\fR 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: 
\v'6p'
.RT
.ce 1000
\fIX
\dt\u\fR = ?71
\d1\u\fIX
\dt\fR \(em1\fI
\u\fR + ?71
\d2\u\fIX
\dt\fR \(em2\fI
\u\fR +
. |  |  + ?71\fI
\dp\uX
\dt\fR \(em\fIp
\u\fR + \fIa
\dt\u\fR 
.ce 0
.ad r
(3\(hy7)
.ad b
.RT
.LP
.sp 1
where
.LP
	\fIa\fR\d\fIt\fR\u\ is white noise at time \fIt\fR ;
.LP
	 ?71\fI\fI\d\fIk\fR\u,\ \fIk\fR \ =\ 1, . |  |  \fIp\fR are the autoregressive 
parameters. 
.LP
The model is denoted by \fIAR\fR (\fIp\fR ) since the order of the model 
is \fIp\fR . 
.PP
By use of regression analysis the estimates of the parameters
can be found. Because of common trends the exogenous variables
(\fIX
\dt\fR \(em1\fI
\u\fR , \fIX
\dt\fR \(em2\fI
\u\fR , . |  |  \fIX
\dt\fR \(em\fIp
\u\fR )
are usually strongly
correlated. Hence the parameter estimates will be correlated. Furthermore,
significance tests of the estimates are somewhat difficult to perform.
.PP
Another possibility is to compute the empirical autocorrelation
coefficients and then use the Yule\(hyWalker equations to estimate the 
parameters [?71\fI\fI\d\fIk\fR\u]. This procedure can be performed when 
the time 
series [\fIX\fR\d\fIt\fR\u]
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, \(sc\ A.1.
.bp
.RT
.sp 1P
.LP
3.4
	\fIAutoregressive integrated moving average (ARIMA) models\fR 
.sp 9p
.RT
.PP
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 \fIq\fR is given by:
\v'6p'
.RT
.ce 1000
\fIX
\dt\u\fR = \fIa
\dt\u\fR \(em \(*h
\d1\u\fIa
\dt\fR \(em1\fI
\u\fR \(em \(*h
\d2\u\fIa
\dt\fR \(em2\fI
\u\fR 
. |  |  \(em \(*h\fI
\dq\ua
\dt\fR \(em\fIq
\u\fR 
.ce 0
.ad r
(3\(hy8)
.ad b
.RT
.LP
.sp 1
where
.LP
	\fIa\fR\d\fIt\fR\u\ is white noise at time \fIt\fR ;
.LP
	[\(*h\fI\fI\d\fIk\fR\u]\ are the 
moving average parameters
.
.PP
Assuming that the white noise term in the autoregressive models in \(sc\ 
3.3 is described by a moving average model, one obtains the so\(hycalled 
ARMA (\fIp\fR , \fIq\fR ) model:
\v'6p'
.ce 1000
\fIX
\dt\u\fR = ?71
\d1\u\fIX
\dt\fR \(em1\fI
\u\fR + ?71
\d2\u\fIX
\dt\fR \(em2\fI
\u\fR +
. |  |  + ?71\fI
\dp\uX
\dt\fR \(em\fIp
\u\fR +
\fIa
\dt\u\fR \(em \(*h
\d1\u\fIa
\dt\fR \(em1\fI
\u\fR \(em \(*h
\d2\u\fIa
\dt\fR \(em2\fI
\u\fR . |  |  \(em \(*h\fI
\dq\ua
\dt\fR \(em\fIq
\u\fR 
.ce 0
.ad r
(3\(hy9)
.ad b
.RT
.PP
.sp 1
The ARMA model describes a stationary time series. If the time
series is non\(hystationary, it is necessary to difference the series. This is
done as follow:
.PP
Let \fIY\fR\d\fIt\fR\ube the time series and \fIB\fR the backwards shift 
operator, then 
\v'6p'
.RT
.ce 1000
\fIX\fR\d\fIt\fR\u= (1 \(em \fIB\fR )\fI\fI\d\fIt\fR\u
.ce 0
.ad r
(3\(hy10)
.ad b
.RT
.LP
.sp 1
where
.LP
	\fId\fR is the number of differences to have stationarity.
.PP
The new model ARIMA (
\fIp, d, q\fR ) is found by inserting
equation\ (3\(hy10) into equation\ (3\(hy9).
.PP
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.
.PP
As indicated in Figure\ 1/E.507 a tentative model is identified. This is 
carried out by determination of necessary transformations and number of 
autoregressive and moving average parameters. The identification is based on
the structure of the autocorrelations and partial autocorrelations.
.PP
The next step as indicated in Figure\ 1/E.507 is the estimation
procedure. The maximum likelihood estimates are used. Unfortunately, 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\ (3\(hy9) and the
process of making forecasts \fIl\fR time units ahead is shown in \(sc\ A.2.
.PP
The forecasting models described so far are univariate forecasting
models. It is also possible to introduce explanatory variables. 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.
.PP
Detailed descriptions of ARIMA models are given in [1], [2], [3], [5], 
[11], [15] and\ [17]. 
.RT
.sp 1P
.LP
3.5
	\fIState space models with Kalman Filtering\fR 
.sp 9p
.RT
.PP
State space models are a way to represent discrete\(hytime process 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].
.bp
.PP
For a stochastic process such a representation may be of the following form:
\v'6p'
.RT
.ce 1000
\fIX\fR\d\fIt\fR\\d+\\d1\u= ?71\fIX\fR\d\fIt\fR\u+ \fIZ\fR\d\fIt\fR\u+ 
\(*w\fI\fI\d\fIt\fR\u 
.ce 0
.ad r
(3\(hy11)
.ad b
.RT
.LP
.sp 1
and
\v'6p'
.ce 1000
\fIY\fR\d\fIt\fR\u= \fIHX\fI\d\fIt\fR\u+ \(*n\fI\fI\d\fIt\fR\u
.ce 0
.ad r
(3\(hy12)
.ad b
.RT
.LP
.sp 1
where
.LP
	\fIX\fR\d\fIt\fR\uis an s\(hyvector of state variables in period \fIt\fR ,
.LP
	\fIZ\fR\d\fIt\fR\uis an s\(hyvector of deterministic events,
.LP
	?71 is an \fIs\fR \(mu\fIs\fR  | ransition matrix that may, in general,
depend on \fIt\fR ,
.LP
	\(*w\fI\fI\d\fIt\fR\uis an s\(hyvector of random modelling errors,
.LP
	\fIY\fR\d\fIt\fR\uis a d\(hyvector of measurements in period \fIt\fR ,
.LP
	\fIH\fR  | s a \fId\fR \(mu\fIs\fR  | atrix called the observation matrix, and
.LP
	\(*n\fI\fI\d\fIt\fR\uis a d\(hyvector of measurement errors.
.PP
Both \(*w\fI\fI\d\fIt\fR\uin equation (3\(hy11) and \(*n\fI\fI\d\fIt\fR\uin 
equation (3\(hy12) are additive 
random sequences with known statistics. The expected value of each sequence 
is the zero vector and \(*w\fI\fI\d\fIt\fR\uand \(*n\fI\fI\d\fIt\fR\usatisfy 
the conditions: 
\v'6p'
.ce 1000
\fIE\fR @ left [ \(*w\fI\fI\d\fIt\fR\u\(*w\fI~$$Ei:T:j_\fR right ] @  = \fIQ\fR\d\fIt\fR\u\(*d\fI\fI\d\fIt\fR\\d\fIj\fR\u
for all \fIt\fR , \fIj\fR ,
.ce 0
.ce 1000
(3\(hy13)
\fIE\fR @ left [ \(*n\fI\fI\d\fIt\fR\u\(*n\fI~$$Ei:T:j_\fR right ] @  = \fIR\fR\d\fIt\fR\u\(*d\fI\fI\d\fIt\fR\\d\fIj\fR\ufor all \fIt\fR , \fIj\fR ,
.ce 0
.LP
.sp 1
where
.LP
	\fIQ\fR\d\fIt\fR\uand \fIR\fR\d\fIt\fR\uare nonnegative definite matrices,
.FS
A matrix A is
nonnegative definite, if and only if, for all vectors\ \fIz, z\fI
 | (>=" | .
.FE
.LP
and
.LP
	\(*d\fI\fI\d\fIt\fR\\d\fIj\fR\uis the Kronecker delta.
.LP
\fIQ\fR\d\fIt\fR\uis the covariance matrix of the modelling errors and 
\fIR\fR\d\fIt\fR\uis the 
covariance matrix of the measurement errors; the \(*w\fI\fI\d\fIt\fR\uand 
the \(*n\fI\fI\d\fIt\fR\uare 
assumed to be uncorrelated and are referred to as white noise. In other
words:
\v'6p'
.ce 1000
\fIE\fR @ left [ \(*n\fI\fI\d\fIt\fR\u\(*w\fI~$$Ei:T:j_\fR right ] @  = 0 for all \fIt\fR ,
\fIj\fR ,
.ce 0
.ad r
(3\(hy14)
.ad b
.RT
.LP
.sp 1
and
\v'6p'
.ce 1000
\fIE\fR @ left [ \(*n\fI\fI\d\fIt\fR\u\fIX\fR~$$Ei:\fIT:\fR~0_ right ] @  = 0 for all
\fIt\fR .
.ce 0
.ad r
(3\(hy15)
.ad b
.RT
.LP
.sp 1
Under the assumptions formulated above, determine \fIX\fR\d\fIt\fR\\d\fI,\fR\\d\fIt\fR\usuch 
that:
\v'6p'
.ce 1000
\fIE\fR @ left [ (\fIX\fR\d\fIt\fR\\d\fI,\fR\\d\fIt\fR\u\(em~\fIX\fR\d\fIt\fR\u)\fI\fI~\u\fIT\fR\d(\fIX\fR\d\fIt\fR\\d\fI,\fR\\d\fIt\fR\u\(em~\fIX\fR\d\fIt\fR\u) right ] @ 
= minimum,
.ce 0
.ad r
(3\(hy16)
.ad b
.RT
.LP
where
.LP
	 \fIX\fR\d\fIt\fR\\d\fI,\fR\\d\fIt\fR\uis an estimate of the state vector 
at time \fIt\fR , 
and
.LP
	\fIX\fR\d\fIt\fR\uis the vector of true state variables.
.LP
.sp 1
.bp
.PP
The Kalman Filtering technique allows the estimation of state
variables recursively for on\(hyline applications. This is done in the 
following manner. Assuming that there is no explanatory variable\ \fIZ\fR 
\fI\fI\d\fIt\fR\u, once a new data point becomes available it is used to 
update the model: 
\v'6p'
.ce 1000
\fIX\fR\d\fIt\fR\\d\fI,\fR\\d\fIt\fR\u= \fIX
\dt,t\fR \(em1
\u
+ \fIK\fR\d\fIt\fR\u(\fIY\fR\d\fIt\fR\u\(em \fIHX\fR \d\fIt,t\fR \(em1
\u)
.ce 0
.ad r
(3\(hy17)
.ad b
.RT
.LP
.sp 1
where
.PP
\fIK\fR\d\fIt\fR\uis the Kalman Gain matrix that can be computed
recursively\ [18].
.PP
Intuitively, the gain matrix determines how much relative
weight will be given to the last observed forecast error to correct it. To
create a k\(hystep ahead projection the following formula is used:
\v'6p'
.RT
.ce 1000
\fIX
\dt\fR +\fIk,t
\u\fR = ?71\fI\fI
\u\fIk\fR\d\fIX\fR\d\fIt\fR\\d\fI,\fR\\d\fIt\fR\u
.ce 0
.ad r
(3\(hy18)
.ad b
.RT
.LP
.sp 1
where
.LP
	\fIX
\dt\fR +\fIk,t
\u\fR is an estimate of \fIX
\dt\fR +\fIk
\u\fR given
observations \fIY\fR\d1\u, \fIY\fR\d2\u,  |  |  | , \fIY\fR\d\fIt\fR\u.
.PP
Equations (3\(hy17) and (3\(hy18) 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. 
.PP
For further studies see [4], [5], [16], [18], [19] and [22].
.PP
The Kalman Filtering works well when the data under examination are
seasonal. The seasonal traffic load data can be represented by a periodic 
time series. In this way, a seasonal Kalman 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]. 
.RT
.sp 1P
.LP
3.6
	\fIRegression models\fR 
.sp 9p
.RT
.PP
The equations (3\(hy1) and (3\(hy2) are typical regression models. In the 
equations the traffic, \fIY\fR\d\fIt\fR\u, is the dependent (or explanatory) 
variable, while time\ \fIt\fR is the independent variable.
.PP
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 parameters.
.PP
A model with several independent variables is called a multiple
regression model. The model is given by:
\v'6p'
.RT
.ce 1000
\fIY\fR\d\fIt\fR\u= \(*b\d0\u+ \(*b\d1\u\fIX\fR\d1\u\fI\fI\d\fIt\fR\u+ 
\(*b\d2\u\fIX\fR\d2\u\fI\fI\d\fIt\fR\u+ |  |  |  +\fR 
\(*b\fI\fI\d\fIk\fR\\d\fIt\fR\u+ \fIu\fR\d\fIt\fR\u
.EF '%	\fI''
.OF '''\fI	%'
.ce 0
.ad r
(3\(hy19)
.ad b
.RT
.LP
.sp 1
where
.LP
	\fIY\fR\d\fIt\fR\uis the traffic at time \fIt\fR ,
.LP
	\(*b\fI\fI\d\fIi\fR\u, \fIi\fR = 0, 1, . |  | , \fIk\fR are the parameters,
.LP
	 \fIX\fR\d\fIi\fR\\d\fIt\fR\u, \fIi\fR\d\fIe\fR\u= 1, 2, . |  | , \fIk\fR 
is the value of the independent variables at time\ \fIt\fR , 
.LP
	\fIu\fR\d\fIt\fR\uis the error term at time \fIt\fR .
.PP
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 \(sc\ 2 \*QBase data for
forecasting\*U in Recommendation\ E.506.
.PP
Detailed descriptions of regression models are given in [1], [5], [7], 
[15] and\ [23]. 
.RT
.sp 1P
.LP
3.7
	\fIEconometric models\fR 
.sp 9p
.RT
.PP
Econometric models involve equations which relate a variable which we wish 
to forecast (the dependent or endogenous variable) to a number of 
socio\(hyeconomic variables (called independent or explanatory variables). The
form of the equations should reflect an expected
.bp
.PP
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.
.PP
There is a wide spectrum of possible models and a number of methods of 
estimating the coefficients (e.g.,\ least squares, varying 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 
subsumed, 
information from separate sections pooled and parameters allowed to vary in
econometric models, to mention a few.
.PP
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.
.PP
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. 
.PP
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 cross\(hysections 
are 
sufficient to support a model used for predictions.
.PP
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 
correlated 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]. 
.RT
.sp 2P
.LP
\fB4\fR 	\fBDiscontinuities in traffic growth\fR 
.sp 1P
.RT
.sp 1P
.LP
4.1
	\fIExamples of discontinuities\fR 
.sp 9p
.RT
.PP
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 arising, 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).
.PP
An illustration of the bearing of discontinuities on traffic growth
can be observed in the graph of Figure\ 4/E.507.
.PP
Discontinuities representing the doubling \(em and even more \(em of traffic 
flow are known. It may also be noted that changes could occur in the growth 
trend after discontinuities.
.PP
In short\(hyterm forecasts it may be desirable to use the trend of the
traffic between discontinuities, but for long\(hyterm forecasts it may be
desirable to use a trend estimate which is based on long\(hyterm observations,
including previous discontinuities.
.PP
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.
.RT
.sp 1P
.LP
4.2
	\fIIntroduction of explanatory variables\fR 
.sp 9p
.RT
.PP
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 concerned 
with. A list of explanatory 
variables is given in Recommendation\ E.506, \(sc\ 2.
.bp
.RT
.LP
.rs
.sp 24P
.ad r
\fBFigure 4/E.507, p.\fR 
.sp 1P
.RT
.ad b
.RT
.PP
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 structure or the process
generating the data often dictate what variables enter the set of explanatory 
variables. For instance, technological relationships may need to be 
incorporated in the model in order to appropriately define the structure.
.PP
Although there are some criteria used in selecting explanatory
variables [e.g.,\ 
\fIR\fR  |
\u2\d, Durbin\(hyWatson\ (D\(hyW)
statistic, root mean square error\ (RMSE), ex\(hypost forecast performance,
explained in the references], statistical problems and/or
availability of data (either historical or forecasted) limit the set of
potential explanatory variables and one often has to revert to proxy variables. 
Unlike pure statistical models, econometric models admit explanatory 
variables, not on the basis of statistical criteria alone but, also, on the
premise that causality is, indeed, present.
.PP
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. 
.PP
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 discontinuity (e.g.,\ the introduction 
of an advertising variable if advertising is judged to be the cause of 
the 
structural change). Once\(hyand\(hyfor\(hyall, or step\(hywise discontinuities, 
cannot be handled by the introduction of explanatory changes: dummy variables 
can resolve this problem. 
.RT
.sp 1P
.LP
4.3
	\fIIntroduction of dummy variables\fR 
.sp 9p
.RT
.PP
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 attribute.
.bp
.PP
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 historical period 
when calls were operator handled and one for the period for which direct 
dial service is available. 
.PP
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. 
.PP
Indiscriminate use of dummy variables should be discouraged for two
reasons:
.RT
.LP
	1)
	dummy variables tend to absorb all the explanatory power
during discontinuties, and
.LP
	2)
	they result in a reduction in the degrees of freedom.
.sp 2P
.LP
\fB5\fR 	\fBAssessing model specification\fR 
.sp 1P
.RT
.sp 1P
.LP
5.1
	\fIGeneral\fR 
.sp 9p
.RT
.PP
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 \(sc\ 3. In particular 
the 
methods relating to regression analysis and time series analysis will be
discussed.
.PP
All econometric forecasting models presented here are described as
regression models. Also the curve fitting models given in \(sc\ 3.1 can be
described as regression models.
.PP
An exponential model given by
\v'6p'
.RT
.ce 1000
\fIZ
\dt\u\fR = \fIae\fR \fI
\ubt
\d\fR  | (mu | fIu\fR\d\fIt\fR\u
.ce 0
.ad r
(5\(hy1)
.ad b
.RT
.LP
.sp 1
may be transformed to a linear form
\v'6p'
.ce 1000
ln \fIZ
\dt\u\fR = ln \fIa\fR + \fIbt\fR + ln \fIu\fR\d\fIt\fR\u
.ce 0
.ad r
(5\(hy2)
.ad b
.RT
.LP
.sp 1
or
\v'6p'
.ce 1000
\fIY
\dt\u\fR = \(*b
\d0\u + \(*b
\d1\u\fIX\fR\d\fIt\fR\u+ \fIa\fR\d\fIt\fR\u
.ce 0
.ad r
(5\(hy3)
.ad b
.RT
.LP
.sp 1
where
.LP
	\fIY\fR\d\fIt\fR\u	=\ ln \fIZ\fR\d\fIt\fR\u
.LP
	\(*b\d0\u	=\ ln \fIa\fR 
.LP
	\(*b\d1\u	=\ \fIb\fR 
.LP
	\fIX\fR\d\fIt\fR\u	=\ \fIt\fR 
.LP
	\fIa\fR\d\fIt\fR\u	=\ ln \fIu\fR\d\fIt\fR\u(white noise).
.sp 1P
.LP
5.2
	\fIAutocorrelation\fR 
.sp 9p
.RT
.PP
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\ \fIr\fR\d\fIk\fR\u, \fIk\fR 
\ =\ 1, 2,\ . |  |  is 
calculated. \fIr\fR\d\fIk\fR\uis the estimated autocorrelation of residuals 
at lag \fIk\fR . A way to detect autocorrelation among the residuals is 
to plot the 
autocorrelation function and to perform a Durbin\(hyWatson test. The
Durbin\(hyWatson statistic
is:
\v'6p'
.RT
.ad r
.ad b
.RT
.LP
where
.LP
	\fIe\fR\d\fIt\fR\u	is the estimated residual at time \fIt\fR ,
.LP
	\fIN\fR 	is the number of observations.
.bp
.sp 1P
.LP
5.3
	\fITest of significance of the parameters\fR 
.sp 9p
.RT
.PP
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 parameters has to be tested.
.PP
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 absolute 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 estimators.
.PP
The 
multiple correlation coefficient
(or 
coefficient
of determination
) may be used as a criterion for the fitting of the
equation.
.PP
The multiple correlation coefficient, \fIR\fR \u2\d, is given
by:
\v'6p'
.RT
.ad r
.ad b
.RT
.PP
If the multiple correlation coefficient is close to 1 the fitting is satisfactory. 
However, a high \fIR\fR \u2\d does not imply an accurate 
forecast.
.PP
In time series analysis, the discussion of the model is carried out in 
another way. As pointed out in \(sc\ 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.
.PP
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. 
.PP
An overall test of the fitting is based on the statistic
\v'6p'
.RT
.ad r
.ad b
.RT
.LP
where \fIr\fR\d\fIi\fR\uis the estimated autocorrelation at lag \fIi\fR 
and \fId\fR is the number of parameters in the model. When the model is 
adequate, \fIQ 
\dN\fR \(em\fId
\u\fR is
approximately chi\(hysquare distributed with \fIN \(em d\fR degrees of 
freedom. To test the fitting, the value \fIQ 
\d\fIN\fR \(em\fId
\u\fR can be compared with fractiles of
the chi\(hysquare distribution.
.sp 1P
.LP
5.4
	\fIValidity of exogenous variables\fR 
.sp 9p
.RT
.PP
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.
.PP
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.
.PP
On the other hand, the values of most of the explanatory variables
cannot be predicted exactly. The confidence of the prediction will then
decrease with the number of periods. Hence, the explanatory 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.
.bp
.PP
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.
.RT
.sp 1P
.LP
5.5
	\fIConfidence intervals\fR 
.sp 9p
.RT
.PP
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, parameter 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
estimates.
.PP
In causal models, another source of uncertainty in the forecast of the 
series under study are the predictions of the explanatory variables. This 
type of uncertainty cannot be handled by confidence 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. 
.PP
For a linear, static regression model, the confidence interval of the forecast 
depends on the reliability of the regression coefficients, the size of 
the residual variance, and the values of the explanatory variables. The 
95% 
confidence interval for a forecasted value \fIY
\dN\fR +1\fI
\u\fR is given
by:
\v'6p'
.RT
.ce 1000
\fI/*^Y\fI\d\fIN\fR\u(1) \(em 2\(*/*^s \(= \fIY\fR\d\fIN\fR\u\d+1
\u \(= \fI/*^Y\fI\d\fIN\fR\u(1) +
2\(*/*^s
.ce 0
.ad r
(5\(hy7)
.ad b
.RT
.LP
.sp 1
where \fI/*^Y\fI\d\fIN\fR\u(1) is the forecast one step ahead and \(*/*^s 
is the standard error of the forecast. 
.PP
This says that we expect, with a 95% probability, that the actual value 
of the series at time \fIN\fR + 1 will fall within the limits given by 
the confidence interval, assuming that there are no errors associated with 
the forecast of the explanatory variables. 
.sp 2P
.LP
\fB6\fR 	\fBComparison of alternative forecasting models\fR 
.sp 1P
.RT
.sp 1P
.LP
6.1
	\fIDiagnostic check \(em Model evaluation\fR 
.sp 9p
.RT
.PP
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. 
.PP
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.
.RT
.sp 1P
.LP
6.2
	\fIForecasts of levels versus forecasts of changes\fR 
.sp 9p
.RT
.PP
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 forecasting 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 elasticities) the collinearity 
of the explanatory variables (known as multicollinearity) renders the results 
of the estimated coefficients unreliable. 
.PP
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. 
.bp
.PP
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. 
.RT
.sp 1P
.LP
6.3
	\fIEx\(hypost forecasting\fR 
.sp 9p
.RT
.PP
Ex\(hypost forecasting is the generation of a forecast from a model
estimated over a sub\(hysample of the data beginning with the first observation
and ending several periods prior to the last observation. In ex\(hypost
forecasting, actual values of the explanatory variables are used to generate
the forecast. Also, if forecasted values of the explanatory variables are 
used to produce an ex\(hypost forecast, one can then measure the error 
associated with incorrectly forecasted explanatory variables. 
.PP
The purpose of ex\(hypost forecasting is to evaluate the forecasting
performance of the model by comparing the forecasted values with the actuals 
of the period after the end of the sub\(hysample to the last observation. 
With 
ex\(hypost forecasting, one is able to assess forecast accuracy in
terms of:
.RT
.LP
	1)
	percent deviations of forecasted values from actual values,
.LP
	2)
	turning point performance,
.LP
	3)
	systematic behaviour of deviations.
.PP
Deviations of forecasted values from actual values give a general idea 
of the accuracy of the model. Systematic drifts in deviations may provide 
information for either re\(hyspecifying 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. 
.sp 1P
.LP
6.4
	\fIForecast performance criteria\fR 
.sp 9p
.RT
.PP
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 comparison of forecasts with
actual observations may give additional information about the quality of the
model. Suppose we have the time series, \fIY\fR\d1\u, \fIY\fR\d2\u,  |  |  |  | , 
\fIY\fR\d\fIN\fR\u, \fI\fR \fIY\fR\d\fIN\fR\\d+\\d1\u,  |  |  |  | , \fIY\fR\d\fIN\fR\\d+\u\fI\fI\d\fIM\fR\u. 
.PP
The \fIM\fR last observations are removed from the time series and the
model building procedure. The one\(hystep\(hyahead forecasting error is given
by:
\v'6p'
.RT
.ce 1000
\fIe\fR\d\fIN\fR\\d+\u\fI\fI\d\fIt\fR\u= \fIY\fR\d\fIN\fR\\d+\u\fI\fI\d\fIt\fR\u\(em 
\fI/*^Y\fI\d\fIN\fR\\d+\u\fI\fI\d\fIt\fR\\d\\u(em\d1\u(1) 
\ \ \ \fIt\fR = 1, 2, |  |  |  |  \fIM\fR 
.ce 0
.ad r
(6\(hy1)
.ad b
.RT
.LP
.sp 1
where
.LP
	 \fI/*^Y\fI\d\fIN\fR\\d+\u\fI\fI\d\fIt\fR\\d\\u(em\d1\u(1) is the one\(hystep\(hyahead 
forecast. 
.sp 1P
.LP
	\fIMean error\fR 
.sp 9p
.RT
.PP
The mean error, ME, is defined by
\v'6p'
.RT
.ad r
.ad b
.RT
.PP
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. 
.sp 1P
.LP
	\fIMean percent error\fR 
.sp 9p
.RT
.PP
The mean percent error, MPE, is defined by
\v'6p'
.RT
.ad r
.ad b
.RT
.LP
.bp
.PP
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. 
.sp 1P
.LP
	\fIRoot mean square error\fR 
.sp 9p
.RT
.PP
The root mean square error, RMSE, of the forecast is defined
as
\v'6p'
.RT
.ad r
.ad b
.RT
.PP
RMSE is the most commonly used measure for forecasting
precision.
.sp 1P
.LP
	\fIMean absolute error\fR 
.sp 9p
.RT
.PP
The mean absolute error, MAE, is given by
\v'6p'
.RT
.ad r
.ad b
.RT
.sp 1P
.LP
	\fITheil's inequality coefficient\fR 
.sp 9p
.RT
.PP
Theil's inequality coefficient is defined as follows:
\v'6p'
.RT
.ad r
.ad b
.RT
.PP
Theil's \fIU\fR is preferred as a measure of forecast accuracy because 
the error between forecasted and actual values can be broken down to errors 
due to: 
.LP
	1)
	central tendency,
.LP
	2)
	unequal variation between predicted and realized changes,
and
.LP
	3)
	incomplete covariation of predicted and actual
changes.
.PP
This decomposition of prediction errors can be used to adjust the model 
so that the accuracy of the model can be improved. 
.PP
Another quality that a forecasting model must possess is ability 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, ex\(hypost forecast analysis can generally detect a model's inability 
to trace closely actuals that display turning points. 
.RT
.sp 2P
.LP
\fB7\fR 	\fBChoice of forecasting model\fR 
.sp 1P
.RT
.sp 1P
.LP
7.1
	\fIForecasting performance\fR 
.sp 9p
.RT
.PP
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.
.PP
The length of the forecast period affects the decision to use one type 
of a model versus another, along with historical data limitations and the 
purpose of the forecasting model. For instance, ARIMA models may be appropriate 
forecasting models for short\(hyterm forecasts when stability is not an 
issue, 
when sufficient historical 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.
.bp
.PP
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 trade\(hyoff between these sources of error.
.PP
Availability of forecasts for explanatory variables and their
reliability record is another issue affecting the choice of a forecasting
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. 
.PP
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 historical data exist. Finally, when the purpose 
of the model is to forecast the effects associated with changes in the 
factors that 
influence the variable in question, time series models may not be appropriate 
(with the exception, of course, of transfer function and multiple time 
series models). 
.RT
.sp 1P
.LP
7.2
	\fILength of forecast period\fR 
.sp 9p
.RT
.PP
For normal extensions of switching equipment and additions of
circuits, a forecast period of about six years is necessary. However, a 
longer forecast period may be necessary for the planning of new cables 
or other 
transmission media or for major plant installations. Estimates in the long 
term would necessarily be less accurate than short\(hyterm forecasts but 
that would be acceptable. 
.PP
In forecasting with a statistical model, the length of the forecast
period is entirely determined by:
.RT
.LP
	a)
	the historical data available,
.LP
	b)
	the purpose or use of the forecast,
.LP
	c)
	the market structure that generates the data,
.LP
	d)
	the forecasting model used,
.LP
	e)
	the frequency of the data.
.PP
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 historical data base can only
support a short prediction interval. For example, with 10 or 20\ observations
.PP
a model can be used to forecast 4\(hy5\ periods past the sample (i.e.\ into the
future). On the other hand, with 150\(hy200\ observations, potentially reliable
forecasts can be obtained for 30 to 50\ periods past the sample \(em\ other 
things being equal. 
.PP
Certainly, the purpose of the forecast affects the number of predicted 
periods. Long range facility planning requires forecasts extending 15\(hy20 
or 
more years into the future. Rate change evaluations may only require forecasts 
for 2\(hy3\ years. Alteration of routing arrangements could only require 
forecasts extending a few months past the sample. 
.PP
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. 
.PP
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
short\(hyrun predictions all models seem to perform equally well.
.PP
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.
\v'1P'
.bp
.RT
.ce 1000
ANNEX\ A
.ce 0
.ce 1000
(to Recommendation E.507)
.sp 9p
.RT
.ce 0
.ce 1000
\fBDescription of forecasting procedures\fR 
.sp 1P
.RT
.ce 0
.LP
A.1
	\fIEstimation of \fR \fIautoregressive parameters\fR 
.sp 1P
.RT
.PP
The empirical autocorrelation at lag \fIk\fR is given by:
\v'6p'
.RT
.ce 1000
\fIr
\dk\u\fR = 
@ { fIv~\dk\u\fR } over { fIv\fR~\d0\u } @ 
.ce 0
.ad r
(A\(hy1)
\v'7p'
.ad b
.RT
.LP
.sp 1
where
\v'6p'
.ad r
.ad b
.RT
.LP
and
\v'6p'
.ad r
.ad b
.RT
.LP
\fIN\fR being the total number of observations.
.PP
The relation between [\fIr\fR\d\fIk\fR\u] and the estimates
[\fI\*^\(*g\fI\d\fIk\fR\u] of [\(*g\fI\fI\d\fIk\fR\u] is given by the 
Yule\(hyWalker
equations
:
\v'6p'
.LP
\fIr\fR \d1\u\ =\ \*^\(*g
\d1\u + \*^\(*g
\d2\u\fIr\fR \d1\u + . |  |  + \*^\(*g\fI
\dp\ur
\dp\fR \(em1\fI
\u\fR 
.LP
\fIr\fR \d2\u\ =\ \*^\(*g
\d1\u\fIr\fR \d1\u + \*^\(*g
\d2\u\fIr\fR \d2\u + . |  |  +
\*^\(*g\fI
\dp\ur
\dp\fR \(em2\fI
\u\fR 
.ad r
\fB\(mu\fR (A\(hy4)
.ad b
.RT
.LP
\fB\(mu\fR 
.LP
\fB\(mu\fR 
.LP
\fIr
\dp\u\fR \ =\ \*^\(*g
\d1\u\fIr\fR \d\fIp\fR \(em1
\u + \*^\(*g
\d2\u\fIr\fR \d\fIp\fR \(em2
\u + . |  |  +
\*^\(*g\fI
\dp\u\fR 
.PP
.sp 1
Hence the estimators [\*^\(*g\fI\fI\d\fIk\fR\u] can be found by solving 
this system of equations. 
.PP
For computations, an alternative to directly solving the equations is the 
following recursive procedure. Let [ \*^\(*g\fI 
\dk\fR , | fIj
\u\fR ]\fI
\dj\u\fR be
estimators of the parameters at lag \fIj\fR \ =\ 1, 2, . |  | ,\ \fIk\fR 
given that the total number of parameters 
are \fIk\fR . The estimators [\*^\(*g\fI
\dk\fR +1, | fIj\fR \fI
\u\fR ]\fI
\dj\u\fR are then found
by
\v'6p'
.RT
.ad r
.ad b
.RT
.PP
Defining \*^\(*g\fI
\dp\fR , | fIj
\u\fR = \*^\(*g\fI
\dj\u\fR ,  | fIj\fR = 1, 2, . |  | , \fIp\fR , the forecast of the traffic 
demand at time \fIt\fR +1 is expressed by: 
\v'6p'
.ce 1000
\fIX
\dt\fR +1\fI
\u\fR = \*^\(*g
\d1\u\fIX
\dt\u\fR + \(*g 
\d2\u\fIX
\dt\fR \(em1\fI
\u\fR + . |  |  +
\*^\(*g\fI
\dp\uX
\dt\fR \(em\fIp
\u\fR 
.ce 0
.ad r
(A\(hy7)
.ad b
.RT
.LP
.sp 1
.bp
.sp 1P
.LP
A.2
	\fIForecasting with ARIMA models\fR 
.sp 9p
.RT
.PP
The forecast \fIl\fR time units ahead is given by:
\v'6p'
.RT
.ad r
.ad b
.RT
.ad r
.ad b
.RT
.LP
which means that [\fIX\fR\d\fIj\fR\u] is defined as a forecast when
\fIj\fR \ >\ \fIt\fR and otherwise as
an actual observation and that [\fIa\fR\d\fIj\fR\u] is defined as 0 when 
\fIj\fR \ >\ \fIt\fR since white 
noise has expectation 0. If the observations are known (
\fIj\fR \ \(=\ \fIt\fR ), then  [\fIa\fR\d\fIj\fR\u] is equal to the residual.
\v'1P'
.ce 1000
ANNEX\ B
.ce 0
.ce 1000
(to Recommendation\ E.507)
.sp 9p
.RT
.ce 0
.ce 1000
\fBKalman Filter\fR \fBfor a linear trend model\fR 
.sp 1P
.RT
.ce 0
.PP
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\ \fIZ\fR\d\fIt\fR\uto zero. Then 
the general state 
space model is:
\v'6p'
.sp 1P
.RT
.ce 1000
\fIX\fR\d\fIt\fR\\d+\\d1\u= \(*f\fIX\fR\d\fIt\fR\u+ \(*w\fI\fI\d\fIt\fR\u
.ce 0
.ce 1000
(B\(hy1)
\fIY\fR\d\fIt\fR\u= \fIHX\fI\d\fIt\fR\u+ \(*n\fI\fI\d\fIt\fR\u
.ce 0
.LP
.sp 1
.LP
where
.LP
	\fIX\fR\d\fIt\fR\u	is an s\(hyvector of state variables in period \fIt\fR ,
.LP
	\fIY\fR\d\fIt\fR\u	is a vector of measurements in year \fIt\fR ,
.LP
	\(*f
	 is an \fIs\fR \(mu\fIs\fR transition matrix that may, in general, depend 
on \fIt\fR , 
.LP
and
.LP
	\(*w\fI\fI\d\fIt\fR\u	is an s\(hyvector of random modelling errors,
.LP
	\(*n\fI\fI\d\fIt\fR\u	is the measurement error in year \fIt\fR .
.PP
For modelling telephone traffic demand, adapt a simple two\(hystate, one\(hydata 
variable model defined by: 
\v'6p'
.ad r
.ad b
.RT
.LP
and
\v'6p'
.ce 1000
\fIy\fR\d\fIt\fR\u= \fIx\fR\d\fIt\fR\u+ \(*n\fI\fI\d\fIt\fR\u
.ce 0
.ad r
(B\(hy3)
.ad b
.RT
.LP
.sp 1
.LP
where
.LP
	\fIx\fR\d\fIt\fR\u	is the true load in year \fIt\fR ,
.LP
	\fI\o'x\(de'\fI\d\fIt\fR\u	is the true incremental growth in year \fIt\fR ,
.LP
	\fIy\fR\d\fIt\fR\u	is the measured load in year \fIt\fR ,
.LP
	\(*n\fI\fI\d\fIt\fR\u	is the measurement error in year \fIt\fR .
.bp
.LP
Thus, in our model
\v'6p'
.ad r
.ad b
.RT
.LP
The one\(hystep\(hyahead projection is written as follows:
\v'6p'
.ad r
.ad b
.RT
.LP
where
.PP
\fIX
\dt\fR +1,\fIt
\u\fR is the projection of the state variable in
period \fIt\fR \ +\ 1 given observations through year\ \fIt\fR .
.PP
The \(*a\fI\fI\d\fIt\fR\uand \(*b\fI\fI\d\fIt\fR\ucoefficients are the 
Kalman gain matrices in year \fIt\fR . Rewriting the above equation yields: 
\v'6p'
.RT
.ce 1000
\fIx\fR\d\fIt\fR\\d\fI,\fR\\d\fIt\fR\u= (1\(em\(*a\fI\fI\d\fIt\fR\u)\fIx
\dt,t\fR \(em1\fI
\u\fR +
\(*a\fI\fI\d\fIt\fR\u
.EF '%	\fI''
.OF '''\fI	%'
.ce 0
.ad r
(B\(hy6)
.ad b
.RT
.LP
.sp 1
.LP
and
\v'6p'
.ce 1000
\fI\o'x\(de'\fI\d\fIt\fR\\d\fI,\fR\\d\fIt\fR\u= (1\(em\(*b\fI\fI\d\fIt\fR\u)\fI\o'x\(de' 
\dt,t\fR \(em1\fI
\u\fR + \(*b\fI\fI\d\fIt\fR\u(\fIy\fR\d\fIt\fR\u\(em \fIx
\dt\fR \(em1,\fIt\fR \(em1\fI
\u\fR )
.ce 0
.ad r
(B\(hy7)
.ad b
.RT
.PP
.sp 1
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\ \(*a\fI\fI\d\fIt\fR\uand \(*b\fI\fI\d\fIt\fR\ucan 
be either fixed or adaptive. In\ [16] Moreland presents a method for selecting 
fixed, robust parameters that provide adequate performance independent 
of system noise, measurement error, and 
initial conditions. For further details on the proper selection of these
parameters see\ [6], [20] and\ [22].
\v'1P'
.ce 1000
ANNEX\ C
.ce 0
.ce 1000
(to Recommendation\ E.507)
.sp 9p
.RT
.ce 0
.ce 1000
\fBExample of an econometric model\fR 
.sp 1P
.RT
.ce 0
.PP
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: 
.sp 1P
.RT
.LP
	a)
	to demonstrate the introduction of explanatory variables,
.LP
	b)
	to point out difficulties associated with models used for
both the estimation of the structure and forecasting purposes, and
.LP
	c)
	to show how transformations may affect the
results.
.PP
The demand of United States billed minutes to Brazil (\fIMIN\fR ) is estimated 
by a 
log\(hylinear equation which includes United States billed messages to Brazil
(\fIMSG\fR ), a real
telephone price index (\fIRPI\fR ), United States personal income in 1972
prices (\fIYP\fR 72),
and real bilateral trade between the United States and Brazil (\fIRTR\fR 
) as explanatory variables. This model is represented as: 
\v'6p'
.ce 1000
ln(\fIMIN\fR )\fI\fI\d\fIt\fR\u= \(*b\d0\u\ +\ \(*b\d1\uln(\fIMSG\fR )\fI\fI\d\fIt\fR\u\ 
+\ \(*b\d2\uln(\fIRPI\fR )\fI\fI\d\fIt\fR\u\ +\ \(*b\d3\uln(\fIYP\fR 72)\fI\fI\d\fIt\fR\u\ 
+\ \(*b\d4\uln(\fIRTR\fR )\fI\fI\d\fIt\fR\u\ +\ \fIu\fR\d\fIt\fR\u 
.ce 0
.ad r
(C\(hy1)
.ad b
.RT
.LP
.sp 1
where \fIu\fR\d\fIt\fR\uis the error term of the regression and where, 
\(*b\d1\u\ >\ 0, 
\(*b\d2\u\ <\ 0, \(*b\d3\u\ >\ 0 and \(*b\d4\u\ >\ 0 are expected values.
.bp
.PP
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:
\v'6p'
.LP
ln(\fIMIN\fR )\fI\fI\d\fIt\fR\u= \(em3.489 + \fB(\fR 0.619\fB)\fR ln(\fIMSG\fR 
)\fI\fI\d\fIt\fR\u\(em \fB(\fR 0.447\fB)\fR ln(\fIRPI\fR )\fI\fI\d\fIt\fR\u+ 
\fB(\fR 1.166\fB)\fR ln(\fIYP\fR 72)\fI\fI\d\fIt\fR\u+ \fB(\fR 0.281\fB)\fR 
ln(\fIRTR\fR )\fI\fI\d\fIt\fR\u 
.ad r
In(\fIMIN\fR )\fI\fI\d\fIt\fR\u= \(em3.489 +
(0.035) 
ln(\fIMSG\fR )\fI\fI\d\fIt\fR\u\(em
(0.095) 
ln(\fIRPI\fR )\fI\fI\d\fIt\fR\u+
(0.269) 
ln(\fIYP\fR 72)\fI\fI\d\fIt\fR\u+
(0.084)
	(C\(hy2)
.ad b
.RT
.LP
	.sp 1
.ce 1000
\fIR\fR \u2\d = 0.985, \fISER\fR = 0.083, \fID\(hyW\fR = 0.922,
\fIk\fR \ =\ 0.10
.ce 0
.ad r
(C\(hy3)
.ad b
.RT
.LP
.sp 1
where 
\fIR\fR \u2\d is the adjusted coefficient of
determination, \fISER\fR is the
standard error of the regression, D\(hyW is the Durbin\(hyWatson statistic, 
and \fIk\fR is the ridge regression constant. The values in parentheses 
under the equation are the estimated standard deviation of the estimated 
parameters \*^\(*b\d1\u, \(*b\(34d2\u, \*^\(*b\d3\u, \*^\(*b\d4\u. 
.PP
The introduction of messages as an explanatory variable in this
model was necessitated by the fact that since the mid\(hyseventies 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. 
.PP
Because the model serves a dual purpose \(em namely, structure estimation 
and forecasting \(em at least one more variable is introduced 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 
\fIR\fR \u2\d and
the Durbin\(hyWatson statistic.
Consequently, the predictive power of the model is reduced somewhat.
.PP
The effect of transforming the variables of a model are shown in the ex\(hypost 
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 variables which were used to obtain a better fit (the estimated 
RMSE for the log\(hylinear regression model is 0.119 | 27). The forecast 
results in level 
and logarithmic form are shown in Table\ C\(hy1/E.507.
.RT
.LP
.sp 2
.ce
\fBH.T. [T1.507]\fR 
.ce
TABLE\ C\(hy1/E.507
.ps 9
.vs 11
.nr VS 11
.nr PS 9
.TS
center box;
lw(30p) | cw(90p) | cw(108p) .
	Logarithms	Levels
.TE
.TS
box center ;
lw(30p) | cw(30p) | cw(30p) | cw(30p) | cw(36p) | cw(36p) | cw(36p) .
	Forecast	Actual	% deviation	Forecast	Actual	% deviation
_
.T&
rw(30p) | cw(30p) | cw(30p) | cw(30p) | cw(36p) | cw(36p) | cw(36p) .
1980:\ 1	14.858	14.938	\(em0.540	2 | 36 | 69	3 | 73 | 97	\(em\ 7.725
2	14.842	14.972	\(em0.872	2 | 91 | 50	3 | 80 | 34	\(em12.234
3	14.916	15.111	\(em1.296	3 | 05 | 37	3 | 54 | 92	\(em17.746
4	14.959	15.077	\(em0.778	3 | 37 | 98	3 | 29 | 16	\(em11.089
1981:\ 1	15.022	15.102	\(em0.535	3 | 41 | 33	3 | 21 | 35	\(em\ 7.731
2	14.971	15.141	\(em1.123	3 | 75 | 77	3 | 62 | 92	\(em15.601
3	15.395	15.261	\(em 0.879	4 | 52 | 78	4 | 44 | 78	\ 14.333
4	15.405	15.302	\(em 0.674	4 | 01 | 46	4 | 21 | 55	\(em 10.844
1982:\ 1	15.365	15.348	\(em 0.110	4 | 09 | 65	4 | 30 | 38	\(em \ 1.702
2	15.326	15.386	\(em0.387	4 | 28 | 47	4 | 07 | 01	\(em\ 5.802
_
.TE
.nr PS 9
.RT
.ad r
\fBTable C\(hy1/E.507 [T1.507] p.\fR 
.sp 1P
.RT
.ad b
.RT
.LP
.bp
.sp 2P
.LP
	\fBReferences\fR 
.sp 1P
.RT
.LP
[1]
	ABRAHAM (A.) and LEDOLTER (J.): Statistical methods for forecasting
\fIJ. Wiley\fR , New York, 1983.
.LP
[2]
	 ANDERSON (O. | .): Time series analysis and forecasting. The Box\(hyJenkins 
approach. \fIButterworth\fR , London, 1976. 
.LP
[3]
	BOX (G. | . | .) and JENKINS (G. | .): Time Series Analysis:
Forecasting and Control, \fIHolden\(hyDay\fR , San Francisco, 1976.
.LP
[4]
	BROWN (R. | .): Introduction to random signal analysis and Kalman
Filtering. \fIJohn Wiley & Sons\fR , New York, 1983.
.LP
[5]
	CCITT: Manual planning data and forecasting methods, Vol.\ I and\ II,
\fIITU\fR , Geneva, 1988.
.LP
[6]
	CHEMOUIL (P.) and GARNIER (B.): An Adaptive Short\(hyTerm Traffic
Forecasting Procedure Using Kalman Filtering. \fIITC\ 11\fR , Tokyo, 1985.
.LP
[7]
	DRAPER (N.) and SMITH (H.): Applied Regression Analysis, Second
Edition, \fIJohn Wiley & Sons\fR , New York, 1981.
.LP
[8]
	DUTTA (M.): Econometric Methods, \fISouth\(hyWestern Publishing Co.\fR ,
Cincinnati, 1975.
.LP
[9]
	 GARDNER (E. | . | r.): Exponential smoothing the state of art. \fIJournal\fR 
\fIof forecasting\fR , 4, pp. 1\(hy28, 1985. 
.LP
[10]
	GILCHRIST W.: Statistical forecasting. \fIJohn Wiley & Sons\fR ,
New York, 1976.
.LP
[11]
	GRANGER (C. | . | .) and NEWBOLD (P.): Forecasting Economic Time
Series, \fIAcademic Press\fR , New York, 1977.
.LP
[12]
	JOHNSTON (J.): Econometric Methods, Second Edition, \fIMcGraw\(hyHill\fR ,
New York, 1972.
.LP
[13]
	JUDGE (G. | .) \fIet\ al.\fR : The Theory and Practice of Econometrics,
\fIJohn Wiley & Sons\fR , New York, 1980.
.LP
[14]
	 KMENTA (J.): Elements of Econometrics, \fIMacmillan Publishing Company\fR 
, New York, 1971. 
.LP
[15]
	MAKRIDAKIS (S.), WHEELWRIGHT (S. | .), McGEE (V. | E.): Forecasting
methods and applications Second Edition. \fIJohn Wiley & Sons\fR , New 
York, 1983. 
.LP
[16]
	 MORELAND (J. | .): A robust sequential projection algorithm for traffic 
load forecasting. \fIThe Bell Technical Journal\fR , Vol.\ 61, No.\ 1, 
1982. 
.LP
[17]
	NELSON (C. | .): Applied Time Series Analysis for Managerial
Forecasting, \fIHolden\(hyDay\fR , San Francisco, 1973.
.LP
[18]
	PACK (C. | .) and WHITAKER (B. | .): Kalman Filter models for network
forecasting. \fIThe Bell Technical Journal\fR , Vol.\ 61, No.\ 1, pp.\ 
1\(hy9, 1982. 
.LP
[19]
	SORENSON (H. | .): Kalman filtering techniques. Advances in control
systems theory and applications. \fIAcademic Press\fR , Vol.\ 3, pp.\ 219\(hy292, 
1966. 
.LP
[20]
	SZELAG (C. | .): A short\(hyterm forecasting algorithm for trunk demand
servicing. \fIThe Bell Technical Journal\fR , Vol.\ 61, No.\ 1, pp.\ 67\(hy96, 
1982. 
.LP
[21]
	THEIL (H.): Principles of Econometrics, \fIJohn Wiley & Sons\fR ,
New\ York, 1971.
.LP
[22]
	TOME (F. | .) and CUNHA (J. | .): Traffic forecasting with a state
space model. \fIITC\ 11\fR , Tokyo, 1985.
.LP
[23]
	WONNACOTT (T. | .) and WONNACOTT (R. | .): Regression.
\fIJohn Wiley & Sons\fR , New York, 1981.
.sp 2P
.LP
	\fBBibliography\fR 
.sp 1P
.RT
.LP
PINDYCK (R. | .) and RUBINFELD (D. | .): Econometric Models and
Econometric Forecasts, \fIMcGraw\(hyHill\fR , New\ York, 1981.
.LP
SASTRI, (T.): A state space modelling approach for time series
forecasting. \fIManagement Science\fR , Vol.\ 31, No.\ 11, pp.\ 1451\(hy1470,
1985.
.bp
.sp 2P
.LP
\fBRecommendation\ E.508\fR 
.RT
.sp 2P
.sp 1P
.ce 1000
\fBFORECASTING\ NEW\ INTERNATIONAL\ SERVICES\fR 
.EF '%	Fascicle\ II.3\ \(em\ Rec.\ E.508''
.OF '''Fascicle\ II.3\ \(em\ Rec.\ E.508	%'
.ce 0
.sp 1P
.LP
\fB1\fR 	\fBIntroduction\fR 
.sp 1P
.RT
.PP
The operation and administration of an international
telecommunications 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 responsive 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 Administrations forecast new international
services within their planning process.
.PP
This Recommendation presents methods for forecasting new services. The 
definitions of some of the characteristics of these services, together 
with 
their requirements, are covered in \(sc\ 2, followed by base data requirements
in \(sc\ 3. \(sc\ 4 discusses research to identify the potential market.
Presentation of forecasting methods are contained in \(sc\ 5. \(sc\ 6 concludes 
with 
forecast tests and adjustments.
.RT
.sp 2P
.LP
\fB2\fR 	\fBNew service definitions\fR 
.sp 1P
.RT
.PP
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.
.sp 9p
.RT
.PP
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.
.sp 1P
.LP
2.2
	\fBenhanced services offered over the existing network\fR 
.sp 9p
.RT
.PP
These are services which are offered over the existing network,
and which offer an enhancement of the original use for which the network was
intended. Services such as the international freephone service, credit card
calling and closed user groups are examples of enhancements of voice services; 
while facsimile, telefax and videotex are examples of non\(hyvoice services. 
These services may be carried over the existing network and, therefore, data
will concern usage or offered load specific to the enhancement. Arrangements
can be established for the measurement of this traffic, such as the use of
special network access codes for non\(hyvoice applications or by sampling 
outgoing circuits for the proportion of non\(hyvoice to voice traffic. 
.RT
.sp 1P
.LP
2.3
	\fBnovel services\fR 
.sp 9p
.RT
.PP
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 definition of bearer services supported by the
.PP
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 services above 2\ Mbit/s are for further study.
.PP
Teleservices may include Group 4 facsimile, mixed mode text and
facsimile, 64\ kbit/s Teletex and Videotex, videophone, videoconferencing,
electronic funds transfer and point of sale transaction 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.
.bp
.RT
.ce
\fBH.T. [T1.508]\fR 
.ce
TABLE\ 1/E.508
.ce
\fBExamples of enhanced and novel services\fR 
.ps 9
.vs 11
.nr VS 11
.nr PS 9
.TS
center box;
cw(90p) | cw(54p) sw(66p) , ^  | c | c.
 {
Enhancement of
existing services
 }	\*QNovel\*U services
	Bearer services	Teleservices
_
.T&
lw(90p) | lw(54p) | lw(66p) .
Teletex	Packet	Group 4 facsimile
.T&
lw(90p) | lw(54p) | lw(66p) .
Facsimile		Mixed mode
.T&
lw(90p) | lw(54p) | lw(66p) .
Videotex		Videophone
.T&
lw(90p) | lw(54p) | lw(66p) .
Message handling systems	Circuit switched services	Videoconferencing
.T&
lw(90p) | lw(54p) | lw(66p) .
International freephone	\(em 64 kbit/s	Electronic funds transfer
.T&
lw(90p) | lw(54p) | lw(66p) .
Credit cards	\(em 2 Mbit/s	Point of sale transactions
.T&
lw(90p) | lw(54p) | lw(66p) .
Closed user groups		 {
Teletex (64 kbit/s)
Videotex (64 kbit/s)
 }
_
.TE
.nr PS 9
.RT
.ad r
\fBTable 1/E.508 [T1.508], p.\fR 
.sp 1P
.RT
.ad b
.RT
.sp 2P
.LP
\fB3\fR 	\fBBase data for forecasting\fR 
.sp 1P
.RT
.sp 1P
.LP
3.1
	\fIMeasurement of enhanced services\fR 
.sp 9p
.RT
.PP
Measurements for existing services are available in terms of
calls, minutes, Erlangs,\ etc. These procedures are covered in
Recommendation\ E.506, \(sc\ 2. In order to measure/identify enhanced service 
data 
.PP
from other traffic data on the same network it may be necessary to establish
sampling or other procedures to aid in the estimation of this traffic, as
described in \(sc\ 4 and \(sc\ 5.
.RT
.sp 1P
.LP
3.2
	\fINovel services\fR 
.sp 9p
.RT
.PP
Novel services, as defined in \(sc 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,
post\(hydialing delay, clipping, bit\(hyerror rate, holding time, set\(hyup 
time, 
error\(hyfree seconds,\ etc.
.RT
.sp 2P
.LP
\fB4\fR 	\fBMarket research\fR 
.sp 1P
.RT
.PP
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 service elasticities. Market research helps 
make 
.PP
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. 
.PP
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 non\(hyresponse 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 
segment is homogeneous while different segments are heterogeneous. 
.bp
.PP
The market research may be useful in forecasting existing services or the 
penetration of new services. The research may be used in forecasting novel 
services or any service which has no historical 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
.PP
and levels. This aspect of market research will aid in redimensioning the
demand upon final determination of the tariff structure and determining the
customers' initial price sensitivity.
.RT
.sp 2P
.LP
\fB5\fR 	\fBForecasting procedures\fR 
.sp 1P
.RT
.sp 1P
.LP
5.1
	\fIGeneral\fR 
.sp 9p
.RT
.PP
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 example, 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.
.RT
.sp 1P
.LP
5.2
	\fISampling and questionnaire design\fR 
.sp 9p
.RT
.PP
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. 
.PP
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 simple random
sampling. The sample can be structured to include specified numbers of
respondents having characteristics that are known, or believed, to affect 
the subject of the research. Examples of customer characteristics would 
be 
socio\(hyeconomic background and type of business.
.PP
The third step is the questionnaire design. A trade\(hyoff exists between 
obtaining as much information as practical and limiting the questionnaire 
to a reasonable length, as determined by the surveyor. Most questionnaires 
have 
three basic sections:
.RT
.LP
	1)
	 qualifying questions to determine if a knowledgeable person has been 
contacted; 
.LP
	2)
	 basic questions including all questions which constitute the body of 
the questionnaire; 
.LP
	3)
	classification questions collecting background on
demographic information.
.PP
The fourth step involves the implementation of the research \(em the actual 
surveying portion. Professional interviewers, or firms specializing 
in market research should be employed for interviewing.
.PP
The fifth and final step is the tabulation and analysis of the survey data. 
\(sc\ 5.3\(hy5.7 describe this process in detail. 
.RT
.sp 1P
.LP
5.3
	\fIConversion ratios for the sample\fR 
.sp 9p
.RT
.PP
Conversion ratios are used in estimating the proportion of
respondents expressing an interest in the service who will eventually
subscribe.
.PP
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: 
.PP
Let
.RT
.LP
	\fIX\fR\d1\u\fI\fI\d\fIi\fR\u	=
	the proportion of firms in market
segment \fIi\fR  | hat are very interested in the service.
.LP
	\fIX\fR\d2\u\fI\fI\d\fIi\fR\u	=
	the proportion of firms in market segment
\fIi\fR  | hat are interested in the service.
.LP
	\fIX\fR\d3\u\fI\fI\d\fIi\fR\u	=
	the proportion of firms in market segment
\fIi\fR  | hat are not interested in the service.
.LP
	\fIX\fR\d4\u\fI\fI\d\fIi\fR\u	=
	the proportion of firms in market segment
\fIi\fR  | hat cannot decide whether they are interested or not.
.PP
The above example has 4 categories of responses. Greater of fewer categories 
may be used depending on the design of the questionnaire. 
.bp
.PP
Notice that
\v'6p'
.RT
.ad r
.ad b
.RT
.LP
where \fIj\fR = the index of categories of responses.
.PP
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 questionnaire and its 
implementation. Conversion ratios applied to the sample will estimate the 
expected proportion of firms \fIin the survey\fR  | hat will eventually 
subscribe, over the planning 
period. For studies related to the estimation of conversion ratios, refer
to\ [1], [3] and\ [5].
.PP
Then,
.RT
.LP
	\fIc\fR\d1\u\fIX\fR\d1\u\fI\fI\d\fIi\fR\u	=
	the proportion of firms in market segment
\fIi\fR  | hat expressed a strong interest and are expected to subscribe.
.LP
	\fIc\fR\d2\u\fIX\fR\d2\u\fI\fI\d\fIi\fR\u	=
	the proportion of firms in market segment
\fIi\fR  | hat expressed an interest and are expected to subscribe.
.LP
	\fIc\fR\d3\u\fIX\fR\d3\u\fI\fI\d\fIi\fR\u	=
	the proportion of firms in market segment
\fIi\fR  | hat expressed no interest but are expected to subscribe.
.LP
	\fIc\fR\d4\u\fIX\fR\d4\u\fI\fI\d\fIi\fR\u	=
	the proportion of undecided firms in
market segment\ \fIi\fR  | hat are expected to subscribe.
.LP
	where \fIc\fR\d\fIj\fR\u	=
	conversion ratio for response \fIj\fR .
.PP
The proportion of firms in market segment \fIi\fR , \fIP\fR\d\fIi\fR\u, 
that are expected to subscribe to the service, equals 
\v'6p'
.ad r
.ad b
.RT
.PP
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. \fIP\fR\d\fIi\fR\u, 
.PP
therefore, can be interpreted as the long\(hyrun proportion of firms in market
segment\ \fIi\fR  | hat are expected to subscribe to the service at some 
future time period,\ \fIT\fR . 
.PP
Two issues arise in the estimation of the proportion of customers that 
subscribe to the service: 
.RT
.LP
	1)
	while \fIP\fR\d\fIi\fR\urefers to the sample surveyed, the results
need to be extrapolated to represent the population.
.LP
	2)
	\fIP\fR\d\fIi\fR\uis the long\(hyrun (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. 
.sp 1P
.LP
5.4
	\fIExtrapolation from sample to population\fR 
.sp 9p
.RT
.PP
To extrapolate the data from the sample to represent the
population, let
.RT
.LP
	\fIN\fR\d\fIi\fR\u\ =
	size of market segment \fIi\fR  | measured for example,
by the number of firms in market segment\ \fIi\fR )
.PP
Then \fIS\fR\d\fIi\fR\u, the expected number of subscribers in the
planning horizon, equals:
\v'6p'
.ce 1000
\fIS\fR\d\fIi\fR\u= \fIP\fR\d\fIi\fR\u\fIN\fR\d\fIi\fR\u
.ce 0
.ad r
(5\(hy2)
.ad b
.RT
.sp 1P
.LP
.sp 1
5.5
	\fIMarket penetration over time\fR 
.sp 9p
.RT
.PP
To determine the expected number of subscribers at various points in time 
before the service reaches maturity, let 
.bp
.PP
\fIp\fR\d\fIi\fR\\d\fIt\fR\u\ =\ the proportion of firms in market segment 
\fIi\fR 
 | hat are expected to subscribe at time\ \fIt\fR .
.PP
Clearly,
\v'6p'
.RT
.LP
	\fIp\fR\d\fIi\fR\\d\fIt\fR\u< \fIP\fR\d\fIi\fR\u
.LP
and
	 \fIp\fR\d\fIi\fR\\d\fIt\fR\u\(ra \fIP\fR\d\fIi\fR\u\ as\ \fIt\fR \(ra | fIT\fR 
.PP
.sp 1
The relation between \fIp\fR\d\fIi\fR\\d\fIt\fR\uand \fIP\fR\d\fIi\fR\ucan 
be explicitly 
defined as:
\v'6p'
.ce 1000
\fIp\fR\d\fIi\fR\\d\fIt\fR\u= \fIa\fR\d\fIi\fR\\d\fIt\fR\u\(mu \fIP\fR\d\fIi\fR\u 
.ce 0
.ad r
(5\(hy3)
.ad b
.RT
.LP
.sp 1
.LP
\fIa\fR\d\fIi\fR\\d\fIt\fR\uis a penetration function, reflecting changing 
market awareness and acceptance of the service over time, in market segment\ 
\fIi\fR . An appropriate 
functional form for\ \fIa\fR\d\fIi\fR\\d\fIt\fR\ushould be bounded in the 
interval\ (0,1). 
.PP
As an example, let \fIa\fR\d\fIi\fR\\d\fIt\fR\ube a logistic function:
\v'6p'
.ce 1000
\fIa\fR\d\fIi\fR\\d\fIt\fR\u=
[Formula Deleted]
.ce 0
.ad r
(5\(hy4)
\v'1P'
\v'1p'
.ad b
.RT
.LP
.sp 1
\fIb\fR\d\fIi\fR\u\(= 0 is the speed with which \fIp\fR\d\fIi\fR\\d\fIt\fR\uapproaches 
\fIP\fR\d\fIi\fR\uin market 
segment\ \fIi\fR , as illustrated in Figure\ 1/E.508.
.PP
For other examples of non\(hylinear penetration functions, refer to the 
Annex\ A. 
.LP
.rs
.sp 12P
.ad r
\fBFigure 1/E.508, p.\fR 
.sp 1P
.RT
.ad b
.RT
.PP
The introduction of a new service will usually differ according to the 
market segment. The rate of penetration may be expressed as a function 
of time, and the speed of adjustment (\fIb\fR\d\fIi\fR\u) may vary across 
segments. 
Lower absolute values of\ \fIb\fR\d\fIi\fR\u, for the logistic function 
will imply 
faster rates of penetration.
.PP
While the form of the penetration function relating the rate of
penetration to time is the same for all segments, the parameter\ \fIb\fR\d\fIi\fR\uvaries 
across segments, being greater in segments with a later introduction of 
the new service. 
.RT
.LP
	Let \fIt\fR\d0\u\fI\fI\d\fIi\fR\u	=
	time period of introduction of service in market segment\ \fIi\fR .
.LP
	Then, \fIt\fR \ \(em\ \fIt\fR\d0\u\fI\fI\d\fIi\fR\u	=
	time period elapsed since
service was introduced in market segment\ \fIi\fR .
.PP
In the diagramatic illustration, of Figure 2/E.508, the service
has achieved the same level of market penetration\ \fIa\fR\d0\u, in\ \fIt\fR\d\fIC\fR\u 
periods after its introduction in market\ \fIC\fR as it did in\ \fIt\fR\d\fIA\fR\uperiods 
after its introduction in market segment\ \fIA\fR . Later introductions 
may not necessarily lead to faster rates of penetration across segments. 
However, within the same market segment, across countries with similar 
characteristics, such an 
expectation is reasonable.
.bp
.LP
.rs
.sp 14P
.ad r
\fBFigure 2/E.508, p.\fR 
.sp 1P
.RT
.ad b
.RT
.sp 1P
.LP
5.6
	\fIGrowth of market segment over time\fR 
.sp 9p
.RT
.PP
The above discussion has accounted for gradual market penetration of the 
new service, by allowing\ \fIp\fR\d\fIi\fR\\d\fIt\fR\uto adjust to\ \fIP\fR\d\fIi\fR\uover 
time. 
The same argument can be extended to the size of market segment\ \fIi\fR  | ver
time.
.PP
Let \fIn\fR\d\fIi\fR\\d\fIt\fR\u= size of market segment \fIi\fR at time 
\fIt\fR . 
.PP
Then, the expected number of subscribers at time \fIt\fR  | n market
segment\ \fIi\fR , equals:
\v'6p'
.RT
.ce 1000
\fIs\fR\d\fIi\fR\\d\fIt\fR\u= \fIa\fR\d\fIi\fR\\d\fIt\fR\u\(mu \fIp\fR\d\fIi\fR\\d\fIt\fR\u\(mu 
\fIn\fR\d\fIi\fR\\d\fIt\fR\u 
.ce 0
.ad r
(5\(hy5)
.ad b
.RT
.LP
.sp 1
and
.LP
	\fIS\fR\d\fIt\fR\u\ =\ 
@ pile { sum above \fIi\fR } @\fIs\fR\d\fIi\fR\\d\fIt\fR\u= expected number of
subscribers across all market segments at time\ \fIt\fR .
.sp 1P
.LP
5.7
	\fIQuantities forecasted\fR 
.sp 9p
.RT
.PP
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
.PP
forecasting method for some of these quantities is to assume constant
relationships such as:
.RT
.LP
	expected\ access\ lines
	=
	(average access lines)
\(mu expected number of subscribers
.LP
	expected\ minutes
	=
	(average use per line) \(mu expected
access lines
.LP
	expected\ messages
	=
	expected minutes/(average length of
conversation)
.LP
	expected\ revenue
	=
	(average rate per minute) \(mu expected
minutes
.PP
The constants, appearing in parentheses, above, can be determined through\ 
1) the process of market research, or\ 2) past trends in similar 
services.
.sp 1P
.LP
5.8
	\fIForecasting with historical data: application analysis\fR 
.sp 9p
.RT
.PP
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 
.PP
supplier contact. Applications of a service vary widely and no single
variable may be an adequate indicator of total demand.
.PP
The following procedure links demand to country characteristics for
forecasting expanded availability of a new service to other countries.
.bp
.RT
.LP
	 Let \fID\fR = (\fID\fR\d\fIi\fR\u, \fID\fR\d2\u, |  |  |  |  |  | , \fID\fR\d\fIn\fR\u)` 
.LP
represent a vector of country\(hyspecific annual demand for the service 
across\ \fIn\fR  | ountries, where the service currently exists. Let \fIC\fR 
\ =\ matrix of\ \fIm\fR 
 | haracteristics relating to each of the\ \fIn\fR  | ountries that are 
reasonable 
explanatory variables of demand. The components of\ \fIm\fR  | ould vary 
depending 
on the nature of the service and its application.
.PP
Some essential components of \fIm\fR  | 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 awareness is one of 
the key determinants of the rate of market penetration of the service. 
Reasonable 
proxies would be advertising expenditures and time (measured
as\ \fIt\fR *\ =\ \fIt\fR \ \(em\ \fIt\fR\d0\u) where\ \fIt\fR * would 
measure time 
elapsed since the service was first introduced at time\ \fIt\fR\d0\u. Market
.PP
awareness can be characterized as some non\(hylinear function of\ \fIt\fR *, as
presented in \(sc\ 5.5. Other components of\ \fIm\fR  | ay include socio\(hyeconomic 
characteristics of the customers, market size and location of customers.
.PP
The model that is estimated is:
\v'6p'
.RT
.ce 1000
\fID\fR = \fIC\fR \(*b + \fIu\fR 
.ce 0
.ad r
(5\(hy6)
.ad b
.RT
.LP
.sp 1
where
.LP
	\fIC\fR is a (\fIn\fR \(mu \fIm\fR ) matrix of country characteristics
.LP
	\fID\fR is a (\fIn\fR \(mu 1) vector of demand
.LP
	\(*b is a (\fIm\fR \(mu 1) vector of coefficients corresponding to each
of the \fIm\fR  | haracteristics
.LP
	\fIu\fR = (\fIn\fR \(mu 1) vector of error terms
.PP
The estimated regression is:
\v'6p'
.ce 1000
\fI\*^D\fR = \fIC\fR \*^\(*b
.ce 0
.ad r
(5\(hy7)
.ad b
.RT
.LP
.sp 1
.PP
Traditional methods of estimating regressions will be applied.
Equation\ (5\(hy7) can be used for predicting demand for any country where the
service is being newly introduced, as long as elements of the matrix\ \fIC\fR 
 | re available. 
.sp 1P
.LP
5.9
	\fIForecasting with limited information\fR 
.sp 9p
.RT
.PP
In the extreme case where no market research data is available (or is uneconomical 
given resource constraints), or country characteristics that 
affect demand are not easily available or quantifiable, other methods of
forecasting need to be devised.
.PP
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 reasonable estimates of the expected 
number of lines: 
.RT
.LP
	a)
	discussions with foreign telephone companies,
.LP
	b)
	discussions with very large potential customers regarding their
future needs,
.LP
	c)
	service inquiries from customers,
.LP
	d)
	customer letters of intent, and
.LP
	e)
	any other similar qualitative information.
.sp 2P
.LP
\fB6\fR 	\fBForecast tests and adjustments\fR 
.sp 1P
.RT
.sp 1P
.LP
6.1
	\fIGeneral\fR 
.sp 9p
.RT
.PP
Forecast tests and adjustments are dependent on the methodology
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
application\(hybased methodology, traditional tests and adjustments applicable 
to regression methods will be employed, as discussed below. 
.RT
.sp 1P
.LP
6.2
	\fIMarket research based analysis\fR 
.sp 9p
.RT
.PP
This section discusses adjustments to forecasts based on the
methodology described in \(sc\(sc\ 5.2 to\ 5.8. The methodology was based on
quantification of responses from a sample survey.
.PP
The forecast was done in two parts:
.RT
.LP
	a)
	extrapolating the sample to the population, using market
size,\ \fIN\fR\d\fIi\fR\u;
.LP
	b)
	allowing for gradual market penetration (awareness),
\fIa\fR\d\fIi\fR\\d\fIt\fR\uof the new service over time.
.bp
.PP
The values attributed to \fIn\fR\d\fIi\fR\\d\fIt\fR\u(which represents 
the size of market segment \fIi\fR at time\ \fIt\fR ) and \fIa\fR\d\fIi\fR\\d\fIt\fR\ucan 
be tracked over time and 
forecast adjustments made in the following manner:
.LP
	a)
	As an example for \fIn\fR\d\fIi\fR\\d\fIi\fR\u, the segments could be
categorized as travel or financial services. The size of the
segment would be the number of tourists, and the number of large
banks. Historical data, where available, on these units of
measurement 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 \fIn\fR\d\fIi\fR\\d\fIt\fR\ushould be
tracked against actual measured values and adjusted for large
deviations.
.LP
	b)
	For \fIa\fR\d\fIi\fR\\d\fIt\fR\u, testing with only a few observations
since the introduction of the service is more difficult.
.LP
	Given that,
\v'6p'
.ce 1000
\fIa\fR\d\fIi\fR\\d\fIt\fR\u=
@ { fIp\fR\d\fIi\fR\\d\fIt\fR\ } over { fIP\fR\d\fIi\fR\ } @ 
.ce 0
.ad r
(6\(hy1)
\v'7p'
.ad b
.RT
.LP
.sp 1
	and \fIP\fR\d\fIi\fR\uis assumed fixed (in the long run), testing
\fIa\fR\d\fIi\fR\\d\fIt\fR\uis equivalent to testing\ \fIp\fR\d\fIi\fR\\d\fIt\fR\u. 
\fIp\fR\d\fIi\fR\\d\fIt\fR\ucan be tracked by observing the proportion of
respondents that actually subscribe to the service at time\ \fIt\fR .
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 cross\(hysections
of the same individuals, over time. This method is commonly used
for household socio\(hyeconomic surveys. Having
observed\ \fIp\fR\d\fIi\fR\\d\fIt\fR\ufor a new period, values of\ \fIa\fR\d\fIi\fR\\d\fIt\fR\ucan 
be plotted against time to study the nature of the penetration
function, \fIa\fR\d\fIi\fR\\d\fIt\fR\u, and the most appropriate functional
.LP
form that fits the data should be chosen. At very early stages
of service introduction, traditional functional forms for market
penetration, such as a logistic function (as illustrated in the
example in \(sc\ 5.5), will be a reasonable form to assume. Other
variations of the functional form depicting market penetration
would be the Gompertz or Gauss growth curves. The restriction is
that the penetration function should be bounded in the
interval\ (0,1). See Annex\ A for an algebraic depiction of
functional forms.
.PP
There are various statistical forms that may be chosen as
representations for the penetration function. The appropriate functional 
form should be based on some theoretical based information such as the 
expected 
nature of penetration of the specific service over time.
.PP
Continuous tracking of \fIn\fR\d\fIi\fR\\d\fIt\fR\u, \fIp\fR\d\fIi\fR\\d\fIt\fR\uand 
\fIa\fR\d\fIi\fR\\d\fIt\fR\uover time will enable adjustments to these 
values whenever necessary and enable greater confidence in the forecasts. 
.RT
.sp 1P
.LP
6.3
	\fIApplication based analysis\fR 
.sp 9p
.RT
.PP
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 hetero\(hyelasticity, serial
correlation and multicollinearity, when suspect. The methodology for performing 
such tests are described in most econometrics text books. In particular, 
references\ [2] and\ [4] can be used as guidelines. Recommendation\ E.507 also
discusses these corrections.
.PP
Adjustments need to be made for variables that should be included in the 
regression model but are not easily quantifiable. For example, market 
.PP
awareness that results from advertising and promotional campaigns 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 measure, 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 particular countries 
will also 
necessitate post forecast adjustments based upon managerial judgement.
.PP
Another important adjustment that may be necessary is the expected
competition from other carriers offering similar or substitutable services.
Competitor prices, if available, may be used as explanatory variables within
the model and allow the measurement of a cross\(hyprice impact. In most
situations, it is difficult to obtain competitor prices. In such cases, 
other methods of calculating competitor market shares need to be developed. 
.PP
Regardless of forecasting methodology, the final forecasts will have to 
be reviewed by management responsible for planning the service as well 
as 
.PP
by network engineers in order to assess the feasibility both from a planning
implementation and from a technical point of view.
\v'1P'
.bp
.RT
.ce 1000
ANNEX\ A
.ce 0
.ce 1000
(to Recommendation E.508)
.sp 9p
.RT
.ce 0
.ce 1000
\fBPenetration functions\fR \fB(growth curves)\fR 
.sp 1P
.RT
.ce 0
.PP
Some examples of non\(hylinear penetration functions are
illustrated below:
.sp 1P
.RT
.sp 1P
.LP
A.1
	\fILogistic curve\fR \v'6p'
.sp 9p
.RT
.ce 1000
\fIa\fR\d\fIi\fR\\d\fIt\fR\u= \(*a / {  + e\uD\dlF261\fI\fI
\u\fIt\fR\ }
.ce 0
.ad r
(A\(hy1)
.ad b
.RT
.LP
.sp 1
.PP
For \(*a = 1, the curve is bounded in the interval (0,1).
Changing\ \fIb\fR will alter the steepness of the curve. The higher the value
of\ \fIb\fR , the faster the rate of penetration. This curve is S\(hyshaped 
and is 
symmetrical about its point of inflection, the latter being where;
\v'6p'
.ce 1000
@ { \u2\d\fIa\fR\d\fIi\fR\\d\fIt\fR\ } over { \fIt\fR~\u2\d } @  = 0
.ce 0
.ad r
(A\(hy2)
\v'7p'
.ad b
.RT
.sp 1P
.LP
.sp 1
A.2
	\fIGompertz curve\fR \v'6p'
.sp 9p
.RT
.ce 1000
\fIa\fR\d\fIi\fR\\d\fIt\fR\u= \(*a exp 
[Formula Deleted]
.ce 0
.ad r
(A\(hy3)
\v'1p'
.ad b
.RT
.PP
.sp 1
As \fIt\fR \(ra \(if \fIa\fR\d\fIi\fR\\d\fIt\fR\u\(ra \(*a, the limiting 
growth. 
.PP
Holding \fIk\fR = 1 and \(*a = 1, higher values of \fIb\fR will imply slower 
rates of penetration. This curve is also S\(hyshaped like the logistic 
curve, but is not symmetrical about its inflection point. 
.PP
When \fIt\fR = 0, then \fIa\fR\d\fIi\fR\\d\fIt\fR\u= \(*ae\uD\dlF261\fI\fI
\u\fIb\fR\d, which is the initial
rate of penetration.
.RT
.sp 1P
.LP
A.3
	\fIGauss curve\fR \v'6p'
.sp 9p
.RT
.ce 1000
\fIa\fR\d\fIi\fR\\d\fIt\fR\u= \(*a 
@ left ( 1~\(em~e\uD\dlF261~\u\fIbt\fR~\u2\d~\d right ) @ 
.ce 0
.ad r
(A\(hy4)
.ad b
.RT
.LP
	As \fIt\fR \(ra \(if, then \fIa
\dit
\u\fR \(ra \(*a
.LP
	As \fIt\fR \(ra 0, then \fIa
\dit
\u\fR \(ra 0.
.PP
.sp 1
Choosing \(*a = 1, the curve is bounded in the interval (0,1).
.sp 2P
.LP
	\fBReferences\fR 
.sp 1P
.RT
.LP
[1]
	AXELROD (J. | .): Attitude measures that predict purchase, \fIJournal\fR 
\fIof Advertising Research\fR , Vol.\ 8, No.\ 1, pp.\ 3\(hy17, New York, 
March\ 1968. 
.LP
[2]
	JOHNSTON (J.): Econometric methods, Second Edition, \fIMcGraw\(hyHill\fR ,
New York, 1972.
.LP
[3]
	KALWANI (M. | .), SILK, (A. | .): On the reliability and predictive
validity of purchase intention measures, \fIMarketing Science\fR , Vol.\ 
1, No.\ 3, 
pp.\ 243\(hy286, Providence, RI, Summer\ 1982.
.LP
[4]
	KMENTA (J.): Elements of econometrics, \fIMacmillan Publishing Co.\fR ,
New York, 1971.
.LP
[5]
	 MORRISON (D. | .): Purchase intentions and purchase behavior, \fIJournal\fR 
\fIof Marketing\fR , Vol.\ 43, pp.\ 65\(hy74, Chicago, Ill., Spring\ 1979. 
.sp 2P
.LP
	\fBBibliography\fR 
.sp 1P
.RT
.LP
BEN\(hyAKIVA (M.) and LERMAN (S. | .): Discrete choice analysis.
.LP
DRAPER (N.) and SMITH (H.): Applied regression analysis, Second
Edition, \fIJohn Wiley & Sons\fR , New York, 1981.
.LP
.bp
.sp 1P
.ce 1000
\v'3P'
SECTION\ 3
.ce 0
.sp 1P
.ce 1000
\fBDETERMINATION\ OF\ THE\ \fR \fBNUMBER\ OF\ CIRCUITS\ IN\fR 
.ce 0
.sp 1P
.ce 1000
\fBMANUAL\ OPERATION\fR 
.ce 0
.sp 1P
.sp 2P
.LP
\fBRecommendation\ E.510\fR 
.FS
This Recommendation dates from the XIIIth
Plenary Assembly of the CCIF (London, 1946) and has not been fundamentally
revised since. It was studied under Question\ 13/II in the Study
Period\ 1968\(hy1972 and was found to be still valid.
.FE
.RT
.sp 2P
.ce 1000
\fBDETERMINATION\ OF\ THE\ NUMBER\ OF\ CIRCUITS\fR 
.EF '%	Fascicle\ II.3\ \(em\ Rec.\ E.510''
.OF '''Fascicle\ II.3\ \(em\ Rec.\ E.510	%'
.ce 0
.sp 1P
.ce 1000
\fBIN\ MANUAL\ OPERATION\fR 
.ce 0
.sp 1P
.PP
\fB1\fR 
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 \(sc\ 3) cannot be satisfied immediately because 
no circuit is free in the relation considered. 
.sp 1P
.RT
.LP
.PP
By 
\fIcall requests satisfied immediately\fR 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. 
.PP
Ultimately, it will be desirable to evolve a corresponding definition based 
on the 
\fIaverage speed\fR 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 \fIsubscriber\fR 
\fIengaged\fR , \fIno reply\fR ,\ etc. But for the moment, in the absence 
of information about the operating time in the European international service, 
such a 
definition cannot be established.
.RT
.PP
\fB2\fR 
The number of circuits it is necessary to allocate to an
international relation, in order to obtain a given grade of service, should 
be determined as a function of the 
\fItotal holding time\fR of the group in
the busy hour.
.sp 9p
.RT
.PP
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
.
.PP
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. 
.PP
The average call duration will be obtained by dividing the total
number of minutes of conversation recorded by the recorded number of effective 
calls. 
.PP
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.
.RT
.PP
\fB3\fR 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. 
.sp 9p
.RT
.PP
Exceptionally busy days, such as those which occur around certain holidays, 
etc., will be eliminated from these returns. The Administrations 
concerned should plan, whenever possible, to put additional circuits into
service for these days.
.PP
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 variations, so that the average established is in
accordance with all the characteristic periods of traffic flow.
.bp
.RT
.PP
\fB4\fR The total occupied time thus determined should be increased by 
a certain amount determined by agreement between the Administrations concerned 
according to the statistics of traffic growth during 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.
.sp 9p
.RT
.LP
.PP
\fB5\fR 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. 
.sp 9p
.RT
.PP
\fB6\fR In the international manual telephone service, the following
Tables\ A and\ B should be used as a basis of minimum allocation:
.sp 9p
.RT
.PP
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.
.PP
Table\ B, corresponding to about 7% of calls deferred, will be used
whenever possible.
.PP
These tables do not take account of the fact that the possibility of
using secondary routes permits, particularly for small groups, an increase 
in the permissible occupation time. 
.RT
.LP
.sp 1
.ce
\fBH.T. [T1.510]\fR 
.ce
TABLE\ 1/E.510
.ce
\fBCapacity of circuit groups\fR 
.ce
(See Supplement No. 2 at the end of this fascicle)
.ps 9
.vs 11
.nr VS 11
.nr PS 9
.TS
center box;
cw(36p) | cw(48p) sw(48p) | cw(48p) sw(48p) , ^  | c | c | c | c.
Number of  circuits	Table A	Table B
	Percentage of  circuit usage	 {
Minutes of circuit usage possible
in the busy hour
 }	Percentage of  circuit usage	 {
Minutes of circuit usage possible
in the busy hour
 }
_
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
\ 1	65.0	\ \ 39	\(em	\(em
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
\ 2	76.7	\ \ 92	46.6	\ \ 56
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
\ 3	83.3	\ 150	56.7	\ 102
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
\ 4	86.7	\ 208	63.3	\ 152
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
\ 5	88.6	\ 266	68.3	\ 205
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
\ 6	90.0	\ 324	72.0	\ 259
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
\ 7	91.0	\ 382	74.5	\ 313
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
\ 8	91.7	\ 440	76.5	\ 367
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
\ 9	92.2	\ 498	78.0	\ 421
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
10	92.6	\ 556	79.2	\ 475
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
11	93.0	\ 614	80.1	\ 529
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
12	93.4	\ 672	81.0	\ 583
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
13	93.6	\ 730	81.7	\ 637
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
14	93.9	\ 788	82.3	\ 691
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
15	94.1	\ 846	82.8	\ 745
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
16	94.2	\ 904	83.2	\ 799
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
17	94.3	\ 962	83.6	\ 853
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
18	94.4	1020	83.9	\ 907
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
19	94.5	1078	84.2	\ 961
.T&
cw(36p) | cw(48p) | cw(48p) | cw(48p) | cw(48p) .
20	94.6	1136	84.6	1015
.TE
.LP
\fINote\fR
\ \(em\ Tables A and B can be extended for groups comprising more than
20 circuits by using the values given for 20\ circuits.
.nr PS 9
.RT
.ad r
\fBTableau 1/E.510 [T1.510], p.16\fR 
.sp 1P
.RT