... Chapter 4 Forecasting (continued from previous page)
4.3 Forecasting Interexchange Traffics
When we are forecasting traffic flows between exchanges, a part of the network or the completer network, it may be possible to extrapolate the past trends from historical traffic records. This would normally yield satisfactory estimates of short to medium term traffic growth. If such historical records are not available or are unreliable, we must use alternative methods to determine the expected growth of traffic between pairs of exchanges during the forecast period. In order to specify this growth we need to determine a scale factor known as a composite growth factor. This factor relates the present traffic flow to the predicted future flow and is generally derived from the growth of the total exchange growth factors for each exchange.
We shall begin this section by considering a number of models for individual growth factors and indicate how they may be used to derive the composite growth factor for exchange pairs. Following this discussion we shall investigate the forecasting of interexchange traffics for a complete network of exchanges using various matrix forecasting methods.
4.2.1 Individual Growth Factors there is a large number of possible models which relate the traffic generated in exchange area i to the number of subscribers Ni (t) at time t. One such model assumes a linear relationship between the traffic Ai (t) and the number of subscribers, i.e. Ai (t)=ko Ni (t) Where ko is a constant in Erlangs per subscriber. The percentage growth of traffic is simply equal to the percentage growth in subscribers.
4.2.2 Composite Growth Factors Once we have obtained individual exchange growth factors we can use them to compute growth factors for traffic flow between pairs of exchanges. A number of different models have been proposed to determine these composite growth factors, and we shall only consider a selection of them in this course.
Rapp gives three different alternative formulae for the prediction of traffic flows between any two exchange areas. All employ the composite growth factor approach, the growth factors themselves being based exclusively on subscriber growth. In the first two formulae, the total traffic per subscriber is assumed to remain constant over the forecast period. In the third formula the traffic per subscriber increases in those exchange areas which grow faster than the network average, and decreases in those areas which lag behind the average network growth.
The three formulae for the prediction of point to point traffics are given below:
i). aij(t) =aij(t0) (Ni(t)Gj(t) + Nj(t)Gi(t))/(Ni(t)+Nj(t))
ii). aij(t) =aij(t0) (Ni2(t)Gj(t) + Nj2(t)Gi(t))/(Ni2(t)+Nj2(t))
iii). aij(t) =aij(t0) . Gi(t)Gj(t))/Gn(t)
where Gi (t) = Ni (t)/Ni (to)
Gj (t) = Nj (t)/Nj (to)
In each case aij (t) refers to the forecasted traffic flow for the year t, and aij (to) refers to the traffic flow in the base year to Gi(t) represent the individual growth factors of subscribers equation (iii) Gn (t) refers to the network growth rate with respect to the base year to. Equation (iii) is identical to a formula used by the Bell System in the USA, except that in the latter case the growth factors refer to originating and terminating traffics of exchanges i and j respectively rather than numbers of subscribers.
Rapp prefers the first two formulas, (i) and (ii) as he believes them to be closer to the observed subscribers’ behaviour. Experience show, however, that this belief is not always justified. For example, equation (i) is based on the assumption that the traffic from one subscriber in area i to all subscribers in area j, and the traffic from one subscriber in area j to all subscribers in area i remains constant during the forecast period. This appears to be valid but preserves the assumption that total originating (and terminating) traffic per subscriber remains unchanged, which is not true in many cases (it is true in exchanges where the connection rate of new – low calling rate – subscribers just matches the increase in the usage rate of the older subscribers). The assumption also implies that the traffic between a subscriber in area i and a subscriber in area j decreases as the total number of subscribers in the two areas grows, which is generally not true (although there is no reliable evidence either to support or reject this assumption).
The three models devised by Rapp which we have described may be expressed solely in terms of traffic growth factors rather than subscriber growth factors. In this case the numbers of subscribers Ni (t) and Nj (t) are replaced by I (t) and J (t) which represent the capacity of exchange i and j to generate traffic, as functions of time t. A further modification to the models involves the use of a factor Kij which express the propensity for exchange i to direct traffic to exchange j. This leads to the following composite growth factors which are analogous to equations (i) and (ii):
iv). Gij = (I(t)Gj(t) + J(t)Gi(t))/(I(t) + J(t))
v). Gij = (I2(t)Gj(t) + J2(t)Gi(t))/(I2(t) + J2(t))
and the forecasted traffic between i and j is given by:
aij(t) =aij(t0)kijGij(t)
kij is the affinity factor.
Equation (iv) is commonly used in Australia for the determination of a composite growth factor between a pair of exchanges. An alternative formula, which was mentioned earlier in this subsection, comes from the Bell System in the USA; it is expressed in terms of the growth factors for traffic in each exchange, together with the growth factor for the network as a whole.
GB =Goi.Gtj/Gn
where
Goi= growth factor of total originating traffic in exchange i.
Gtj= growth factor of total terminating traffic in exchange j.
Gn= growth factor of the total network traffic (originating and terminating).
4.2.3 Matrix Methods While the methods described in the previous subsection are satisfactory for the forecasts of traffic streams associated with individual exchanges, matrix methods are preferred for forecasting the simultaneous growth of all interexchange traffics. Although composite growth factor or trend extrapolation methods could be used to predict the growth (or decline) of every interexchange traffic stream in the network, an iterative matrix balancing and forecasting process leads to a more accurate and consistent overall forecast for the network. This method, however, involves a great deal of computational effort and is only practicable if a computer is available. The following data are required.
i). Interexchange traffics at the base year for every pair of exchanges in the network (aij) taken at the network busy hour.
ii). Base year total originating and terminating traffics for every terminal exchange in the network (Aoi, Ati).
iii). Base year total network traffic (Ab).iv). Originating and terminating growth factors for every exchange in the network (Goi, Gti).
v). Network growth factor Gn.
The data are assembled into a matrix, where each row contains outgoing traffics from a particular exchange to all other exchanges in the network, and each column contains incoming traffics from exchanges in the network to a particular exchange. This means that for a network containing N exchanges the matrix would consist of N2 elements, some of which may be zero. If we systematically lay out the matrix, the main diagonal will contain intra-exchange traffics.
The first step in the matrix forecasting process is to balance the base year matrix i.e. the sum of the originating traffic is equal to the sum of the terminating traffics. Having done this we proceed to adjust the elements of the point to point traffic matrix, so that the sum of the elements in each row equals the total adjusted originating traffic of that respective exchange, and the sum of the elements in each column equals the adjusted total terminating traffic of the corresponding exchange. The algorithm for achieving this adjustment is knows as the “double-factor” transformation, devised by Kruithof. The process is iterative and will (in general) converge.
The next step is to multiply the total network traffic and all originating and terminating traffics by their respective growth factors for the forecast year. Having done this, we check to ensure that the sum of projected originating traffics and the sum of projected terminating traffics both equal the projected total network traffic. If either sum is not equal to the projected network traffic, the individual exchange projected traffics are adjusted by a factor necessary to achieve agreement.
We now take the base year traffic matrix and multiply the elements of each row by the ratio between the (adjusted) projections of exchange originating traffics and the sum of the traffic elements in the corresponding rows, etc., in the manner described above. The base year matrix is thus scaled to the new projected totals, to give a matrix of forecasted between each pair of exchanges in the network.
The double factor method has a number of desirable properties which make it suitable for forecasting purpose. i). Uniqueness There is only one final matrix having the specified row and column totals, obtainable from a given initial matrix by this method. ii). Reversibility The final matrix can be transformed into the initial matrix by the same procedure.
4.2.4 Practical Considerations
The way in which we define an element of the base year traffic matrix is of considerable practical importance as it may significantly affect the dimensioning of the network. For example, we could define an element busy hour in which each elements of the matrix represents the traffic flow as the busiest hour for that origin-destination pair. Such a definition is necessary in mesh networks with no alternative routing, since it requires each route to be dimensioned separately. On the other hand, filling up the matrix with traffics taken at individual element busy hours would be inappropriate if we required data to design a network employing alternative routing. Apart from the fact that such a matrix would be difficult to balance against the total originating and terminating traffics (typically recorded at the exchange busy hour), the use of its data for dimensioning high usage and final choice routes would lead to over-provision. An alternative to the element busy hour would be to use a matrix consisting of traffics recorded at the matrix or network busy hour. Clearly, this can have repercussions in situation where OD(origin-destination) pair traffics peak at markedly different times from the network busy hour, and consequently, such OD pairs will be represented in the matrix by a value which is smaller than its own busy hour traffic. If a network trunking design were based solely on the data contained in such a matrix, routes peaking outside the network busy hour could be under dimensioned and cause excessive congestion. Thus the use of this type of matrix may cause unbalances in the dimensioned network at times other than the network busy hour.
4.4 Other Methods
In this chapter we have devoted most space to time series projection and iterative methods because they are the most useful and, next to intuitive forecasts, the most widely used methods in traffic engineering work. Normative methods are somewhat less useful, as there are few exogenous variables (e.g. leading economic time series) which have a clearly defined relationship to traffic data. Nevertheless, normative methods are very important in long term forecasting, where trend extrapolation becomes unreliable or leads to unacceptably wide confidence limits. Comparison methods, which were mentioned in passing at the beginning of this chapter, may be very useful when to direct data are available and a parallel situation exists in another part of the network. For example, if we need to forecast traffic growth in a new exchange for which no historical records are yet available, we can use a projection based on the average growth trends of adjacent older exchanges, provided they are serving areas with similar characteristics to the new one. Dynamic modelling is a relatively new forecasting technique which is being applied to estimate the effects of man-made environment changes on complex ecosystems. A multi-variable computer model of the system is constructed, defining all relationships and interactions, which is then used to simulate the response of the system to predetermined progressive changes in some of the variables. It is not known, whether this technique has been used anywhere for traffic forecasting. One area, where it could be applied in traffic engineering is in the prediction of changes in offered traffic volume in response to variation of tariffs, service quality, and the introduction of new facilities at predetermined times in the future Generally, it is unwise in forecasting work to rely on a single method. In particular, where investment of large sums of money depends on the results of a traffic forecast, it is advisable to make independent forecasts of the same variable using two or three different methods and average the results. A word of caution, however: if the independently obtained forecasts are significantly different from each other (i.e. differences exceed the confidence limits of the forecasts), one, or more of the forecasts contains a large error and averaging will only compound it. In such cases a through check of the calculation and the data used is strongly advised. Many telecommunications administrations use what is known as the “top- down”, bottom-up’ approach to forecasting. That is, traffic forecasts are independently made at the “micro” (individual exchanges, traffic routes) and the “macro” (total area, network traffics) levels and, after summing appropriate “micro” forecasts they are compared with the “macro” forecasts. If there is substantial difference between the two, both forecasts are reviewed to reconcile the differences. This principle can be applied to forecasts of exchange lines, calls, as well as offered traffics. The “top-down, bottom- up” process can involve several stages of summation and reconciliation: individual traffic route forecasts are added and compared with the total exchange traffic forecast, exchange forecasts are added and compared with the local switching area forecast, the sum of local switching area forecasts is compared with the secondary area forecast, and so on. The introduction of further stages leads to additional opportunities for detecting errors eliminating them before issuing the forecasts. Forecasting should be a continuing process. It would be foolish make a series of forecasts and then ignore the avoidance of new data being acquired subsequently. If the new data deviate significantly from the forecast, the forecast and the parameters of the forecasting model – or even the model itself – must be changed. This continuous adjustment of the forecasting model as new data comes in is sometimes referred to as “adaptive forecasting”. In trend forecasting this model adjustment takes place each time we repeat the least squares curve fitting process after more data have been added the time series. In concluding the discussion of forecasting methods it should be pointed out that forecasting is a very inexact “science” (many practitioners prefer to regard it as an art, rather then a science). While it is highly desirable to develop rigorous mathematical forecasting models which would automatically produce future traffics from a set of key data, no method has been found that would replace or substantially reduce our dependence on sound judgement and careful evaluation of basic assumptions. The most sophisticated mathematical techniques will not produce accurate forecasts if we choose an inappropriate model or make wrong assumptions about the appearance of new factors in the forecast period.
We have seen in this chapter methods that telecommunications engineers can employ to forecast subscriber demand and traffic. For dimensioning access networks subscriber demand in sections and blocks within an exchange area become important. In both, planning of copper based and fibre based access network, block(cabinet area) forecasts help in deciding the sizes of copper or optical cabinets and sizes of the cable connecting these to the central office. Section forecasts help in dimensioning the distribution network. As mentioned earlier, traffic forecasts are currently used to dimension the transport part of the network but they would play increasingly important role in the dimensioning of the access network in future.
1. Introduction to Telecommunications Networks
2. Structure of the Access Network
4. Planning Access Networks
5. New Technologies in the Access Network
