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Trafic Monitoring and Forecasting Having explained the underlying principles of electrical communication and the statistical ‘laws’ oftelecommunicationstraffic, we cannowconsiderthepracticaldesignandoperationof networks. A prime concern is to ensure that there are adequate resources to meet the traffic demand, or to prioritize the use of resources when shortfalls are unavoidable. Two things have to be done to keep abreast of demand.

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