Conclusion

The hereby suggested function derives from repeated experiments which have set the format of the equation as presented. The parameters have been selected after a few millions of comprehensive calculations to determine a low error and minimize the effort to form a reasonable complex relation. As described in Leonard Euler’s work

“Mathematics by experiment” in book [Ngu2011], patterns can be detected using “The Scientific Method” or “The Mathematical Method” by keeping a good record of the data

to allow relations to be easily observed. Part of those ideas have been as well applied for this thesis but it is reasonable to say that perhaps there is another similar relation which might produce better results – fitted and forecasts – and which has not come to my attention yet. The extremely large number of computations may have missed optimum choices to assign specific numbers to variables and constants, but this is reasonable too.

When actual data prove or disprove the proposed model, any further updates must realize even slight variations of the involved parameters can affect the exponent and final results seriously. A general representation of possible updates in the future may be given with the following expression:

(48)

ΔP is to indicate a certain degree of variation when new measurements can be compared with the historical traffic by that time. ΔT(ε–1) refers to the changing trend that each of the historical years exhibit by the time when new studies, if any, are to revise the proposed model. Even in the ideal situation where forecasts will be proved to be absolutely successful, an extensive revision of the formula using updated facts can give even more precise figures. A possible change in Internet users’ behaviour, especially for the high-impact global traffic sources, would mean some level of improvement. Some degree of change in the trend over the years may add or remove existing variables and/or constants in (45). Alternatively, the nature of this change may prioritize different operations to be considered for the new formula or simply altering specific numbers in the exponent. At this point it is reminded that any updates on new data must take into consideration all four criteria exactly as described in the third chapter.

The static characteristics of the proposed studies exclude forecasts that have severe fluctuations. However, the presence of many numbers in the main equation allows certain changes to take effect, even though there is some complexity in the exponent.

Apart from the year, most of the selected constants give more detail to results and may set the slope of the graphical representation at varying levels. Minor modifications of such numbers have been put into experimentations and it has been observed that those changes could be sufficient. According to the past trend of 2005 to 2011, an appropriate extrapolation of a possible future trend within some limited range can be represented by updating the decay rate using constant 90 in the formula. Numerical results have

indicated a good response, in the meaning that extrapolated figures do not significantly deviate from the initial calculations. Thus, equation (45) could be of the following general format:

(49)

More specific, the constant term 90 has been replaced with variable ΔF and has been selected over other numbers from the exponent mainly because it does not produce undesired results. ΔF has been assigned a suitable range and is between 82.5 to 92.5, which has slightly changed the slope of the original curve – the upper and lower limit – as shown in the following figure:

Figure 31: Extrapolated estimates for 2012-2015

The expansion of the range falls within ±10% of the initial graph included in an earlier section and is an alternative representation and at the same time it is a potential update of the original formula. However, the exact value of ΔF is to be determined in the future, if applicable, and according to the new traffic figures.

5.7 Evaluation

The ultimate target of coming up with a prediction error of no more than 10% (and ideally less than 5%) when some new measurements would be known, is now a fact.

Global figures for 2012, 2013 and 2014 have been made available to the public from Cisco Systems. The following table highlights successful predictions by the time respective studies were conducted. Again, relation (5) from section 2.9 is employed to calculate yearly forecast error rates.

Table 26: Demonstration of the goodness of the proposed model

The average error rate at 4.12% is of excellent accuracy so far and seems to be the most accurate result in all relevant studies for predictions focused on the long term. In anticipation to 2015 global IP traffic, the average rate will most likely stay below 5%, however this is not guaranteed mainly because the proposed prediction timeframe has been set to only three years, i.e. 2012 to 2014 inclusive. The additional fourth year of 2015 has been included only to test the predictability of the model beyond the proposed period which, in general, is not advised. And as already demonstrated, forecasts targeted to the very long run are susceptible to excessive errors and should be avoided.

CHAPTER 6