Verification of Results

A large telecommunication network may be difficult to analyze directly with analytical models because of its complexity; however small sections of the network may be analyzed with these models and the results, possibly, extended to the whole network. The analytical models can serve as performance bounds and can be used to verify the simulation algorithms. The verification can prove that the simulation model is a true representation of the scenario investigated and that the results are reliable. Two approaches for result validation are considered in this work: analytical performance bound derivation and confidence interval estimation.

3.6.1 Analytical Bounds

Queuing theory provides analytical tools for evaluating systems which attempts to serve randomly arriving user requests with limited system resources [135]. It is a quite popular analytical model and has been applied in telecommunication systems as early as 1917 by the Danish mathematician, Erlang, who proposed the Erlang B and Erlang C formulae [129]. In telephone systems, these models provide mathematical relationship between the traffic load offered to the system, the desired grade of service, and the number of channels needed to achieve this grade of service [134].

The Erlang B model assumes that a user’s call blocked by the system is lost completely and a reattempt from the same user is seen as a new call [133]. In the work presented in this thesis, the assumption is that users will reattempt their request after waiting for a period of time which follows an exponential distribution with the mean equivalent to the mean arrival rate of users into the system. The Erlang B model is a best case approximation of this approach since blocked calls are cleared and would not contribute to further congestion. It is important to note that although the Erlang B model was originally derived and used in the context of voice calls and telephone channels, it is relevant to the data service considered in this work when utilised in the light of channel occupancy by transmitted data.

The Erlang B model is used to derive a lower bound on the number of active ABSs required to serve a particular traffic load. However, interference is not considered

with this model but interference has been considered in all the simulations in this thesis to approach the practical conditions in wireless systems as much as possible. Nevertheless, the lower bound derived based on the Erlang B model provides a theoretical limit on the minimum number of active ABSs required for a given traffic load. Further details about the Erlang B lower bound and the derivation are provided in Chapter 7. The derived bound is used to verify the performance of an enhanced version of the proposed RRM and TM schemes which mitigates interference properly across all traffic loads.

3.6.2 Confidence Intervals

The confidence interval estimates a population parameter, such as mean, with a range of values the parameter will most likely fall within at a predefined probability of success referred to as confidence level [136, 137]. Also, a confidence interval usually constitutes a range of value above or below a point estimate of the population parameter. In addition, it provides a means of specifying the level of accuracy that should be expected from the method used for estimation of the population parameter. Estimation of a confidence interval at high confidence level makes it possible to state that the population parameter has been evaluated with a high degree of confidence. In this work, the blocking probability and average file transfer delay have been evaluated after repeated sampling of the user population in order to obtain parameters that are truly representative of the system considered. The average file transfer delay is analogous to the mean of a population and the confidence interval is estimated for the average file transfer delay based on the assumption of normal distribution of this parameter. This is explained in the following.

When the sample size is large enough, usually greater than 30, the point estimate of mean given by is an approximate normal distribution with mean, , and variance, [138]. The confidence interval estimation, , for such large sample case at a confidence level of , , is given by [138]:

where is a particular point estimate of the mean, is the size of the sample and is the variance of the population which can be approximated with the variance of the collected sample without significant loss of accuracy [138]. is the upper 100 ( percent point of the standard normal distribution [136]. The standard normal distribution is obtained by normalizing the mean random variable,

In the case where the outcome of an evaluation (or experiment) has a binary outcome e.g. failure or success, the probability of success based on several trials can be modelled with a binomial distribution. The binomial distribution is characterised by the probability of success, , and the number of trials or samples considered, [139]. The blocking probability is assumed to have a binomial distribution and confidence interval estimation for this case is as follows.

The confidence interval estimate, , of for a large sample case at a confidence level of is given by [138]:

(3.18)

where is a point estimate of the population parameter, . varies across confidence levels, high confidence value ranging from 90% to 99.9% can be used to provide high degree of accuracy of estimation of population parameters. The values of for difference confidence levels are shown in Table 3.1. The confidence interval has been evaluated at 99% for the delay in Chapter 4 to validate the reliability of the results. The confidence interval has also been used in a novel way to develop an adaptive RRM and TM scheme in Chapter 6. As a result, confidence interval and its novel application are explained in greater detail in Chapter 6.

Table 3.1 Confidence Interval Parameters

Confidence Level 90% 95% 99% 99.9%

3.7 Conclusion

In this chapter, the system modelling and the performance evaluation techniques utilised in this work are discussed. A simulation model is chosen as the major tool for modelling the system and evaluating performance on the modified BuNGee Architecture considered in this work due to the complexity associated with analytical modelling. The choice of MATLAB as the programming language for the simulation model is as a result of its flexibility, ease of coding and rich database of built-in mathematical and graphical tools. Furthermore, blocking probability, average file transfer delay and throughput are considered as the QoS performance metrics for evaluation in the subsequent chapters. The energy efficiency metrics are energy reduction gain (ERG) and effective energy saving (EES). Finally, the derivation of a performance bound and confidence interval estimation as the approaches used to validate simulation results are also presented.