First, the motivation behind the proposition of a methodology for the evaluation of MINs is presented and then a distance function called the Universal Perfor- mance Factor (UPF) is defined and its suitability for the performance evaluation of MINs is studied.
3.4.1 Motivation
The proposed methodology aims to introduce a mathematical function regroup- ing a number of performance metrics in order to compare MINs. The number as well as the choice of the metric is supposed to be a conceptual decision, i.e. before the use of the function, evaluated factors are chosen.
The proposed function tries to answer the following question: given a number of MINs with different architectural characteristics, how can they be compared with respect to a number of different performance metrics?
After defining the evaluation function it can be utilized in two ways: simply as a distance function used to compare different MINs, or as an optimization function aimed at defining a set of Pareto optimal networks.
3.4.2 The Universal Performance Factor
In this section, the proposed methodology used to combine a number of perfor- mance factors in order to get a universal performance factor is explained.
In order to define the UPF, let us suppose that the factors to be evaluated as well as the importance attended to them are a part of the designing pro- cess (i.e. the performance factors to be evaluated are chosen). In general, per- formance evaluation factors can be divided into two major groups: factors to be maximized and factors to be minimized. We call the group of factors to
be maximized pmax = {pmax
1 , pmax2 , . . . , pmaxk } and the factors to be minimized pmin = {pmin
1 , pmin2 , . . . , pminl }, where k is the number of factors to be maximized and l is the number of factors to be minimized.
The beginning of this study of the universal factor will concentrate on factors to be minimized only. The definition will be generalized later for the case where
the two types of factors are involved. For a group pmin of performance metrics, the universal performance factor is defined by the Euclidean distance of the per- formance projection value into an n dimensions vector space, where dimensions represent different performance factors.
Definition 16. Given a MIN and a groups of performance factors pmin. The universal
performance factor (UPF) can be broadly defined by:
U P F = v u u t l X i=1 (pmin i )2 (3.2)
Definition 17. Given two networks µ1and µ2and their UPFs, UP F1and UP F2respec-
tively. We say that µ1is more powerful than µ2if UP F1 < U P F2.
By this definition, the MINs multi-criteria performance evaluation and com- parison is transformed into the evaluation of a unique function, for which the parameter of comparison is the distance between the value of the UPF and an ideal (non-realistic) network, for which the UPF value is equal to 0. In order to clarify the idea behind this definition consider an example concerning the per- formance evaluation of two MINsusing two performance factors p0 and p00. We assume that these two factors are both to be minimized. Figure 3.1 presents in two dimensional space the performance of the two MINs µ1and µ2. Let p10(resp. p20) and p100(resp. p200) be the calculated values of these factors for µ
1(resp. µ2). From Figure 3.1 one can notice that µ1is more powerful than µ2as µ1gives smaller values than those of µ2. Note that the UPF is the length of the vector having for coordinates (p0
i, p00i). One can notice that the smaller the value for UPF, the better the network performance.
p’ p" p1’ p2’ p1" p2" UPF1 UPF2
Figure 3.1: An example of the use of the UPF factor.
Usually different parameters have different types of measures and scaling. In order to solve this problem, values can be normalized to a certain value, this may
be the average value, or the maximum value for each factor. The normalization procedure using the maximum value provides the possibility of giving all factors the same importance, and thus of being able to compare different scaling metrics. The equation 3.2 can be replaced by the following equation:
U P F = v u u t l X i=1 pmin i M AX(pmin i ) !2 (3.3) Equation 3.3 can be further improved by including the importance aspect of each factor in the design process of a MIN. This can be done by multiplying each term by a factor called theweight (w). The weight wiexpresses the importance of the performance parameter pi. This leads to the following equation.
U P F = v u u t l X i=1 pmin i M AX(pmin i ) !2 wi (3.4)
Now, the UPF formula (equation 3.4) will be generalized to the case where both factors to be maximized and factors to be minimized are to be evaluated simultaneously. To introduce this aspect it is a good idea to note that normalizing the performance factors using the maximum value permits us to consider the
maximizing of a factor as being equivalent to the minimizing of 1 − pmax
M AX(pmax
). This leads to the following final formula for the UPF:
U P F = v u u u t l X i=1 pmin i M AX(pmin i ) !2 wi+ k X j=1 1 − p max j M AX(pmax j ) !2 wj (3.5)
Two conditions have to be considered in order to use the UPF as a performance factor. First, it is assumed that MAX(p) 6= 0. Second, the measured factors are assumed to be inter-independent.
Defining the UPF, measurable metrics that it can take in account have to be evaluated. The evaluation of these factors is done by the simulation of the net- works and the routing of certain communication patterns. The simulation tool is presented in the following section.