Orthogonal Array

An OA is an array which contains a reduced set of Nexp parameter combinations to be tested from the full search space Ω [HSS99]. The total number of parameters,

62 Chapter 4: Manual Optimization of Handover Thresholds

serving and target handover thresholds of all 3G and LTE cells, is indicated by Np = 2 _{· N}c. Every parameter xp has a set of testing values corresponding to a set of levels L = {1, . . . , ℓ, . . . , Nv} where Nv is the total number of levels. For instance, if a parameter xp can take three values 5, 6 and 7, level 1 refers to value 5, level 2 to value 6 and level 3 to value 7. Each row e = 1, . . . , Nexp of the OA describes a possible combination of parameter levels to be tested in a corresponding experiment. Hence, an OA determines the testing level of each parameter in each experiment. To perform the experiments, each level of a parameter determined by the OA is mapped to a corresponding testing value. The optimization function y is evaluated for each parameter combination determined by row e of the OA resulting in a measured response ye. In every iteration of TM, the levels of each parameter are mapped to different testing values based on the candidate solution found in the previous iteration. Hence, a new set of Nexp parameter combinations is tested in each iteration. The properties of the OA are described in the following.

By definition, an Nexp× Np matrix A, having elements fromL, is said to be an orthog- onal array OA(Nexp, Np, Nv, S) with Nv levels, strength S and index λ if every Nexp×S sub-array of A contains each S-tuple based on L exactly λ times as a row [HSS99]. Thus, λ denotes the number of times each S-tuple based on _{L is tested. The higher} the strength S, the more the OA considers the interactions among the configuration parameters. In this study, each column in the OA corresponds to a handover threshold value xp. The first Nc columns can be assigned for the serving cell thresholds and the rest for target cell thresholds. For illustration, an example of an OA(9,4,3,2) having Nexp = 9 which is 9 times smaller than all 34 = 81 possible combinations, Np = 4 configuration parameters, Nv = 3 levels and strength S = 2 is depicted in Table 4.1.

Table 4.1. An illustrative OA(9,4,3,2) with the measured responses and their corre- sponding SN ratios. Experiment x1 x2 x3 x4 Measured SN response ratio 1 1 1 1 1 y1 SN1 2 1 2 2 3 y2 SN2 3 1 3 3 2 y3 SN3 4 2 1 2 2 y4 SN4 5 2 2 3 1 y5 SN5 6 2 3 1 3 y6 SN6 7 3 1 3 3 y7 SN7 8 3 2 1 2 y8 SN8 9 3 3 2 1 y9 SN9

4.3 Cell-Specific Optimization of Handover Thresholds 63

In any 9 × 2 sub-array of the OA in Table 4.1, the following nine row combinations (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) are found and each pair appears the same number of times, i.e., λ = 1. In other words, every level of a parameter xp is tested with every other level of a parameter xp′ 6= x_p exactly λ = 1 times. This property

of the OA accounts for the interactions that might exist between the parameters. Therefore, the OA depicted in Table 4.1 does not only analyze the individual impact of each parameter on the performance, but also the effect of the combination of any two parameters.

A basis property of the OA is that each parameter is tested at each level the same number of times. This allows for a fair and balanced manner of testing the values of the parameters. In Table 4.1, each level is tested three times for every parameter. Moreover, any sub-array of A is also an OA. Therefore, a new OA having a smaller number of parameters can be obtained from an existing one by removing one or more columns. This property is especially useful when the number of parameters of an optimization problem is smaller than Np. In this case, an OA can be directly obtained from A without the need to construct it.

Another fundamental issue is the construction and existence of an OA. Many tech- niques are known for constructing OAs based on Galois fields and finite geometries. More details about how to construct an OA are found in [HSS99]. Besides, it is not always possible to construct an OA with the desired number Nexp of experiments. The higher Nexp, the higher is the computational complexity. If the values of Np, Nv, and S are specified, there is a lower bound on the minimum number Nexp of experiments so that an OA exists. The Rao’s bounds, defined in [Rao47] for an OA of strength two and three, set a restriction on the number Nexp of experiments and, therefore, the com- putational complexity of the algorithm. The parameters of the OA(Nexp, Np, Nv, S) should satisfy the following inequalities

Nexp ≥ s X g=0  Np g  (Nv− 1)g, if S = 2s, s > 0 (4.2) Nexp ≥ s X g=0  Np g  (Nv− 1)g+  Np− 1 s  (Nv− 1)s+1, if S = 2s + 1, s≥ 0. (4.3)

In principle, Nexp is much smaller than the number NvNp of all possible parameter combinations, i.e., Nexp ≪ NvNp.

Many OAs having different number Np of configuration parameters have been already constructed and archived in the database maintained in [Slo]. Thus, the required

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OA can be directly selected from this database if found, otherwise, it needs to be constructed. The construction of an OA having high number Np of parameters might be challenging or even not possible. This is because an OA would exist only if it satisfies the inequalities of (4.2). Moreover, even if the OA exists, it is not always obvious how to construct it. Thus, the TM based on OA can be used only for a limited number of optimization problems whose OA exists and can be constructed.