Determinación de la dimensionalidad dela escala: unidimensional-

Having validated that the proposed analytical modelling method is accurate and both mathemati- cally and numerically simple to apply, it is then exploited to resolving the network planning and dimensioning issue.

3.5.1 Problem Formulation

In this context, network dimensioning is formulated as an optimisation problem, with the power consumption cost being the objective function. The primary task is to obtain the optimal network parameter settings which support service provisioning with a target QoS level, while also minimis- ing the overall power usage. In this regard, the optimisation algorithm needs to address two major issues: constraints satisfaction and optimisation. Thereby, the optimisation formula is defined as follows,

• Given : the switch size N ; the input link load ρ; the mean packet length t; the FDL base delay DG; the number of allowable FDL re-circulations R.

• Optimisation Variables : K, S and L. K is the number of transmission channels (FWCs) per switch output port, S is the number of TWCs per FDL, and L is the number of FDLs. • Performance Constrains : the blocking probability Plimit, and the overall latency Tlimit.

• Objective function : minimise the overall power consumption of the switch, denoted by E(K, S, L).

The above defines the optimisation problem. The optimisation variables are K, S and L, hence the state space, denoted by ∆, is defined as ∆ = {m|m = (K, S, L)}, a three-dimensional search space. The objective function is the overall power consumption E(K, S, L) which is formulated as a function of the variables K, S and L. The calculation of E(K, S, L) includes the major consumers of the power, including TWCs at switch inputs, FWCs at switch ouputs and TWCs in FDLs. [25] indicates that a 10GBASE SFP+ transceiver consumes 1W power. The supply power of the TWC is set to be 1.5W [26], and the power of the FWC is set to be 0.6W [27]. The network performance measured in terms of contention probability P (K, S, L) and overall delay T (K, S, L), which are defined in Section 3.3, is taken as constraints. The overall aim is to find a state (K, S, L) which min- imises the power consumption E(K, S, L), and also satisfies the performance criterion. As this op- timisation search space is most likely complex and non-convex, given the nature of the performance equations, a heuristic search procedure is implemented to address the optimisation problem.

3.5.2 Optimisation Algorithm

The detailed steps of the heuristic optimisation algorithm used here are summarised in Algo- rithm 1. The first step is to identify an initial point m0= (K0, S0, L0) where the network per-

formance quantities, predicted by the analytical model, meet the conditions. Suppose that the step size of the search algorithm is taken as 1, then the neighborhood state space is described by Z = {(K, S, L)|K0 − 1 ≤ K ≤ K0 + 1, S0− 1 ≤ S ≤ S0 + 1, L0 − 1 ≤ L ≤ L0 + 1}. Next,

the algorithm explores the neighborhood region Z to check for minimum. This way, the algorithm allocates the optimal point in the neighborhood area, which achieves the minimum power usage and also ensures the guaranteed performance. The least power consumption Eminis then updated. Sub- sequently, the algorithm proceeds to search the neighborhood region of the current optimal point obtained in the previous search. The procedure comes out of the loop until there is no further im- provement for the power consumption. Obviously, the proposed algorithm proceeds in a descent direction, and eventually converges to an optimum. This provides a practical and computationally efficient way to estimate the network parameters. However, the heuristic has a main drawback that the outcome may be a local optima. Additionally, different initial starting points m0may result in different outcomes, which is confirmed by the numerical results in Tables 3.3 and 3.4. In order to obtain reasonably accurate dimensioning, multiple runs with different starting points are made, and

Algorithm 1 Min. Energy Heuristic Search Algorithm Initialization :m0 ← (K0, S0, L0)

if P (m0) < Plimitand T (m0) < Tlimit, then i ← 0 goto loop else goto Initialization end if loop: current state mi = (Ki, Si, Li) neighborhood state space Z :

Z = {(K, S, L)|Ki− 1 ≤ K ≤ Ki+ 1, Si− 1 ≤ S ≤ Si+ 1, Li− 1 ≤ L ≤ Li+ 1} check the neighborhood state space, and obtain the neighborhood feasible region Z : Z = {m|m ∈ Z, P (m) < Plimit, T (m) < Tlimit}

search the feasible state space Z find the state point mj, which satisfies E(mj) ≤ E(m), for all m ∈ Z if mi6= mj, then i ← j (mi ← mj) goto loop. else return mi; end if

the best solution is chosen as the final solution.

3.5.3 Optimal Solutions

Table 3.2: A collection of conditions for the optimisation

N 64 256 Given ρ 0.8 0.8 DG 0.1 0.1 Constraints P (K, S, L) < 10−8 < 10−8 T (K, S, L) < 60ns < 80ns Kmax 6 6

Search Limits Smax 10 12

Lmax 24 32

The heuristic procedure described above is now deployed to seek the optimal networking param- eters for two different network configurations, using the proposed analytical modelling solution method. Table 3.2 lists the performance constraints with respect to the contention probability and the overall delay. To facilitate the analysis, it is assumed that the maxima Kmax, Smaxand Lmaxare imposed on the variables K, S and L. As a result, the state space involving K, S and L becomes

Figure 3.15: N = 64. The feasible state space of S and L in the scenarios: K = 4 and K = 6. Note that the blank space represents the area in which the state point breaks the constraints.

Figure 3.16: N = 256. The feasible state space of S and L in the scenarios: K = 4 and K = 6. Note that the blank space represents the area in which the state point breaks the constraints.

a constrained three-dimensional area. The heuristic procedure searches for the optimal solution in the feasible region of the state space which is determined by the analytical framework. As an example, in Figures 3.15 and 3.16, the feasible state space of variables S and L is plotted in the case of K = 4 and K = 6 for two different switch sizes: (i) N = 64 and (ii) N = 256. The graphs show that increasing K expands the state space for variable S and L, and the same is expected to apply to variables S and L. Note that sufficiently many replications have been conducted such that different local optima are obtained. During the experiments, it is noticed that the proposed optimisation algorithm converges quickly. Experimental results are shown in Tables 3.3 and 3.4. Observing that the resulting optimal solutions collected from the experiments are quite different, due to the fact that the heuristic procedure starting with a different initial state point may converge to a different local optimum. For N = 64, three sets of parameters are obtained, and the parameter setting (4, 2, 23) is chosen as the final optimal solution. Alternatively, in the case of N = 256, four distinct optimal points are collected, and the optimal configuration (4, 5, 29), which achieves the Table 3.3: Optimal solutions for N = 64. Emin(K, S, L) represents the minimum power consump- tion (W). Emin(K, S, L) represents the minimum power consumption per switch port, defined by E(K, S, L) =E(K,S,L)_N

N=64

Optimal Solutions K S L Emin(K, S, L) Emin(K, S, L)

Solution 1 4 2 23 222.6 3.478

Solution 2 4 3 19 239.1 3.736

Solution 3 3 5 17 242.7 3.792

Table 3.4: Optimal solutions for N = 256. Emin(K, S, L) represents the minimum power consump- tion (W). Emin(K, S, L) represents the minimum power consumption per switch port, defined by E(K, S, L) =E(K,S,L)_N

N=256

Optimal Solutions K S L Emin(K, S, L) Emin(K, S, L)

Solution 1 4 5 29 831.9 3.250

Solution 2 4 6 26 848.4 3.314

Solution 3 4 7 24 866.4 3.384

Solution 4 4 10 20 914.4 3.572

Global Optimum 4 5 29 831.9 3.250

least power consumption, is selected.

Finally, to measure the performance of the proposed optimisation algorithm, a comparison has been performed between the optimal solutions obtained in Tables 3.3 - 3.4 and the worst-case solutions computed using the same heuristic algorithm except that the objective is to maximise the power usage of the network. The worst-case solutions also fulfill the target performance requirements. In Table 3.5, the numerical comparison shows that the obtained optimal solution not only enhances the energy efficiency, but also lowers the deployment cost significantly, as the number of required opti- cal components in the network is greatly reduced. Specifically, in the scenario of N = 64, the power usage in the network is reduced by 42% in comparison with the power requirement of the worse-case solution. For N = 256, the optimal solution results in a reduction of 30% in power consumption. This indicates that the proposed optimisation procedure is successful in finding a good optimum, given the simplicity of the formulation concerning computational time and resources.

Table 3.5: Power consumption comparison. The worst-case solutions are obtained by exploring the feasible state space for the maximum, and the best-case solutions are the global optimal solutions in Tables 3.3 - 3.4

N = 64 N = 256

Worst-case Best-case Worst-case Best-case

Solution Solution Solution Solution

K 6 4 6 4

S 10 2 12 5

L 10 23 15 29

E(K, S, L) 380.4 222.6 1191.6 831.9