Numerical results

In order to observe the accuracy of the WDMM algorithm, we compared it with the Mul-tifiber Link-Load Correlation model MLLC introduced in [52] and with simulations. We present numerical results on the blocking probability of optical channel requests in func-tion of the network load in different scenarios. The scenarios differ in topologies, fiber capacities and traffic patterns. Another basis for model comparison is the computation time, which refers the complexity of the algorithm.

Although our model accepts also traffic models that are better suited to WDM net-works, in the studied scenarios the connection requests were assumed to arrive following a Poisson process and with exponential holding time. Considering, that as nearly all pub-lished solution, MLLC supports only the Poisson traffic model this is the only choice that allows the model comparison.

In these illustrative studies the load is set to get blocking probabilities in the range of 10⁻⁶to10⁻¹. This range is too high for a traditional connection based service but can be accepted in the IP over WDM scenario as we mentioned in Section 2.3.2. On the figures the network load is indicated in Erlangs. To accommodate the accepted requests the fixed shortest path routing is applied with the random wavelength assignment described in Section 3.2.

We used the ASONCLES [38] tool for the simulation task. This simulator consid-ers Automatic Switched Optical Networks and allows the investigation of different RWA solutions. The simulations ran until the blocking probability as significant statistical vari-able reached the confidence level0.99with0.01accuracy around the point of estimate.

In the computations of WDMM the iteration was stopped when the difference between the blocking probability values got in two successive steps decreased under10⁻⁶.

3.2.3.1 Uniform ring topology

Figure 3.2: Blocking probability in the uniform 13-node ring: for uniform traffic (left) and non-uniform traffic (right)

The first results were obtained using a 13 node ring with Cj = 24optical channels on each link. The number of wavelengthsCwas set to 24. This network is a single fiber uniform ring that is a very appropriate topology for the model validation because of the high dependence of paths. The numerous common links in the paths imply also the high dependence of the traffic coming from and going to different network nodes. We used a uniform traffic pattern. Due to the uniformity this scenario simplifies the study of basic model properties.

The comparison of the theoretic models WDMM and MLLC with simulation results can be seen on the left plot of Figure 3.2. One can see that WDMM estimates accurately the blocking probability, while the accuracy of the MLLC model is very good in the light-load cases, but it overestimates the blocking rate in cases of higher network load.

This behaviour may come from the fact, that MLLC does not include any fixpoint search mechanism.

In the next study case we setCto6and, concurrently,Mjto4for each link using the same uniform ring topology. Here we applied a random generated non-uniform traffic

pattern. For this multifiber network we obtained the results shown on Figure 3.2 in the right plot.

The large number of fibers causes that in the low-load area our model underestimates the simulated results of the blocking probability, while it is still accurate at high loads.

MLLC works similarly as in the above case.

Let us introduce now the comparison of the computation times measured for the mod-els that were used in the above study cases. All computations were done on an Intel P4 system running at 2 GHz. The results given in seconds are presented in Table 3.1.

Table 3.1: Computation times for the uniform 13 node ring

Load WDMM MLLC WDMM MLLC

28.08 1 263 <1 78

33.70 3 262 1 77

37.44 5 261 <1 78

42.12 7 262 1 77

46.80 10 262 1 78

51.48 15 262 1 78

56.15 21 262 2 78

60.84 31 263 4 78

Uniform traffic,Mj = 1 Non-uniform traffic,Mj = 4 The computation times of the model WDMM grows with the growth of the load. This comes from the fact that using the same accuracy limit in the fixpoint search, the iteration takes more steps to stop. It is easy to observe that the algorithm MLLC consumes much more time than our method. The complexity of this method is not less thanO(HC⁵M³) due to the use of recursive steps. This is considerably larger than the complexity of WDMM presented in Section 3.2.2.

3.2.3.2 Non-uniform ring topology

The next set of results were obtained using a ring of 13 nodes with different Cj link capacities. The total sum of the capacities in this network is the same as in the case of the previously studied uniform ring, i.e.,24∗13 = 312optical channels, the link capacities

24

can be seen on Figure 3.3. For this topology the MLLC model can not be used due to its constraints and WDMM is compared only to the simulation results.

1e-06

Figure 3.4: Blocking probability in the non-uniform 13-node ring: for uniform traffic (left) and non-uniform traffic (right)

First we applied uniform traffic pattern and the value C was set to 12, hence the number of fibers on link j wasMj = Cj/C accordingly. Left plot of Figure 3.4 shows the results. For the right plot of Figure 3.4 the results were obtained by settingCto4and applying a random generated non-uniform traffic pattern. In both cases we can observe that the values computed by the WDMM model fit very well the simulation results.

Berlin

After the evaluation of the proposed model on regular topologies let us present results using a meshed network. The hypothetic Central-West European network shown on Fig-ure 3.5 consists of 11 nodes and 19 optical links. The numbers on the links are the length derived from the distance in kilometres and the capacity in wavelengths. Its design was based on previous publications [60] and [61] that studied pan-European optical network opportunities. The link capacities and the applied traffic pattern were determined consid-ering a population-distance-based traffic relation matrix.

The number of wavelengthsC was set to32in the first case and to 16in the second case. The results are presented on Figure 3.6 left and right plot respectively. We can observe the robust accuracy of the WDMM model in both cases.