Multi-start results

The multi-start method discussed in Section 4.3 is investigated for some of the experimentations. The Bayesian stopping rule in which the termination loss is related to total relative volume of the unobserved regions of attraction was used. This rule was shown to be the fastest in [30].

In the uniprocessor setting, partition offsets were generated uniformly (ti ∈ Ti) to form sample

solutions. For each sample point the algorithm was run to obtain an equilibrium specified by the sys- tem evolution margin α. Interestingly applying the above indicated Bayesian stopping rule lead to a 0.25% and 0% average relative error on α for the examples of Figures 4.4a and 4.4b respectively, along with a mean execution time of 3 minutes which remains faster than the exact method. This means that the multi-start method succeeded in finding an optimal equilibrium in most of the cases. As indicated earlier, the elevated relative error for instance 13 in Figure 4.4b was due to the consideration of a bad starting point.

To demonstrate the difference between both termination losses, Figure 4.6 represents, for a given instance of the uniprocessor examples, the number of discovered and estimated number of equilibria as more runs are performed. The stopping rule related to the regions of attractions stopped the procedure after 37 runs (5 major equilibria in 96 seconds), whereas the one related to finding all equilibrium points stopped the procedure after 289 runs (12 equilibria in 743 seconds). Both found an optimal equilibrium point, with the difference that the former stopped at 5 equilibrium points with a 97.74% estimated total relative volume for the regions of attraction. This means that in case there are a colossal number of equilibria, stopping after the discovery of a sufficient total region of attraction might be more appropriate.

In the multiprocessor setting, partition offsets and module allocations were initialised uniformly in a manner respecting the various resource constraints (e.g. memory). Fifty instances of 4M20P-H and 4M20P-NH from Table 4.2 were considered. The single runs for these two sets of instances gave a mean relative error of 16.28% and 17.64% respectively, as compared to the exact method. The application of the aforementioned multi-start method reduced this relative error to 0.58% and 0.09% respectively in about 15 minutes. The relative error distributions are represented in Figure 4.7. For some instances the Bayesian stopping rule guaranteed discovering almost all of the regions of attraction, such as instance 20 of Figure 4.7b where a 99.8% total volume for the regions of attraction was estimated. Plotting the

110 4. ABEST-RESPONSE SCHEDULING ALGORITHM 0 2 4 6 8 10 12 14 0 50 100 150 200 250 300 Number of equilibria Runs Discovered Estimated

Figure 4.6: Evolution of the discovered and estimated number of equilibria for a uniprocessor instance. evolution of the estimated and discovered number of equilibria showed that almost all of equilibrium points were discovered (cf. Figure 4.8a). However, for instance 16 of Figure 4.7b, the runs where stopped after a 97% total estimated volume for the regions of attraction while providing a solution at about 6% relative error. As can be seen in Figure 4.8b, this is due to the fact that, although a relatively high probability for the discovered regions of attraction was estimated, several equilibrium points lying in small regions remain to be discovered. This can be overcome by continuing the runs until the esti- mated and discovered number of equilibria converge.

Finally, for the benchmark presented in Table 4.4, the application of the multi-start method on ‘48M636P’ yielded an ameliorated solution at α = 2.0833 instead of α = 1.56. This solution was dis- covered in early runs but the method terminated after about 3 days with a total estimated volume of the regions of attraction that was greater than 99%. It should be noted that the lower bound on the number of required modules is 19 as inferred from3

i∈ΠTbii. Hence dividing the number of actual modules with

this lower bound gives an upper bound on the evolution coefficient, that is α = 2.564. In other words, if all time budgets are multiplied by this coefficient, the best-case scenario is to to have a schedule with 100% loads on the 48 modules.

As a conclusion on the application of multi-start methods, and although a single run of the best- response algorithm yields interesting results in most of the cases, the solution quality can be amelio- rated through further exploration of the solution space. As was noticed, optimal solutions are not always guaranteed to be found. Nevertheless, some probabilistic measures on the optimality of discovered solu- tions can be supplied, such as an estimation on the number of equilibria that remain undiscovered. In all, the application of multi-start methods appeared extremely beneficial given the associated computation times.

4.4. RESULTS 111 0 5 10 15 20 25 30 35 40 45 5 10 15 20 25 30 35 40 45 50

Relative error on evolution coefficient [%]

Instance

Single run

(a) Single runs for 50 instances of 4M20P-H.

0 5 10 15 20 25 30 35 40 45 5 10 15 20 25 30 35 40 45 50

Relative error on evolution coefficient [%]

Instance

Multi-start

(b) Multi-start on the 50 instances of 4M20P-H.

0 10 20 30 40 50 60 5 10 15 20 25 30 35 40 45 50

Relative error on evolution coefficient [%]

Instance

Single run

(c) Single runs for 50 instances of 4M20P-NH.

0 10 20 30 40 50 60 5 10 15 20 25 30 35 40 45 50

Relative error on evolution coefficient [%]

Instance

Multi-start

(d) Multi-start on the 50 instances of 4M20P-NH.

Figure 4.7: The application of multi-start methods on multiprocessor examples.