Processor minimization context

The adaptability of the best-response algorithm, in a module minimization context, is superficially demonstrated. Instead of seeking a maximum evolution coefficient on predefined modules, the problem is transformed into minimizing the number of required modules for scheduling the partitions, i.e. de- termine minimum P such that α ≥ 1. Without indulging into much detail, the modified best-response algorithm (MBR) starts with one module and tries to obtain α ≥ 1. If it fails, an extra module is activated and the procedure is redone. The algorithm terminates with P modules and an evolution co- efficient α ≥ 1. The algorithm is compared to a fast ILP-based Bin-Formulation [56] (BF), and the approximation algorithm proposed in [91] for constrained processor scheduling (KS). Since BF is lim- ited to harmonic periods, the comparison is only demonstrated for harmonic examples.

Three experiment sets, each consisting of 100 instances, where generated. Instances of the 3 sets are constituted of 20, 40 and 60 partitions respectively with harmonic periods and with no exclusion constraints defined. The various attributes were generated as indicated previously.

112 4. ABEST-RESPONSE SCHEDULING ALGORITHM 0 5 10 15 20 25 30 35 40 45 50 0 100 200 300 400 500 Number of equilibria Runs Discovered Estimated

(a) An instance where almost all equilibrium points were discovered. 0 10 20 30 40 50 60 70 80 0 50 100 150 200 250 300 Number of equilibria Runs Discovered Estimated

(b) An instance where several equilibrium points remain to be discovered.

Figure 4.8: Evolution of the discovered and estimated number of equilibria for multiprocessor instances. Table 4.5 represents the optimality of the three methods, in addition to their respective computation times. Since BF is an exact formulation, a solution supplied within a 30 minutes time limit is specified as optimal. For the other two methods, their optimality is determined from BF whenever it is considered optimal (does not exceed time limit).

Table 4.5: Comparison between MBR, BF and KS. Experiment set Optimality Average CpuTime

BF MBR KS BF MBR KS 20P 100% 67% 42% 0.08s. 2.38s. 0.04s. 40P 97% 16% 13% 0.97s. 6.22s. 0.35. 60P 93% 11% 12% 3.4s. 33s. 1.51s.

As can be seen in this table, the best-response algorithm, although not representing the best results with its simple implementation, can be also adapted to processor minimization problems. Throughout all of the experimentations carried out the obtained number of modules was in worst cases 2OPT, where OPT denotes the optimal number of processors.

4.5 Conclusion

In this chapter a best-response algorithm for the non-preemptive and strictly periodic multiprocessor scheduling problem has been proposed. This game theoretic approach appeared to be extremely effi- cient in solving the scheduling problem, especially when the exact method fails. The quality of the supplied solutions appeared promising and execution times were highly convenient given the complex- ities dealt with.

4.5. CONCLUSION 113

Numerical experiments have shown that the solution space is separated into a certain number of regions of attraction. Started in any point of such a region, the best-response algorithm will converge to the associated equilibrium. In order to ensure a better solution quality, multi-start methods with Bayesian stopping rules can be implemented. This enhanced greatly the quality of the algorithm, where the chances of finding an optimum increases. In addition, statistical information, such as an estimation on total volume of the regions of attraction, can be supplied.

CHAPTER 5

Virtual Link routing

After the allocation of resources to partitions, which was handled in previous chapters, routing tables for data and message transmission need to be completed, as was indicated in Section 1.5.2. For this purpose, the Virtual Link (VL) routing problem is addressed. In other words, approaches for routing the VLs through AFDX switches are investigated. These approaches should take into consideration the congestion in the AFDX network and the end-to-end transmission delays of the VLs.

Some characteristics of Virtual Links are first discussed in Section 5.1 to then indicate the problem- atic and related work in Section 5.2. A formal definition for the problem is indicated in Section 5.3. In Section5.4, an exact method is discussed. A two-level heuristic is then proposed in Section 5.5. Some experimentations are presented in Section 5.6 whereas Section 5.7 concludes this chapter.

5.1 Virtual Links

In AFDX networks, Virtual Links, which can be thought of as multicast trees [146], have the interest of routing isolated packets of data from a unique End System (interfacing a source Avionic Subsystem) to a predetermined set of other End Systems (destined Avionic Subsystems). In other words, the AFDX switches are configured to direct frames with the same Virtual Link ID through a predefined tunnel in the network. This data tunneling requires a reserved amount of bandwidth, which is defined by system integrators, on each link it traverses. Figure 5.1 demonstrates a Virtual Link originating at End System 1 and delivering data to destination End Systems 2, 3 and 4.

As multiple VLs share the physical links’ bandwidth, preventing the traffic of one VL from interfer- ing with traffic of other VLs on the same physical link is essential. This is done by limiting the rate at which Ethernet frames can be transmitted and their respective frame sizes on a virtual link. Each virtual link is thus assigned two parameters (cf. Figure 5.2):

• Bandwidth Allocation Gap (BAG): it defines the minimum time interval between the starting bits of two successive AFDX frames, assuming zero jitter. BAG values range from 1 ms to 128 ms

116 5. VIRTUALLINK ROUTING

Figure 5.1: A virtual link originating at one End System and sending data to 3 others.

(values that are a power of 2). In other words, the BAG defines the maximum frequency at which frames are sent in a VL.

• Maximum Frame Size (MFS): it describes the maximum size, in bytes, of the transmitted Ethernet frames. MFS values can range from 64 bytes to a maximum of 1518 bytes.

The Virtual Link bandwidth can be hence calculated from the following equation, bw = (67 + M F S)

BAG , (5.1)

where 67 bytes are added to the MFS to represent the frame header.

Figure 5.2: A VL can send frames with a maximum size of M F S bytes every BAG ms.