Formal description

The proposed algorithms employ three ordering rules according to the weighting metrics and path costs introduced in Section 4.2.4:

H applies the costc^H

RES applies the cost

c^RES(π) = 1

minei∈πw_i^RES (4.5)

ABW applies the costc^ABW

Using these ordering rules letP_G^γ(s, d)be equal to the setPG(s, d)ordered by the cost metricc^γ. We introduce the functionχ(n),n ∈ N. It returns an even number computed as the quotient ofn/2, without remainder, and incremented by one. Ifn ≤ 2, it returns 1. More precisely:

χ(n) =

( 2∗ dn/4e ifn >2

1 otherwise (4.6)

Theχ(·)function will be used in determining the number of paths to drop from the ordered set before applying a different filtering. LetΞ(P^γ), be the filtering function of the ordered set of pathsP^γ:

Ξ(P^γ) ={πk|πk∈ P^γ, k ≤χ(||P^γ||)} (4.7) whereπkis thek^thelement of ordered setP^γ. The returned setQ= Ξ(P^γ)is unordered.

An MSF algorithm performs a sequence of combined steps that include both filtering and ordering by different cost metrics. Due to the included filtering, this method differs significantly from the simple concatenation of cost metrics introduced before.

To model the routing functions working in a network that considers the periodic distribution of link state information we need to use the link weights RES(tu) and ABW(tu, fina)instead ofRES andABW respectively.

Using these formalisms, we give now the description of four different flavours of Multimetric Sequential Filtering algorithms.

4.4.2.1 Algorithm MSF1

Given a set of pathsPG(s, d) withm0 paths between a source s and a destination d in graphG:

1. Considering the hop-ordered setP_G^H(s, d), let us first restrict our scope to the first m1paths; thus, we will define:

Q_G(s, d) ={πk|πk∈ P_G^H(s, d), k ≤m₁} (4.8) (m1 is just an upper bound to the initial number of paths in the set: in the simula-tions we usedm1 = 32).

2. RG(s, d) = Ξ(Q^H_G(s, d))

given them1 paths inQ^H_G(s, d), let us select the firstm2 =χ(m1)and assign them to the unordered setRG(s, d).

3. SG(s, d) = Ξ(R^RES_G (s, d))

given them2 paths in the residual-bandwidth ordered setR^RES_G (s, d), let us select the firstm₃ =χ(m₂)and assign them to the unordered setS_G(s, d).

4. Best-effort flows betweensanddwill be routed over the pathπin setSG(s, d)that is the “lightest” by the cost metric ABW. In other words, the choice is the first path of the ordered setS_G^ABW(s, d).

As discussed above, steps 1, 2 and, if only one traffic class is present then even step 3, of the MSF1 algorithm can be executed off-line. In step 4 the selection of the path that has the largest amount of bandwidth available to a single flow must be performed on-line.

4.4.2.2 Algorithm MSF2

Version 2 of the MSF routing algorithm further filters paths according to the hop count.

As a result, steps 1 through to 3 are the same as in MSF1. The further steps are:

4. TG(s, d) = Ξ(S_G^ABW(s, d))

given them₃paths in the available-bandwidth ordered setS_G^ABW(s, d), let us select the firstm4 =χ(m3)and assign them to the unordered setTG(s, d).

5. Best-effort flows betweensanddwill be routed over the pathπin setTG(s, d)that is the “lightest” by the cost metricH, i.e., the choice is the first path of the ordered setT_G^H(s, d).

4.4.2.3 Algorithm MSF3

Version 3 of the MSF routing essentially captures the spirit of the Load-Dependent (LD) algorithm that was introduced in [22] and outlined in Section 4.2.4. The algorithm chooses the same π path as MSF1 if the available bandwidth on it is large enough for a new connection, i.e., if _cABW¹ (π) < kBM, and the FSP choice otherwise. However, note that this solution is not identical to the LD algorithm.

4.4.2.4 Algorithm MSF4

The last version is a Load-Dependent extension of the MSF2 algorithm: specifically, the comparison with the maximal bandwidthBM is carried out after algorithm MSF2 has selected a path π, and the routing decision is taken following the guidelines given for algorithm MSF3 too.

Figure 4.9: Relative gainη(left) and starvation probability (right) of the MSF1 algorithm for different update periods

Figure 4.10: Relative gain η (left) and starvation probability (right) of the MSF2 algo-rithm for different update periods

To observe the robustness of the Multimetric Sequential Filtering algorithms from the link state update frequency point of view, we considered the same network topology and traffic scenario as in Section 4.3.3. Regarding Figures 4.9 and 4.10 we can state that MSF1 resists hardly to the “bad” effects of the large update period, while MSF2 suits better to our aim and looses less of its low-load gain, even if it does not stand to our expectations. When the offered load grows over about 1000 Mbps η starts to decrease strongly and in higher load regions even the starvation probability of MSF1 and MSF2 algorithms is larger than that of FSP.

0.95

Figure 4.11: Relative gain η (left) and starvation probability (right) for continuously update

In the second set of results we present a comparison of the relative per connection throughput gain and the starvation probability achieved by algorithms MSF3 and MSF4 to those achieved by WS and LD. Figures 4.11 and 4.12 present an overview of the gains and probabilities fortu = 0s andtu = 100s respectively.

As we can see, the adaptive algorithms on the one hand outperform FSP and the MSF ones when the traffic load is low and the link state information is distributed continuously or with high frequency. On the other hand MSF3 and MSF4 do not loose to much of their performance even if the information are wide out of date. As we discussed earlier, MSF algorithms inherently lessen the impact of outdated information by restricting their scope to a limited set of paths, namely those with a smaller hop count and a larger capacity. It is very important to stress that algorithms like MSF3 and MSF4 do not perform worse

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Figure 4.12: Relative gain η (left) and starvation probability (right) for 100 s update period

than FSP, even whentu is comparable with the flow average minimal duration.

Simulation results show that some MSF routing algorithms can be resilient to out-of-date link state information, while offering a non marginal gain when the update is reasonably frequent compared to the average duration of flows and the network load is moderate.