Cost of using resources

This function accounts the cost C_e of using hop-by-hop connections (electronic switching) and the cost C_o of using end-to-end ligthpaths (optical switching) which, for a decision vector ~e, is given by:

C_T(~e) = C_e(~e) + R_costC_o(~e) (4.3)

where R_costis the relative cost of using the optical and electronic resources. In other words, an optical lightpath is R_costtimes more expensive than the same connection in the electronic layer. Note that R_cost is not a monetary cost but a metric that helps network operators decide how valuable their optical resources are with respect to the already deployed IP layer.

Cost computation has been chosen to follow the next design premises:

1. LSPs should be switched in the electronic domain while their utility perceived is correct, hence the cost of using electronic resources is cheaper than that of using optical resources, for the same amount of traffic (routers are already deployed).

2. If an optical bypass is to be set up, the longer it is, the better (less cost), that is, the cost of long connections should be lower than short optical by-pass connections.

Thanks to this cost model, only the necessary e2e optical connections are created, and this occurs when the IP layer do not provide the necessary utility to the traffic.

Following these premises, we define the cost of transmitting an LSP optically per hop as ^k+1_k , where k is the length of the optical by-pass (that is, a lightpath created from node j to the destination node is of length k = M + 1 − j). Note that this series is strictly decreasing since ^k+1_k > ^L+1_l , ∀k < L, giving a cheaper cost per hop the longer the lightpath is, thus promoting the creation of long e2e by-pass optical connections in the network. It is worth noticing that, in the scenario proposed in Figure 4.1, the longest (and cheapest) lightpath possible is of cost ^{M +1}_M , and it is the cheapest one since ^{M +1}_M < ^k+1_k , ∀k < M . In conclusion, the optical cost of sending e LSPs through k hops is ^k+1_k × k × e = (k + 1) × e.

The definition of the electronic and optical cost functions are as follows:

Ce(~e) = XM

j=1

2ej (4.4)

C_o(~e) = Ã

(M + 1)(N₁− e₁) +

M −1X

j=2

(M − j + 2)(N_j − e_j + e_j−1)

!

(4.5)

According to the previous definitions (eq. 4.3, optical resources are Rcost times more expensive than electronic in the case of one-hop switching. The one-hop electronic cost is 2 (k = 1), and ^{(M +1)}_M is the cheapest optical cost per hop in this scenario, since the maximum optical path length is M. According to this, R_cost must satisfy R_cost> _{M +1}^2·M to ensure that

the cheapest optical lightpath is more expensive than its electronic counterpart. Otherwise, if optical resource utilization is very cheap and they add no delay to the packets, there is no reason to send the traffic using the IP layer.

Figure 4.2(a) shows a multi-hop scenario with three hops (M = 3). In such scenario there are three possible end-to-end paths from node 1 to the destination node. If a LSP is sent through the hop-by-hop connection, the associated cost is due to the electronic cost, since no utilization of the optical resources is done. Its cost would be 2 × M. If the end-to-end connection is used, the cost is just optical cost and it is (M + 1) × R_cost = 4 × R_cost. Concerning the hybrid case, the cost is one hop electronic and two hops in the optical domain, so its cost is 2 + M × R_cost = 2 + 3 × R_cost. Figure 4.2(b) depicts the cost of sending one LSP using the hop-by-hop connection, the end-to-end lightpath or a hybrid connection, when Rcost = {1, 1.5, 2, 2.5, 3}. According to the previous designed rule, Rcost

should be greater than _{M +1}^2·M , in this case 1.5. This is the reason why the cost per LSP of a hop-by-hop connection is more expensive when R_cost= 1 or equal when R_cost = 1.5 than the end-to-end path. When R_cost > 1.5, the cost per LSP is cheaper in the electronic than in the optical domain. The cost of the hybrid connection is intermediate, since the first hop is done in the IP layer and the rest in the optical domain. Let us remark that the cost is not the only value to make the decision. The LSPs sent through the IP layer suffer a delay, which increase their risk. Therefore, although the cost is lower the traffic can be routed in the optical domain.

Destination Node 1 Node 2 Node 3 node

e2e Hybrid Hop-by-hop

(a) Multi-hop scenario with three hops (M = 3) and possible end-to-end paths

1 1.5 2 2.5 3

0 2 4 6 8 10 12

R_cost

Cost per LSP

Hop−by−hop connection Hybrid connection e2e connection

(b) Cost of routing one LSP in each path

Figure 4.2: R_cost designed rule example

Once the definition of the cost function is stated, replacing eq. 4.4 and 4.4 in eq. 4.2,

The utility function applied to a decision vector ~e gives a metric for the delay experienced by the electronically-switched LSPs, such that, the more delay experienced by them, the less utility achieved. The electronically-switched LSPs are assumed to experience some degree of delay, since they must traverse several hops with their respective electronic queues. On the other hand, the delay experienced by the optically-switched LSPs is assumed negligible compared to the electronic delay, since optical LSPs are provided a dedicated e2e path.

Such an electronic delay is calculated based on the load level of a queue fed with self-similar traffic, as explained in section 3.1.2. Once the e2e electronic delay is obtained, the utility function operates to derive a utility metric following one of these Class of Service (CoS) utility models (see section 3.2.2: average delay, hard real-time and elastic utilities, as follows:

Average delay-based utility (U_mean)

This utility is defined as: Umean(x^e2e_j ) = −x^e2e_j which, after applying the expectation operator E_xof eq. 4.2, provides a utility function based on the average e2e delay experienced by the electronically-switched LSPs. This value is computed as:

E_x[U_mean(x^e2e_k )] = E_x[−x^e2e_k ] = −

As we previously explained, this utility function can be used for best-effort services, whereby great service interactivity provides high utility values, but this utility function does not excessively penalize if such interactivity is low. In this multi-hop scenario, the U_mean