A Hierarchical Waveband Switching Algorithm

We combine above three factors and propose a hierarchical waveband switching (HWA) algorithm. HWA divides the RWA problem into the routing and the wavelength/waveband assignment subproblems. The routing subproblem is resolved by choosing the path among k-shortest paths while balancing the load over all the links. The wavelength/waveband assignment process gives higher priorities to Fiber-tier and Band-tier traffic while grouping the lightpaths with higher overlapping and bypass traffic. The following notations are used for describing the HWA algorithm.

X: The number of fibers per link;

F : The number of wavelengths per fiber;

B: The number of wavelengths per band;

W : The minimum overlapping hops among routing paths of

any grouped traffic demands;

P aths,d: The routing path of the node-pair (s, d);

D[i]: The degree of node i;

T [s][d]: The traffic demand between the node-pair (s, d); F T [s][d]: The Fiber-tier traffic between the node-pair (s, d); NF T [s][d]: The Non-Fiber-tier traffic between the node-pair (s, d).

We assume that for each node-pair (s, d), the amount of traffic demands are T [s][d] lightpath(s). The amount of the Non-Fiber-tier and the Fiber-tier traffic is specified in Eq.

(3.1) and Eq. (3.2), respectively. In the proposed two-stage algorithm, we accommodate as much the Fiber-tier traffic as possible in Stage 1. Then the Non-Fiber-tier traffic (i.e., including the Band-tier and the Wavelength-tier traffic) is satisfied in Stage 2.

NF T [s][d] = T [s][d]%F (3.1)

F T [s][d] = (T [s][d] − NF T [s][d])/F (3.2)

Stage 1: The Fiber-tier traffic assignment. Since the number of fibers and wavelengths per link is limited and large node size is generally not encouraged, we select the routes while balancing the load on all the links in the network.

1. Path generation. Use the k-shortest path algorithm [41] to generate the k (k >= 1)

path(s), namely Ph

s ,d, where h = 1, 2, . . . , k, for each node-pair (s, d).

2. Path selection. The link load is defined as the sum of all the traffic on the link, and the maximum load M is defined as the largest link load among all the links in the network. Starting from the node-pair with the largest traffic demands, assign one of the k paths to the node pair while minimizing M, until all the node-pairs with traffic demands are

considered. We denote the selected path for the node-pair (s, d) as P aths,d.

3. Fiber-tier traffic accommodation. Following the same sequence as in the previous step, assign bypass fiber(s) along the selected routing path to satisfy all the Fiber-tier traffic. For the node-pair (s, d), if only part of the Fiber-tier traffic can be accommodate along P aths,d, we accommodate the remaining part of the Fiber-tier traffic along another

candidate path in Ph

s ,d.

Stage 2: The Non-Fiber-tier traffic assignment. In this step, we accommodate the Non- Fiber-tier traffic by using the routing path selected in Stage 1. Assume that the set of the nodes along P aths,d for the node-pair (s, d) is s, s1, s2, . . . , si, d, we use NodeSets,dto denote this set. To take advantage of the bypass traffic in wavebanding, we calculate the sum of the bypass traffic at each node i using Eq. (3.3). Then we calculate the sum of the bypass-traffic

node-degree ratio along the path, say BDSums,d, for each selected routing path P aths,d as shown in Eq. (3.4). bypassT [i] = s!=d,s!=i,d!=i_X i²N odeSets,d NF T [s][d] (3.3) BDSums,d = X i²N odeSets,d bypassT [i]/D[i] (3.4)

For the Non-Fiber-tier traffic, Algorithm 1 is adopted for routing and wavelength assign- ment. Specifically, the lines 2-11 are to accommodate the traffic demands as bypass bands and the lines 13-22 are to accommodate the traffic demands one by one while considering wavelength grouping. The lines 24-29 are to handle the exception when the traffic cannot be accommodated using the current selected path. The variables w and b (both are initialized to be 0) in this algorithm are the index of the respective wavelength and waveband from which to start the search for an available wavelength and waveband.

In this scheme, we adopt the intermediate wavelength grouping policy. Since one-hop overlapping does not help in wavebanding, we require that traffic demands which are to be grouped with the current traffic demand should have at least W -hop (W ≥ 2) overlapping with the current traffic demand along their routing paths. There are four steps in this stage.

1. Starting with the node-pair (s, d) that has the largest BDSums,d, use Algorithm 1 to

accommodate the traffic demand for the node-pair (s, d).

2. Use Algorithm 1 to accommodate the traffic for every node-pair along P aths,dthat has

the source s, and at least W overlapping hops with P aths,d, starting with the node-pair (s, si), node-pair (s, si−1), ..., until the node-pair (s, s2).

3. Use Algorithm 1 to accommodate the traffic for every node-pair along P aths,d that

has the destination d, and at least W overlapping hops with P aths,d, starting with the

node-pair (s1, d), node-pair (s2, d), ..., until the node-pair (si−1, d).

Algorithm 1 Wavelength Assignment for the Traffic Demands Between Node Pair (s, d)

1: Begin:

2: while NF T [s][d] >= B do

3: Find a free band m starting with b along P aths,d;

4: if the band m exists then

5: b ← (m + 1)%B

6: else

7: Break

8: end if

9: Assign the band m to this traffic;

10: NF T [s][d] ← NF T [s][d] − B

11: end while

12:

13: while NF T [s][d] > 0 do

14: Find a free wavelength n starting with w along P aths,d;

15: if the wavelength n exists then

16: w ← (n + 1)%F

17: else

18: goto Exception

19: end if

20: Assign the wavelength n to this traffic;

21: NF T [s][d] ← NF T [s][d] − 1

22: end while

23:

24: Exception:

25: if all the k paths for (s, d) has been tried then

26: Block and Exit

27: else

28: Select an alternative path for the node-pair (s, d) from P_s,dh as P aths,d and goto Begin

29: end if

whose routing paths overlap with P aths,d by at least W hops. Update BDSums,d and goto Step 1, until all traffic demands are satisfied.

It is worth noting that HWA does not separate the Non-Fiber-tier traffic further into the Band-tier and Wavelength-tier traffic to accommodate them in different stages. For the purpose of comparison, we consider another scheme, namely, Hierarchical Waveband Algo- rithm with Full Separation (HWAF), which accommodates the Band-tier traffic (using steps similar to lines 2-11 of Algorithm 1) and the Wavelength-tier traffic (using steps similar to lines 13-22 of Algorithm 1) of a node-pair (s, d) separately. Figure 3.2 shows the comparison of port count for HWA and HWAF based on simulations on a random 6-node network with X = 2, F = 100, B = 5. The X-axis is the size of the uniform traffic (t) for each node-pair, and the Y-axis is the port count for both algorithms. As shown in Fig. 3.2, when the traffic size is less than the band size 5, there is no difference between HWA and HWAF since the Band-tier traffic is 0 for both algorithms. When the traffic size is a multiple of the band size (e.g., 5, 10), the difference is small since only the Band-tier traffic exists in both HWA and HWAF. As a result, one can observe a steep drop when the size of the traffic demand is a multiple of the band size. For all the remaining traffic requests, we can see that HWA out- performs HWAF. The advantage of HWA over HWAF can be explained as follows: (i). The probability that we fail to find free bands to accommodate the Band-tier traffic in Algorithm 1 is very small compared to the probability of failing to find free fibers to accommodate the Fiber-tier traffic. (ii). The coupling of the Band-tier and Wavelength-tier traffic can facilitate the formation of bypass fibers.