Mathematical Programming

After the tunnels has been allocated off-line in the network. Now we want to find a pair of link-disjoint lightpaths for each incoming request. The most popular issue on finding the backup path is the sharing concept. That is, several backup paths can utilize the same fiber link with the same wavelength plane as long as their corresponding working paths are disjointed.

In this section, we formulate the static routing problem in MG-OXC model. This problem can be described more clearly as follows: Given a traffic matrix and a direct auxiliary graph. The auxiliary graph includes wavelength-switching layer edges and some tunnel layer edges. The term wavelength-switching layer edge and tunnel layer edge are simplified to wavelength edge and tunnel edge in the following paragraph.

The added tunnel edges are decided by TPP or TSP. The objective here is to satisfy as many as possible connection requests in the traffic matrix. Each request has to find a pair of working and backup path from the source node to the destination node. The wavelength in each link may be occupied to form a part of working or backup path.

We assume the network have full wavelength conversion.

There is something need to be noticed. Since a tunnel physically includes several wavelength edges. Some tunnel edges in auxiliary graph look like separated, but they may have some common edges on wavelength switching layer. That is, some separated edges may failure at the same time. Hence, we quote the share risk link group (SRLG) concept to solve this problem. The original definition of SRLG is a group of network links that shared a common resource whose failure will cause the failure of all the links of the group. In this chapter, we only take some tunnels which include the same link to form a SRLG group such as tunnel AE and BF in fig.11.

A. TPP Based Formulation

Base on TPP tunnel allocation, we want to find a pair of SRLG-disjoint path for each connection request. Let N, E be the set of nodes and edges. E , ^T E be the set ^W of tunnel edges and wavelength edges. E =E^T ∪E^W. request is the set of requests.

is one if the working path of n-th lightpath request passes through link ij, and zero

otherwise. is one if the backup path of n-th lightpath request passes through link ij, and zero otherwise. The set g(i,j) includes all links that may shared a common physical link. is one if the working path and backup path of n-th lightpath request travel link ij and link uv respectively and zero otherwise. The variable ,

and are 0/1 variable. and are positive integer. means the backup capacity on link uv because a SRLG g broken. is the backup capacity on link uv. The following is our formulation for static RWA problem, and the objective function is to find as many successful working and backup path for each request as possible.

E

The equation (18) is the objective function. We want to find as many successful working and backup path for each request as possible. Equation (19) and (20) are the flow constraint for working and backup path. The equation (21) is the SRLG constraint. If link ij is a part of working path, the links which belong to the same SRLG can’t be the part of backup path for someone lightpath request. The objective of equation (22) is to confirm the variable is one if the working path and backup path of n-th lightpath request travel link ij and link uv respectively and zero otherwise. The equation (23) computes the demand of backup capacity on link uv if someone SRLG g broken together. The backup capacity on link uv can be shared by different SRLG g, so we only to choose a large enough number as the backup capacity on link uv. Then, the backup capacity on link uv must greater than the maximum

n uv

Sij ^,

demand of all groups. These are described in equation (24). After tunnel allocation, the capacity of every links (tunnel link and wavelength link) and input/output ports on every node are all assigned. Constraint (25), (26) and (27) show the resource constraint.

B. TSP Based Formulation

Base on TSP tunnel allocation, the working tunnel and its corresponding backup tunnel have already been allocated. Hence, we don’t leave the backup tunnel edges occurred in the auxiliary graph. For a lightpath request, if its working path passed through a working tunnel, the corresponding backup tunnel should be traveled naturally by its backup path. Then, we only need to find a pair of link-disjoint paths on the wavelength layer. Each pair of working and backup tunnel should be already SRLG-disjoint in TSP scheme. These following constraints only have a little different with the TPP one. Equation (27) constraint the working path and backup path must be link disjoint in wavelength switching layer. Equation (28) restricts the backup path to go through the same working tunnel edges to replace backup tunnel. Other’s constraints are the same with the TPP one.

∑

W

4.4 Simulation Results

We evaluate the performance of TPP and TSP via simulation using a 6-node and 16-node topology (Fig15 (a) and (b)). Because the high complexity of ILP formulation, we only use the 6-node topology to examine. The simulation environment is 1F1L and W=4. For the static traffic, Fig. 16 shows the simulation result. The horizontal axle is the number of request, and the vertical axle is the blocking probability. The TSP always outperforms TPP in all cases. And the ILP always outperforms the heuristic which follows the Dijkstra’s algorithm.

(a)

(b)

Fig. 15 Network topologies adopted in this simulation. (a) 6-node network topology. (b) 16-node network topology.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

10 15 20 25

num. of request

blocking rate TSP_heu

TSP_for TPP_heu TPP_for

Fig. 16 the blocking rate among different protection schemes