Rerouting requests in WDM networks†

Usually, when connexion requests are added or removed from a network, for instance a WDM network, the routing of older connexions is not modified. Hence it is likely that after some additions and removings, the overall use of resources is far from optimal. In particular, a new request may be rejected, even if it could be added up to a whole rerouting of older requests. So operators have to reorganise regularly the routing of all requests so as to make better use of the resources. Here we are interested in the problem of going from one routing to another without loss of services. Given a network, a set of requests I and two different routings for it in the network, R1 and R2, we want to switch from routing R1 to routing R2. Let u and v be two requests. We denote by Ri(u) (resp. Ri(v)) the routing of request u (resp. v) in Ri,1 ≤ i ≤ 2. If R2(u)∩R1(v) 6= / 0, i.e. the routing of request u in R2 uses resources already used by the routing of request v in R1, then obviously the request v has to be rerouted before we can reroute request u. However, a request might be switched to an intermediate route, that uses available resources. For instance, the operator may reserve a dedicated wavelength in the network for temporary routes. We assume that each request cannot be switched to more than one temporary route, that is the next routing of a request routed on a temporary route has to be its final routing. When a request that was previously switched to a temporary route reaches its final routing, then the freed resources can be used again, for another request. While independent switching of requests can be made simultaneously, we will consider, for matter of exposition, that only one request is switched per unit of time. The problem is modelled as follows: we construct a directed graph D = (V,A), where each vertex u corresponds to one request, and there is an arc from vertex u to vertex v if and only if R2(u)∩R1(v) 6= / 0. A vertex is said to be processed as soon as its corresponding request has been rerouted. We introduce the notion of Temporary Memory Unit (TMU): routing the request u on an intermediate route corresponds to putting the vertex u in a TMU. So a vertex can be processed if and only if all its outneighbours are either processed or in TMU’s. Note that a vertex without any outneighbour can be processed at any time. There are two basic operations: process a vertex according to the preceding rule ; put a vertex in a temporary memory unit. Figure 1 shows the processing steps of a graph using one TMU. We underline the fact that once placed in a TMU, a vertex cannot recover its original state: it has to be processed. Nevertheless, it can occupy its TMU as long as desired. Processing a vertex which occupies a TMU frees the TMU, so that it can immediately be used by another vertex. The digraph is said to be processed when all its vertices have been processed. The problem is hence to find a suitable order to