## 3 Applications to Communication Network Design  
### Background Concepts  
We should note that the mathematical formulation of a routing model (envisaged as a logical-mathematical system encompassing the assumptions and mathematical entities which enable an unequivocal specification of the route/path calculation and selection problem in a given network environment) typically considers a network representation in terms of a capacitated graph and, in many models, a matrix of node-to-node expected (average) offered demand. If we wish to have a more complete representation of the factors of the network environment associated with routing methods, then traffic flows (in fact of a stochastic nature), a more general representation, through a 'teletraffic network', composed of several mathematical and other logical entities (see e.g., [1; 142]), should be used. Furthermore, the nature of real communication networks is very complex, since they are organized, from the point of view of the functional, operational and control/management aspects, in several interrelated layers, which leads to the necessity of considering them as multilayer network structures. An Internet network, for example, in a limited national area, has at least three layers, namely the physical infrastructure (or _physical layer_, including coaxial cables, optical fiber cables and microwave links), the router topology layer (corresponding to the _logical layer_) and the third layer (corresponding to the representation of application level and possibly social network flows). This often raises difficult modelling issues in the development of adequate/realistic routing models in a given application context which can have an impact on the mathematical formulation of the path problem(s) defined as a nuclear part of the routing model. The formulation of the routing models is strongly dependent on various aspects, mainly (i) basic features of the network communication technology (for example, whether it is circuit-switched or equivalent, packet-switched, as in technologies based on IP, or hybrid); (ii) the underlying routing principles (such as static, dynamic, fully distributed, centralized, on-line or off-line routing methods); (iii) the network architecture (for example, IP-over MPLS over WDM); (iv) the QoS features of provided services and service classes, including resilience/protection requirements; (v) the computing capability and network information provided to the route calculation devices (such as switches in circuit-switched networks, routers in IP-based networks or optical-cross connectors in GMPLS optical networks); (vi) the network functional layer where the routing procedure is applied (for example, the logical layer, physical/transport layer or an intermediate layer); and (vii) end-to-end traffic management mechanisms (associated with bandwidth allocation throughout each path for different end-to-end service demands, as in routing methods with explicit bandwidth allocation or traffic splitting). A general class of routing methods other than the basic class (usually designated as _unicast_ routing where only one node-to-node route/path is calculated for each service demand) is _multipath routing_ that involves the simultaneous calculation of several paths, between two nodes or between two sets of originating and terminating nodes. An important type in this class refers to routing models with explicit reliability requirements/optimization objectives, in which an active/primary path and a backup path/protection path (which will be used for actually carrying the end-to-end communication messages in the event of failure of the active path) have to be computed simultaneously for each pair of origin-destination nodes; this should not be confused with _alternative routing_ (where a secondary path has to be computed and used when the primary or first choice path is blocked due to traffic congestion, typical of circuit-switched networks or of equivalent logical type).