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Network operators have understood the importance of migrating their backbone networks to IP over WDM architectures, whereby an underlying optical infrastructure can automatically set up and tear down direct optical connections (lightpaths), yet keeping electronic IP routers on top of it. In such multi-layer networks, it is necessary to efficiently combine the resources available in both electronic and optical layers, providing the necessary quality of service (QoS) to end-users at the minimum possible cost. This work defines a multi-layer Bayesian decisor in a multi-hop scenario which finds a compromise between the utility perceived by the users in terms of delay and the utilisation costs of the optical and electronic resources. The mathematical formulation of such a Bayesian decisor is formulated and its behaviour is further analysed for different configurations in realistic scenarios. Its behaviour shows that the algorithm is capable of adapting its decision according to traffic characteristics, while utilising only the necessary optical and electronic resources.