[ca] Fog computing ´es b'asicament una extensi´o del cloud computing, on els recursos de computaci´o es troben al l´ımit de la xarxa, la qual cosa permet un millor rendiment pel que fa a lat'encia i ample de banda. Per tant, els centres de dades (DC) que s'utilitzen en fog computing han de ser molt m´es petits, aix´ı com el nombre de hosts i switches instal􀀀�lats. En aquest context, aquesta dissertaci´o de tesi aborda el modelatge d'alguns dissenys de DC per fog computing, posant el focus nom´es en topologies de xarxa simples, encara que altres m´es complexes puguin aconseguir millor rendiment. Aquestes topologies poden modelar-se de diferents maneres, exposant la ruta m´ınima, o rutes m´ınimes del mateix cost, a trav´es de les quals poden tenir lloc les migracions de recursos entre un host origen i un altre dest´ı. Aix´ı, un usuari movent-se per un domini fog tindr'a els seus recursos de computaci´o associats seguir-lo el m´es a prop possible amb un interval de temps m´ınim, que ´es el punt clau en els entorns de mobilitat IoT. S'ha de destacar que els models van a ´esser exposats seguint un enfocament anal´ıtic, mitjan¸cant la combinaci´o de diferents branques matem'atiques per preparar els models, com la geometria i la topologia per obtenir els dissenys adequats, l'aritm'etica per dur a terme el moviment dels recursos de computaci´o a la destinaci´o adequada, la l'ogica per implementar aquestes accions en diagrames de flux i pseudocodi, o l''algebra per a proporcionar una descripci´o formal de tot el model. La principal contribuci´o en aquesta dissertaci´o de tesi es basa en obtenir models de rutes 'optimes per a migracions de VM en topologies de DC relacionades amb desplegaments fog computing mitjant¸cant un 'algebra de processos abstracta anomenada ACP, amb la qual s'han especificat i verificat formalment aquests models, ja que permet raonar sobre termes de processos de forma anal´ıtica, tornant als or´ıgens en el camp de les tecnologies de la informaci´o. En resum, pel que respecta a cada topologia, es proposen diferents models, como ara diagrames de flux, pseudocodi i un model algebraic formal, per a posteriorment presentar un model algebraic formal per a un sistema complet de fog/cloud.

[en] Fog computing is basically an extension of cloud computing where the computing resources are located on the edge of the network, allowing for better performance regarding latency and bandwidth. Hence, data centres (DC) being used in fog computing may be far smaller and so may the number of hosts and switches in use. In this context, the present thesis dissertation undertakes the modelling of some DC designs for fog computing, setting the focus on just simple network topologies, even though more complex ones might achieve better performance. Such topologies may be modelled in different ways, exposing the minimal path, or equal-cost multiple paths, through which a live Virtual Machine (VM) migration of computing assets may take place from a source to a destination host. This way, a user moving throughout a fog domain will have their computing assets following it as close as possible with a minimal time interval, which is the key point in Internet of Things (IoT) moving environments. It is to be stressed that the models are going to be exposed by following an analytical approach, with the combination of different mathematical branches to get the models ready, such as geometry and topology to obtain the appropriate designs, arithmetic to forward the moving assets to the proper destination, logic to implement those actions in flow charts and pseudocode, or algebra to exhibit a formal description of the whole model. The main contribution in this thesis dissertation goes about obtaining models of optimal paths for VM migrations in DC topologies related to fog computing deployments by means of an abstract process algebra called Algebra of Communicating Processes (ACP), which has been used in order to formally specify and verify such models, as it allows to reason about process terms on an analytical basis, getting back to basics regarding the information technology field. In summary, regarding each topology, different models may be proposed, such as by means of flow charts, pseudocode and a formal algebraic model, followed by the presentation of a formal algebraic model for a whole fog/cloud system.