Actually the whole show is run by the Central Flow Management Unit (CFMU) of EuroControl in Brussels.
Reading between the lines they use some sodding great mainframe computer to manage the ATC 'flow', i.e. because if you want to connect to it you have to use an SNA compliant connection - where SNA = mainframe type networking.
Now in one of my other guises (i.e. I've got two hats) I'm presently involved in connecting my airlines Ops Department to EuroControl's Slot Management computer - and you would not believe what a nause it is to get the network hardware to function (or perhaps you might).
That said, EuroControl will soon be making available (about a month, or so) connection via TCP/IP (the internet protocol - which you're using right now to access this site).
According to the CFMU literature it initially should be able to handle the viewing & requesting of slots, but it won't be until next year (apparently) that you will be able to ask "if I re-route, can I get a better slot ?" type questions - via this TCP/IP browser based connection.
Of course with browser based TCP/IP connectivity you could use a laptop on the flight deck to connect to the web via a mobile phone, or better still a wireless LAN, login to CFMU with your corporate account, and then manage the slot / routing yourself - to say nothing of having a quick surf of PPRuNe too !