I originally speculated on whether there might be a "deep theory of telecoms". Roy Simpson wrote to me as follows.
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Ok, well here are some comments on that 2008 post based on the original question.
First there is no reference to the ideas of new physics directly applied to telecoms which was my original point, e.g. nothing on quantum routers (even quantum computers).
More generally the problem with the post is that because the axiomatisation mentioned by A. Wheen never happened in the book we don't actually know what you count as "Telecoms Mathematics" i.e. what is in scope and what is out of scope.
We have several axes to consider and the impression I have from that post is the following:
1. Operations vs. development axis
On this axis I guess that you are referring to the mathematics as used in current telecoms operations (which I agree is relatively graph- and statistics-based but not a huge conceptual leap anywhere), rather than what might be required in the development process of new types of structures, connections, equipment and technology.
2. Telecoms as existing system vs software development
Basically further depth on the point above. Two thirds of a telecoms system is software - an area in which reliability, real-time behaviour modelling, enhancement is underdeveloped and requires mathematical models (maybe) of all kinds.
3. Current vs. new axis
Again are we refering to what is currently done to model and develop systems or what could/should be done to make it all cheaper/better/smarter. Remember that some have advocated e.g. learning systems to help the telecoms network do its network management better.
4. Maths vs. Physics axis
In true STL systems fashion are we referring to just a maths solution or also a physics based solution? Any physics (e.g. the network timings using relativistic satellites) brings in its own maths and maybe its own new maths challenges.
And lets not forget the NP-completeness corner. So many published network algorithms seem to falter "in general" as the underlying problem is NP-complete. Again any solution here might bring in unknown new maths/logic.
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Response
Well, several of the problems you note above are common to any large-scale software system. To concern oneself with telecoms per se, we need to formalise what we mean by a telecoms system in some useful way.
I guess my intuitions tell me that a telecoms system is a network structure with endpoint nodes and interior nodes such that messages can be routed node-to-node between arbitrary endpoints. I guess everybody can have their own definition of course.
As I've defined it, the analysis of telecoms systems is a subdivision of classical graph-theory - especially in the design of routing protocols as you might expect. Small networks never pose any problems, so requirements for new technologies could be expected when dealing with scaling problems.
Telecom architects and designers are, as you might expect, highly-sensitised to scalability issues just because they are so important - the IETF for example normally has scalability as a major protocol/architecture design constraint, see any RFC.
As to whether quantum theory could be in any way relevant, it seems to me that today we see the relevance in three forms.
a. Classical devices (e.g. integrated circuits) which internally rely upon explicitly quantum phenomena to work at all. These are found in all telecom equipment
b. Quantum network functionality - most likely involving entanglement - which explicitly might add novel functionality to improve telecom network operation. There might I suppose be applications in routing, considered as a distributed form of quantum computing, although it seems a long way off. Ditto for quantum teleportation. As you suggest, this will be in SF stories before it gets to research papers and then into 22nd century networks!
c. End-to-end issues, most notably quantum encryption and key distribution, where it's a constraint upon the network that the end-to-end requirement (typically not to collapse entanglement) should be met. Here we obviously have things happening right now.
