At times it has been possible to make some progress on the complexity by regarding each application, virtual machine, server, server cluster, or server farm as a network of queues and apply queuing theory and/or Monte Carlo simulation. In addition some of the old work in optimization under uncertainty likely applies.
Network of queues can become hrrendoulsy difficult to analyze unless one assumes memoryless arrivals etc etc. Network calculus and stochastic network calculus simplifies things a great deal by going after bounds rather than exact answers. You might like it
For "going after bounds rather than exact answers", for analyzing network performance, there is a hugely simplifying approach: Just look at the bottlenecks and largely f'get about the rest; maybe that is related to what you mentioned.
For network queuing, really I was suggesting using that as a paradigm to formulate an analysis; I accept that analytic solutions are unpromising -- more generally, the exact probabilistic calculations out of queuing theory research are complicated even for simple cases. So, for solutions, i.e., actionable information, use Monte Carlo.
Once I knew a guy at IBM's Watson lab who was big on such things. There was a claim that at one time his work was useful in designing some of an IBM mainframe I/O subsystem. He had some software that I used once. His software wanted to collect the usual descriptive statistics, but I wanted the sample paths from the Monte Carlo, got those, and did more analysis of those.
...and now you have :)
> Certainly no such topics were in any of the applied math grad courses I took.
Yeah they are a newer development. Stochastic network calculus more so. Network calculus is older, it even has a wikipedia page https://en.wikipedia.org/wiki/Network_calculus
A key idea is convolution but in the max-plus algebra and martingale large deviations.
> For network queuing, really I was suggesting using that as a paradigm to formulate an analysis;
Indeed. SNCs can be a helpful tool there. Yes they are bounds, but a lot of progress has been made to make them tight https://arxiv.org/pdf/1303.4114.pdf they arent quite there yet numerically but tight enough to give an intuition. NC bounds are a lot looser.