Queue machines and belt machines are different, and both are different from stack machines. I was aware of some of the prior work on queue machines that canedo cites in his thesis, but not all, and his compiler work was completely new to me.
The key insight in a queue machine is that if the dependencies form a complete tree then a level-based schedule permits access to be at only two points, the front and the back of the queue, and maximal parallelism falls out automatically. I suspect there is a relationship between what a queue machine can do vs. what a general-register machine can do that is mathematically related to the difference between what a FSM can do and what a CFG can do, but am not sure. Perhaps those of you with CS skills might find a thesis topic in there 
Canedo’s work relaxes the pure-tree requirement of a classic queue machine toward more realistic code, and his thesis addresses how to compile for that model. I haven’t finished the thesis yet, but it’s clear that his scheduler takes dataflow graphs as input just like ours does, but the scheduler works very differently than ours. Of course, we have to deal with additional real-world issues that he doesn’t: an exposed pipeline with varying latency, for example. But even without those complications. I’m not sure that his approach to turning the graph into sequential code will work even theoretically for a belt machine like the Mill, no more than it would produce reverse Polish for a stack machine. Still reading