Queue automaton

A queue machine or queue automaton is a finite state machine with the ability to store and retrieve data from an infinite-memory queue. It is a model of computation equivalent to a Turing machine, and therefore it can process the same class of formal languages.
Theory.
A queue machine can be defined as a six-tuple
A machine "configuration" is an ordered pair of its state and queue contents formula_8, where formula_9 denotes the Kleene closure of formula_4. The starting configuration on an input string formula_11 is defined as formula_12, and the transition formula_13 from one configuration to the next is defined as:
where formula_15 is a symbol from the queue alphabet, formula_16 is a sequence of queue symbols (formula_17), and formula_18. Note the "first-in-first-out" property of the queue in the relation.
The machine accepts a string formula_19 if after a finite number of transitions the starting configuration evolves to exhaust the string (reaching a null string formula_20), or formula_21
Turing completeness.
We can prove that a queue machine is equivalent to a Turing machine by showing that a queue machine can simulate a Turing machine and vice versa.
A Turing machine can be simulated by a queue machine that keeps a copy of the Turing machine's contents in its queue at all times, with two special markers: one for the TM's head position, and one for the end of the tape; its transitions simulate those of the TM by running through the whole queue, popping off each of its symbols and re-enqueing either the popped symbol, or, near the head position, the equivalent of the TM transition's effect.
A queue machine can be simulated by a Turing machine, but more easily by a multi-tape Turing machine, which is known to be equivalent to a normal single-tape machine.
The simulating queue machine reads input on one tape and stores the queue on the second, with pushes and pops defined by simple transitions to the beginning and end symbols of the tape. A formal proof of this is often an exercise in theoretical computer science courses.
Applications.
Queue machines offer a simple model on which to base computer architectures, programming languages, or algorithms.