We consider a discrete-time queue with general service distribution and characterize a class of arrival processes that possess a large deviation rate function that remains unchanged in passing through the queue. This invariant rate function corresponds to a kind of exponential tilting of the service distribution. We establish a large deviations analogue of quasireversibility for this class of arrival processes. Finally, we prove the existence of stationary point processes that have a probability law that is preserved by the queueing operator and conjecture that they have large deviation rate functions which belong to the class of invariant rate functions described above.
"Invariant rate functions for discrete-time queues." Ann. Appl. Probab. 13 (2) 446 - 474, May 2003. https://doi.org/10.1214/aoap/1050689588