We derive a law of large numbers for a tagged particle in the one-dimensional totally asymmetric simple exclusion process under a scaling different from the usual Euler scaling. By interpreting the particles as the servers of a series of queues we use this result to verify an open conjecture about the scaling behavior of the departure times from a long series of queues.
"A scaling limit for queues in series." Ann. Appl. Probab. 7 (4) 855 - 872, November 1997. https://doi.org/10.1214/aoap/1043862414