Some ideas from the theory of weak convergence of probability measures on function spaces are modified and extended to show that the queue-length of the GI/G/s system converges in distribution as time passes, for the case of atomless interarrival and service distributions. The key to this result is the concept of the uniform $\sigma$-additivity of certain sets of renewal measures on a space endowed with incompatible topology and $\sigma$-field.
"Translated Renewal Processes and the Existence of a Limiting Distribution for the Queue Length of the GI/G/s Queue." Ann. Probab. 3 (3) 424 - 439, June, 1975. https://doi.org/10.1214/aop/1176996350