Limits for queues as the waiting room grows

We study the convergence of finite-capacity open queueing systems to their infinite-capacity counterparts as the capacity increases. Convergence of the transient behavior is easily established in great generality provided that the finite-capacity system can be identified with the infinite-capacity system up to the first time that the capacity is exceeded. Convergence of steady-state distributions is more difficult; it is established here for the GI/GI/c/n model withc servers,n-c extra waiting spaces and the first-come first-served discipline, in which all arrivals finding the waiting room full are lost without affecting future arrivals, via stochastic dominance and regenerative structure.