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 infinitecapacity system up to the first time that the capacity is exceeded. Convergence of steady-state distribution is more difficult; it is established here for single-facility models such as GI/GI/c/n with c servers, n − c extra waiting space and the first-come first-served discipline, in which all arrivals finding the waiting room full are lost without affecting future arrivals, via stability properties of generalized semi-Markov processes.