Sharp results on convergence rates for the distribution of GI/M/1/K queues as K tends to infinity

In this paper, we investigate how fast the stationary distribution π^(K) of an embedded Markov chain (time-stationary distribution q^(K)) of the GI/M/1/K queue converges to the stationary distribution π of the embedded Markov chain (time-stationary distribution q) of the GI/M/1 queue as K tends to infinity. Simonot proved certain equalities. We obtain sharper results than these by finding limit values limK→∞ σ^−K‖π^(K) − π ‖ and limK→∞ σ^−K‖q^(K) − q ‖ explicitly.