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 (1997) 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.