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

Abstract

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.