Sharp results on convergence rate 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 lim_K→∞σ^-K||π^(K) - π|| and lim_K→∞σ^-K||q^(K) - q|| explicitly.