In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/1/n and GI/M/1 queuing systems. We show that, if the inter-arrival cumulative distribution function H is non-lattice with mean value λ-1, and if the traffic intensity ρ = λμ-1 is strictly less than one, then the convergence rate of the stationary distributions of GI/M/1/n to the corresponding stationary distributions of GI/M/1 is geometric. Moreover, the convergence rate can be characterized by the number ω, the unique solution in (0, 1) of the equation z = ∫∞0 exp{ – μ(1 – z)t } dH(t). A similar result is established for the M/GI/1/n and M/GI/1 queueing systems.
