Convergence rate for the distributions of GI/M/1/n and M/GI/1/n as n tends to infinity

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 queueing systems. We show that, if the interarrival c.d.f. H is non-lattice with mean value lambda(-1), and if the traffic intensity rho=lambda mu(-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 omega, the unique solution in (0, 1) of the equation z=integral(0)(infinity)exp{-mu(1-z)t}dH(t). A similar result is established for the M/GI/1/n and M/GI/1 queueing systems.