Queueing Systems 44, 125–136, 2003  2003 Kluwer Academic Publishers. Manufactured in The Netherlands. Exact Convergence Rate for the Distributions of GI/M/c/K Queue as K Tends to Infinity ∗ BONG DAE CHOI, BARA KIM, JEONGSIM KIM and IN-SUK WEE Department of Mathematics, Korea University, 1, Anam-dong, Sungbuk-ku, Seoul, 136-701, Korea Received 15 September 2002; Revised 10 January 2003 Abstract. We obtain the exact convergence rate of the stationary distribution π (K) of the embedded Markov chain in GI/M/c/K queue to the stationary distribution π of the embedded Markov chain in GI/M/c queue as K →∞. Similar result for the time-stationary distributions of queue size is also included. These generalize Choi and Kim’s results of the case c = 1 by nontrivial ways. Our results also strengthen the Simonot’s results [5]. Keywords: GI/M/c/K queue, GI/M/c queue, stationary distribution 1. Introduction We consider GI/M/c/K queue with general interarrival time distribution A(x) with mean λ -1 and Laplace–Stieltjes transform A ∗ (s). The service times of customers are exponentially distributed with mean μ -1 . The traffic intensity ρ is given by ρ = λ/cμ, and is assumed to be strictly less than 1. Let π (K) = (π (K) 0 ,...,π (K) K ) (respectively π = (π 0 ,π 1 ,π 2 ,...)) be the stationary distribution of the embedded Markov chain (EMC) of the GI/M/c/K (respectively GI/M/c) queue just before the epochs of arrivals. We assume that A(x) is non-lattice. Then there exists the time-stationary distribution q (K) = (q (K) 0 ,...,q (K) K ) (respectively q = (q 0 ,q 1 ,q 2 ,...)) of the GI/M/c/K (respectively GI/M/c) queue [1, p. 255]. Simonot [5] proved that lim K→∞ r K   π (K) - π   = lim K→∞ r K   q (K) - q   = 0 if r<σ -1 , (1) lim K→∞ r K   π (K) - π   = lim K→∞ r K   q (K) - q   =∞ if r>σ -1 , (2) where σ is the unique solution in (0, 1) of A ∗ (cμ - cμz) = z, and   π (K) - π     j 0   π (K) j - π j   and   q (K) - q     j 0   q (K) j - q j   ∗ This work was supported by Korea Research Foundation Grant (KRF-2001-042-D00009).