A comparison of the stationary distributions of GI/M/c/n and GI/M/c

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/c/n and GI/M/c queueing systems as n tends to infinity. We show that earlier results established for GI/M/1/n and GI/M/1 remain true. Namely, it is proved that if the interarrival time c.d.f. H is non lattice with mean value lambda(-1) and if the traffic intensity rho = lambda/mu c is strictly less than one, then the convergence rates in l(1) norm of the arrival and time stationary distributions of GI/M/c/ to the corresponding stationary distributions of GI/M/c are geometric and are characterized by omega, the unique solution in (0, 1) of the equation z = integral(0)(infinity) exp(-mu c(1 - z)t) dH(t).