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In this paper we consider an irreducible random walk in Z+defined by X(m+1) = max(0, X(m) + A(m+1)) with E{A} < 0 and for an s= 0 where a+= max(0,a). Let p be the stationary distribution of X. We show that one can find probability distributions pnsupported by {0,n} such that ||pn- p||1= Cn-s, where the constant Cis computable in terms of the moments of A, and also that ||pn- p||1= o(n-s). Moreover, this upper bound reveals exact for s= 1, in the sense that, for any positive e, we can find a random walk fulfilling the above assumptions and for which the relation ||pn- p||1= o(n-s-e) does not hold. This result is used to derive the exact convergence rate of the time stationary distribution of an M/GI/1/nqueueing system to the time stationary distribution of the corresponding M/GI/1 queueing system when ntends to infinity.

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/1/nand GI/M/1 queueing systems. We show that, if the inter-arrival c.d.f. His non-lattice with mean value ? –1,and if the traffic intensity ? = ?µ –1is strictly less than one, then the convergence rate of the stationary distributions of GI/M/1/nto the corresponding stationary distributions of GI/M/1 is geometric. More-over, the convergence rate can be characterized by the number ?, the unique solution in (0, 1) of the equation . A similar result is established for the M/GI/1/nand M/GI/1 queueing systems.

In this note, we compare the arrivaland time stationary distributionsof the number of customers in the GI/M/c/nand GI/M/cqueueing systems as ntends to infinity. We show that earlier results established for GI/M/1/nand GI/M/1 remain true. Namely, it is proved that if the interarrival time c.d.f. His non lattice with mean value ?-1and if the traffic intensity is strictly less than one, then the convergence rates in l1normof the arrival and time stationary distributions of GI/M/c/nto the corresponding stationary distributions of GI/M/care geometric and are characterized by ?, the unique solution in (0,1) of the equation z= ?80exp{-µc(1-z)t}dH(t).