Moments based approximation for the stationary distribution of a random walk in Z+with an application to the M/GI/1/nqueueing system

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.