Moments based approximation for the stationary distribution of a random walk in Z(+) with an application to the M/GI/1/n queueing 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 E{A(+)(s+1)} < +infinity for an s greater than or equal to 0 where a(+) = max(0, a). Let pi be the s tationary distribution of X. We show that one can find probability distribu tions pi(n) supported by {0, n} such that \\pi(n) - pi\\(1) Cn(-s), where t he constant C is computable in terms of the moments of A, and also that \\p i(n) - pi\\(1) = o(n(-s)). Moreover, this upper bound reveals exact for s g reater than or equal to 1, in the sense that, for any positive epsilon, we can find a random walk fulfilling the above assumptions and for which the r elation \\pi(n) - pi\\(1) = o(n(-s-epsilon)) does not hold. This result is used to derive the exact convergence rate of the time stationary distributi on of an M/GI/1/n queueing system to the time stationary distribution of th e corresponding M/GI/1 queueing system when n tends to infinity.