We consider the renewal process Rt = 1 ζj < t , where ζi are mutually independent random variables,
n=0 j=0
and P{ζi s} = F(s) for i ∈ N (i.e. the distribution function of ζ0 can be not F(s)). For t ∈ (ti,ti+1), where
def i def def
ζj, we denote xt = t − ti = t − tRt (backward renewal time). The backward renewal process Xt = (xt) is
It is well known, that the backward renewal processes and a knowledge about its convergence rate to the sta- tionary distribution are very important in the queueing theory. Asymptotic estimates of the convergence rate of the distribution of the backward renewal process are well known. Namely, if the condition (i) {Eζk < ∞, k > 1} or (ii) {E exp(aζ) < ∞, a > 0} is satisfied, then (correspondingly) for all κ < k − 1 or α < a the following inequality is true for A ∈ σ(X) and all t > 0: (i′) |P{Xt ∈ A}−P(A)| < K(κ)t−κ or (ii′) |P{Xt ∈ A}−P(A)| < K(α)e−αt, where P is the stationary distribution of Xt, and K(·) are some (unknown) constants. Our goal is to find strong bounds for this constants.