Stability in the mean of the characterization of queueing models (Q1115339)

The present study is a continuation of previous papers of the authors [in: Problems of stability of stochastic models, Proc. Semin., Moskva 1984, 61-89 (1984; Zbl 0571.60092); Adv. Appl. Probab. 17, 868-886 (1985; Zbl 0582.60084)] and it is based on the general scheme of characterization of models proposed in the first paper cited above. Any queueing model is considered in this scheme as a transformation F of the input random variables \(U\in {\mathcal U}\) into output random variables \(V\in {\mathcal V}\). This study differs from the above mentioned papers in two aspects: 1) In the above mentioned papers we studied continuity of F when \({\mathcal U}\) and \({\mathcal V}\) are endowed with metrics guaranteeing weak convergence or convergence in variation. In this paper \({\mathcal U}\), \({\mathcal V}\) are metrized by distances which induce Bernstein convergence, guaranteeing weak convergence and convergence of moments of certain order. 2) Mathematical models used in applications are based on a collection of empirical observations. The problem of characterization of queueing systems and its stability is thus naturally stated in terms of sample characteristics of observations. In Section 4 we examine the problem of stability of the characterization of queueing systems in the terms of empirical distributions of the random inputs, i.e., investigate the consistency of the theoretical output distributions with empirical distributions for these systems.