Correlation formulas for Markovian network processes in a random environment. (March 2016)

• Record Type: Journal Article
• Authors: Daduna, Hans; Szekli, Ryszard
• Is Part Of: Advances in applied probability. Volume 48:Number 1(2016)
• Page Start: 176
• Page End: 198
• Publication Date: 2016-03
• Subjects: Product-form network, -- space-time correlation, -- spectral gap, -- asymptotic variance, -- Peskun ordering; Primary 60K25, -- Secondary 60J25, -- 60K37; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2015.12
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: We consider Markov processes, which describe, e.g. queueing network processes, in a random environment which influences the network by determining random breakdown of nodes, and the necessity of repair thereafter. Starting from an explicit steady-state distribution of product form available in the literature, we note that this steady-state distribution does not provide information about the correlation structure in time and space (over nodes). We study this correlation structure via one-step correlations for the queueing-environment process. Although formulas for absolute values of these correlations are complicated, the differences of correlations of related networks are simple and have a nice structure. We therefore compare two networks in a random environment having the same invariant distribution, and focus on the time behaviour of the processes when in such a network the environment changes or the rules for travelling are perturbed. Evaluating the comparison formulas we compare spectral gaps and asymptotic variances of related processes.