Jackson networks in nonautonomous random environments. (10th June 2016)

• Record Type: Journal Article
• Authors: Krenzler, Ruslan; Daduna, Hans; Otten, Sonja
• Is Part Of: Advances in applied probability. Volume 48:Number 2(2016)
• Page Start: 315
• Page End: 331
• Publication Date: 2016-06-10
• Subjects: Randomised random walk, -- Jackson network, -- skipping, -- processes in random environment, -- reflection, -- product form steady state, -- breakdown of nodes, -- degrading service, -- speed-up of service; Primary 60K37, -- Secondary 60K25, -- 90B22, -- 90B25; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2016.2
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: We investigate queueing networks in a random environment. The impact of the evolving environment on the network is by changing service capacities (upgrading and/or degrading, breakdown, repair) when the environment changes its state. On the other side, customers departing from the network may enforce the environment to jump immediately. This means that the environment is nonautonomous and therefore results in a rather complex two-way interaction, especially if the environment is not itself Markov. To react to the changes of the capacities we implement randomised versions of the well-known deterministic rerouteing schemes 'skipping' (jump-over protocol) and `reflection' (repeated service, random direction). Our main result is an explicit expression for the joint stationary distribution of the queue-lengths vector and the environment which is of product form.