A stochastic process on a network with connections to Laplacian systems of equations. (21st March 2022)

• Record Type: Journal Article
• Authors: Bagchi, Amitabha; Gillani, Iqra Altaf; Vyavahare, Pooja
• Is Part Of: Advances in applied probability. Volume 54:Number 1(2022)
• Page Start: 254
• Page End: 278
• Publication Date: 2022-03-21
• Subjects: Queueing networks -- random walks -- geometric ergodicity; 60G10 -- 60G50 -- 60K25; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2021.27
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: We study an open discrete-time queueing network. We assume data is generated at nodes of the network as a discrete-time Bernoulli process. All nodes in the network maintain a queue and relay data, which is to be finally collected by a designated sink. We prove that the resulting multidimensional Markov chain representing the queue size of nodes has two behavior regimes depending on the value of the rate of data generation. In particular, we show that there is a nontrivial critical value of the data rate below which the chain is ergodic and converges to a stationary distribution and above which it is non-ergodic, i.e., the queues at the nodes grow in an unbounded manner. We show that the rate of convergence to stationarity is geometric in the subcritical regime.